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L

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Page 1: L-functions, elliptic curves and the parity conjecturetomlr.free.fr/Math%E9matiques/Elliptic%20curves%20and%20Parity%20... · Paris 6, Pierre et Marie Curie, pendant l'année scolaire

L-functions, elliptic curves

and the parity conjecture

Benjamin Wagener

Université Pierre et Marie Curie

Paris VI

June 15, 2008

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Ce texte est un mémoire de Master de Mathématiques eectué à l'UniversitéParis 6, Pierre et Marie Curie, pendant l'année scolaire 2007-2008. Ce travail estcomposé essentiellement de deux parties. Premièrement, je présente le contextegénéral des fonctions L. Ensuite, après avoir fait quelques rappels sur les courbeselliptiques, je présente un article ([14]) de Tim et Vladimir Dokchitser. J'aichoisi de réaliser ce travail en anglais principalement car je pense que c'est unbon exercice dans le contexte de la recherche actuelle.

Presentation

This work has been done in the context of a training course at the MasterAlgèbre et Géométrie of the University Paris 6, Pierre et Marie Curie, duringthe year 2007-2008. The primary goal of this course is to prepare students todo research in Mathematics by giving them research articles to study. I wasdirected by the Professor Marc Hindry of Paris 7 University who gave me thearticle [14] of Tim and Vladimir Dokchitser about a special case of the parityconjecture.

The parity conjecture says that the sign of the functional equation of the L-function of an elliptic curve E over a number eld K is equal to (−1)rank E(K),the group of rational points of E, E(K), been a nitely generated abelian groupby the Mordell-Weil theorem. This conjecture is a consequence of the Birch andSwinnerton-Dyer conjecture.

L-function of algebraic varieties are not simple objects and the related theoryremains conjectural. That's why I wanted to have some insight into the theo-retical framework that underpins this subject. This is what I did in the rstpart of this work. As such, beginning with classical zeta functions, I motivatethe introduction of L-functions of algebraic varieties and their related objectssuch as l−adic representations, ε−factors and root numbers. I think that it is agood way to place the article of Tim and Vladimir Dokchitser in its conceptualcontext. This part is more general that what is needed to [14], however it isonly with these ideas in mind that one may fully understand it in my opinion.

The second part of this text is devoted to elliptic curves and the article [14].First I deal with elliptic curves, I presented what is necessary to understandthe text, Weierstrass equations, reduction of elliptic curves, L-functions... Thenext chapter deals with the article itself. I didn't want to reproduce the article,so I had proofs of facts that could appear not obvious at a rst reading. AlsoI motivated the way the articled is hanged together: p−Selmer ranks, rootnumbers, parity conjecture with a 2-isogeny.

I am grateful to the Professor Marc Hindry who has permitted me to dothis work and for his willingness teaching the courses on elliptic curves and onabelian varieties during this academic year.

Benjamin Wagener

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Contents

Presentation 3

I L Generalities 7

Introduction 9

1 Dirichlet L-functions 11

2 Dedekind zeta functions and Hecke's L-functions 15

2.1 The Dedekind zeta function . . . . . . . . . . . . . . . . . . . . . 152.2 Hecke Grössenchrakters . . . . . . . . . . . . . . . . . . . . . . . 18

3 From classical to contemporary number theory 23

3.1 Tate's Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.1.1 From ideals to idèles . . . . . . . . . . . . . . . . . . . . . 233.1.2 Some Fourier theory . . . . . . . . . . . . . . . . . . . . . 28

3.2 Artin L-functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.2.1 Some usual facts about representations . . . . . . . . . . . 313.2.2 Basic properties of Artin L-functions . . . . . . . . . . . . 323.2.3 Artin vs Hecke . . . . . . . . . . . . . . . . . . . . . . . . 323.2.4 The functional equation . . . . . . . . . . . . . . . . . . . 36

3.3 Weil L-functions . . . . . . . . . . . . . . . . . . . . . . . . . . 383.3.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . 383.3.2 Weil Groups, according to Tate in [55] . . . . . . . . . . . 393.3.3 Special cases . . . . . . . . . . . . . . . . . . . . . . . . . 413.3.4 L-functions and ε-factors . . . . . . . . . . . . . . . . . . . 42

4 The geometric objects case 47

4.1 The general picture . . . . . . . . . . . . . . . . . . . . . . . . . . 474.1.1 Étale and l-adique cohomology . . . . . . . . . . . . . . . 474.1.2 Zeta functions for schemes over nite elds . . . . . . . . 494.1.3 The global case . . . . . . . . . . . . . . . . . . . . . . . . 50

4.2 L-functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.2.1 Weil groups again: The Weil-Deligne group . . . . . . . . 54

5

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6 CONTENTS

4.2.2 Conductors and ε-factors . . . . . . . . . . . . . . . . . . 574.2.3 L-? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

II Parity conjecture for elliptic curves 61

5 Elliptic Curves 63

5.1 Basic facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665.1.1 Elliptic curves over local elds . . . . . . . . . . . . . . . 675.1.2 Formal groups . . . . . . . . . . . . . . . . . . . . . . . . 695.1.3 Elliptic curves over p-adic elds . . . . . . . . . . . . . . . 71

5.2 L-functions of elliptic curves . . . . . . . . . . . . . . . . . . . . . 735.3 Selmer and Shafarevich-Tate groups . . . . . . . . . . . . . . . . 75

5.3.1 Basic group cohomology . . . . . . . . . . . . . . . . . . . 755.3.2 Selmer and Shafarevich-Tate groups . . . . . . . . . . . . 775.3.3 The Cassels-Tate pairing . . . . . . . . . . . . . . . . . . . 79

5.4 BSD and parity conjectures . . . . . . . . . . . . . . . . . . . . . 805.4.1 L-functions again . . . . . . . . . . . . . . . . . . . . . . . 805.4.2 The Birch and Swinnerton-Dyer conjecture . . . . . . . . 815.4.3 The parity conjecture . . . . . . . . . . . . . . . . . . . . 82

5.5 Néron Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 835.5.1 Algebro-geometric preliminaries . . . . . . . . . . . . . . . 835.5.2 The bers of Néron models . . . . . . . . . . . . . . . . . 85

6 Parity conjecture with a cyclic isogeny 87

6.1 Presentation of the article of Tim and Vladimir Dokchitser . . . 876.2 The p−Selmer rank . . . . . . . . . . . . . . . . . . . . . . . . . . 88

6.2.1 Basis for computation of parity of p−Selmer ranks . . . . 886.2.2 The computation of p−Selmer ranks . . . . . . . . . . . . 94

6.3 The root number . . . . . . . . . . . . . . . . . . . . . . . . . . . 946.3.1 Potential multiplicative reduction . . . . . . . . . . . . . . 956.3.2 Potential good reduction and p ≥ 5 . . . . . . . . . . . . . 966.3.3 Potential good reduction and p = 3 . . . . . . . . . . . . . 97

6.4 The case of a 2−isogeny . . . . . . . . . . . . . . . . . . . . . . . 986.4.1 Hilbert symbols . . . . . . . . . . . . . . . . . . . . . . . . 986.4.2 Discussion about the proof of theorem 6.1.2 . . . . . . . . 99

7 Conclusion 103

III Bibliography 105

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Part I

L Generalities

7

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9

Preliminary remark: This chapter is a general presentation of L-functions.For this reason there are only few proofs but I tried to add references in thetext.

The history of zeta and L-functions goes back to Euler ,at least, with theRiemann zeta function dened, for <(s) > 1, by

ζ(s) =∑n≥1

1ns.

The zeta function has two properties which are veried by similar functions.First of all, it has an Euler product:

ζ(s) =∏

p prime

11− p−s

, for <(s) > 1.

Secondly, it has an analytic continuation to the whole complex plane with asimple pole at s = 1. Furthermore if we dene ( almost the Mellin transform ofζ) :

ξ(s) = π−s/2Γ(s/2)ζ(s),

then ξ satises the following functional equation:

ξ(s) = ξ(1− s).

These properties link the zeta function with the properties of the set of primenumbers, the crucial conjecture being the still unproved Riemann hypothesis.

There are many generalizations of this, the rst to have appeared are knownas Dirichlet L-functions, they were generalized to number eld and the proof ofthe corresponding relation was given by Hecke and then the correct conceptualpoint of view was given by Tate in his Ph.D thesis. This together with a pointof view developed by Artin is the origin of the modern standpoint of the sub-ject. The generalization of this to algebraic geometry is still conjectural even ifGrothendieck and Deligne, among others, have made the ground work for it.

In this part we are going to present some essential concept of the theoryof L-functions, our principal goal being to motivate the notion of L-functionassociated to elliptic curves and the related concepts.

This part is essentially descriptive.

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Chapter 1

Dirichlet L-functions

Series of the form: ∑n≥1

anns

(an, s ∈ C)

are called Dirichlet series. Dirichlet used these series in the special case wherean = χ(n) for χ a modular character to prove a conjecture of Gauss that thereare innitely many primes in any sequence of the form an+b(n = 1, 2, . . .) whena and b are relatively prime integers.

More precisely, a Dirichlet (modular) character modulo m is dened in thefollowing way:

Denition 1.0.1 Dirichlet charactersLet m be a positive integer, a Dirichlet character χ modulo m is a group

homomorphism from (Z/mZ)∗ to the multiplicative group C∗. In other terms

it is an element of the dual group (Z/mZ)∗. As (Z/mZ)∗ is a nite group,χ takes its values in the group of roots of unity modulo φ(m) (φ is the Eulertotient function). If χ is a Dirichlet character modulo m we dene by extensionχ : Z→ C∗ by

χ(n) :=χ(nmod[m]) if gcd(m,n) = 10 otherwise

A Dirichlet character modulo m is called primitive if it is not induced fromany character to a modulus n with n ≤ m. If χ is primitive modulo m we callm the conductor of χ and denote it by fχ.

The inverse of χ is the character χ which is the complex conjugate map ofχ : χ(a) = χ(a), a ∈ Z

Let m ≥ 1 and χ a Dirichlet character modulo m, we dene the Dirichlet L-serierelative to m and χ as :

L(s, χ) =∑n≥1

χ(n)ns

11

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12 CHAPTER 1. DIRICHLET L-FUNCTIONS

Due to the multiplicative property of Dirichlet characters we have the followingproposition ([45]):

Proposition 1.0.1 If χ 6= 1, L(s, χ) is convergent in the right half plane<(s) > 0 and absolutely convergent in the right half plane <(s) > 1. More-over it has the following Euler product expansion :

L(s, χ) =∏p

11− χ(p)p−s

,<(s) > 1,

where the product is taken over every prime number.

Dirichlet used this to prove that if gcd(a,m) = 1 :∑p

p ≡ amod[m]

1ps∼s→1

1φ(m)

log1

s− 1,

proving in a non trivial manner that there are innitely primes p such thatp ≡ amod[m].

Dirichlet L-functions admit a meromorphic continuation to the whole com-plex plane, furthermore if χ = 1 it has a unique pole of order 1 and residue 1at s = 1 and if χ 6= 1 it is holomorphic everywhere.

Dirichlet L-functions also satisfy a functional equation which is given in thefollowing theorem (see [22] for a proof).

Theorem 1.0.1 Let χ be a Dirichlet character with conductor f, we dene theGauss sum of χ by:

r(χ) =f∑a=1

χ(a)e2πiaf

and dene the modied L-function of χ by

Λ(s, χ) =(f

π

)s/2Γ(s+ δ

2

)L(s, χ),

where δ = 0 if χ(−1) = 1 and δ = 1 if χ(−1) = −1 and Γ is the usual gammafunction. Then

Λ(s, χ) = W (χ)Λ(1− s, χ).

W (χ) being given by :

W (χ) =r(χ)√fiδ

, |W (χ)| = 1, i =√−1

The behavior or value of an L-function near s = 1 is important in numbertheory. In the special case of a Dirichlet L-series it is :

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13

Proposition 1.0.2 If χ = 1 then

lims→1

L(s, 1) = 1.

If χ 6= 1

L(1, χ) = −r(χ)f

∑1 ≤ a ≤ f

gcd(a, f) = 1

χ(a)log(1− e−2aiπ/f )

, where f and r(χ) are dened in the previous theorem.

The following is a special case which is very important from a historical pointof view ([4]).

Proposition 1.0.3 Let K be a quadratic imaginary eld, χ the quadratic char-acter attached to it, such that χ(−1) = −1, f the conductor of χ and dK thediscriminant of K. Let h be the class number of K and ω be the number of rootsof unity in K. Then

L(1, χ) =2πh

ω√|dK |

.

From this it can be inferred that

h = − ω

2|dK |

f∑m=1

χ(m)m (Dirichlet's class number formula)

The next step was to generalize this to number elds in general.

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14 CHAPTER 1. DIRICHLET L-FUNCTIONS

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Chapter 2

Dedekind zeta functions and

Hecke's L-functions

2.1 The Dedekind zeta function

With the development of algebraic number theory, it was natural (and needed)to generalize these notions to arbitrary number elds. Instead of Q we considera number eld K and instead of Z, we consider the ring OK of integers of K.We denote by JK the group of fractional ideals, PK the group of principal idealsand Pic(K) = JK/PK the ideal class group.

If a ⊂ OK is an ideal the quotient group OK/a is nite and we dene thenorm of a as the index N(a) = Card(OK/a). Then the Dedekind zeta functionof the number eld K is dened by :

ζK(s) =∑

06=a∈I

1N(a)s

for <(s) > 1.

The reason why ζK has the same half plane of convergence as ζ is that thereare at most [K : Q] ideals in OK of given norm. It what follows, we denoten := [K : Q].

Furthermore, the Dedekind zeta function has an Euler product over everyprime ideals, a functional equation, and a meromorphic continuation to theentire complex plane.

ζK(s) =∑

p

11−N(p)−s

for <(s) > 1

If dK is the absolute discriminant of K and if r1 and r2 are the number ofreal and complex places of K respectively, then the function

ξK(s) =

( √|dK |

2r1πn/2

)sΓ(s

2

)r1Γ(s)r2ζK(s)

15

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16CHAPTER 2. DEDEKIND ZETA FUNCTIONS ANDHECKE'S L-FUNCTIONS

satisesξK(s) = ξK(1− s).

The factors π−s/2Γ(s2

)and (2π)1−sΓ(s) (r1 + 2r2 = n) are interpreted as

the missing factors at real and complex innite places respectively.The functional equation was proved by Hecke using a generalization of what

has been done about the Riemann zeta function.Following [29], let us present some steps of the proof.The rst step is to transform the denition of the Dedekind zeta function

into something computable. For this, for each ideal class R ∈ Pic(K) we choosean ideal aR ∈ R−1. Then the map R 3 b 7→ aRb = (ξ) ⊂ aR denes a bijectionbetween ideal in R and elements of aR modulo units. We call C(aR) a set ofrepresentatives of elements in aR under the equivalence relation for which twoelements of aR are equivalent if they dier by a unit.

ThenζK(s) =

∑06=I∈I

1N(I)s =

∑R∈Pic(K)

∑b∈R

1N(b)s

=∑

R∈Pic(K)

∑ξ∈C(aR)

1

(N(a−1R (ξ)))s

=∑

R∈Pic(K) N(aR)s∑ξ∈C(aR)

1NK/Q(ξ)s

.

Now we sum norms of algebraic numbers. We denote by σν the canonicalembeddings of K at innite places ordered so that σ1, . . . , σr1 : K → R andσr1+1, . . . , σr1+r2 , σr1+1, . . . , σr1+r2 : K → C and for ξ ∈ K and i = 0, . . . , n weput ξσi = σi(ξ).

The next step is to use the following formula:

Γ(s/2)as

=∫ ∞

0

exp(−a2y)ys/2dy

y, a > 0

which gives for 1 ≤ i ≤ r1 + r2:(π−1/2d

1/2K N(aR)1/n

Nσi

)sNσi Γ(sNσi/2)|ξσi |sNσi

=∫ ∞

0

exp(−πd−1/na Nσi |ξσi |2y)ysNσi

dy

y,

where we put da = N(aR)2dK (the absolute value of the discriminant of aR) andNσi = 1 or 2 provided that σi is a real or a complex embedding respectively.

In order to perform the product we have to introduce some notations whichis due to the fact that there is a dierence between real and complex em-beddings. The product will give an integral over r1 + r2 copies of R+∗. Fory = (yi) ∈

∏r1+r2i=1 R+∗ we dene N(y) = y1y2 · · · yr1y2

r1+1 · · · y2r1+r2 and we

dene the measure dyy on

∏r1+r21 R+∗ as the product measure of the Haar mea-

sures dtt on R+∗ (for details see the paragraph on higher-dimensional gamma

functions in the next section). Then

(2−r2d1/2K π−n/2)sΓ(s/2)r1Γ(s)r2 N(aR)s

NK/Q(ξ)s

=∫∞

0· · ·∫∞

0exp(−πd−1/n

a

∑1≤i≤r1+r2

|ξσi |2yi)N(y)s/2 dyy

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2.1. THE DEDEKIND ZETA FUNCTION 17

The nal step is to sum over ξ and to transform the expression into somethingwhich involves theta functions as for the Riemann zeta function.

For this purpose, we dene theta function in the following way: we choosec = ci|1 ≤ i ≤ r1 + r2 a list of n strictly positive real numbers enumeratedsuch that cr1+i = cr1+r2+i for 1 ≤ i ≤ r2 and we dene for a fractional ideal aof K:

Θ(c, a) =∑ξ∈a

exp

−πd−1/na

∑1≤i≤r1+r2

ci|ξσi |2

where da = Na2dK is the absolute discriminant of a.Hecke proved that the following functional equation holds:

Θ(c, a) =1√∏

1≤i≤r1+r2ci

Θ(c−1, a′),

where c−1 = c−1i |1 ≤ i ≤ r1 + r2 and a′ is the ideal dual to a with respect to

the bilinear form given by the trace of K over Q, a′ is also given by a′ = (da)−1,where d is the dierent of K over Q.

With some manipulations (see [29] or [37]), it can be shown that the functionΞ(s,R) dened by:

Ξ(s,R) = (2−r2d1/2K π−n/2)sΓ(s/2)r1Γ(s)r2N(aR)s

∑ξ∈C(aR)

1NK/Q(ξ)s

,

satises the following equation:

Ξ(s,R) =∫∞

1

∫E

(Θ(t1/nc, a)− 1

)dc ts/2 dtt −

2µ(E)sω

. . .+∫∞

1

∫E

(Θ(t1/nc, a′)− 1

)dc t(1−s)/2 dtt −

2µ(E)(1−s)ω

Here R′ = (dR)−1 is the dual to R with respect to the trace and ω is thenumber of roots of unity in K. The terms t1/N c come from the fact that we canwrite (yσi) ∈

∏i R as t1/N c with

∏i yσi = t and

∏i ci = 1. Moreover, E is an

adequate fundamental domain and µ(E) is the measure of E with respect to dc.This show that Ξ(s,R) = Ξ(1 − s,R′) and summing over R the functional

equation for ζK(s) follows.Furthermore, the value of µ(E) can be computed as something depending

on the regulator Reg(K) of the eld and the residue of ζK(s) in the simple poles = 1 follows:

ress=1ζK(s) =2r1(2π)r2Reg(K)

d1/2K ω

As we can see, the proof is quite complicated and we could expect it to beeven more if we consider the equivalent for number elds of Dirichlet characters,this was done by Hecke in 1920 and conceptually improved later by Tate.

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18CHAPTER 2. DEDEKIND ZETA FUNCTIONS ANDHECKE'S L-FUNCTIONS

2.2 Hecke Grössenchrakters

We keep the notations of the previous section for K, n = [K : Q], OK , JK (idealgroup), Cl(K) (ideal class group)...

We could consider generalizations of Dirichlet characters on JK . We wouldconsider an integral ideal m ∈ JK , the group Jm

K of fractional ideal relativelyprimes to m and the subgroup Pm

+ of all those principal ideals (x), x ∈ K∗ forwhich x is totally positive and x− 1 ∈ m. Then a Dirichlet character modulo m(also called a periodic character) is a character on Jm

K/Pm+ .

Actually, Hecke considered more general characters in such a way that afunctional equation still exists.We want to dene characters on OK modulo anintegral ideal m ∈ JK . So we dene a group homomorphism χm : (JK/m)∗ →C∗. Equivalently, we consider a character on OK which is trivial on m.

The problem is that we would like to extend it to a character on JK ? Wemay try to dene it on principal ideals at least by dening it on generators.However there is no reason for such a function to be trivial on units.

Hecke succeeded in doing this but the properties of the characters are notsimple ; he called the corresponding characters Grössencharakters.

Denition 2.2.1 GrössencharaktersLet m be an integral ideal of the number eld K and let Jm

L be the group of allideals of K which are relatively prime to m.A Grössencharakter mod m is a group homomorphism

χ : JmK → S1 = z ∈ C||z| = 1,

for which exists a pair of characters:

χf : (OK/m)∗ → S1, χ∞ : R∗ → S1,

such thatχ((a)) = χf (a)χ∞(a)

for every algebraic integer a ∈ OK relatively prime to m.A Grössencharakter χ mod m is a called primitive if it is not the restriction ofa Grössencharakter χ′ mod m′ for any proper divisor m′ of m. It can be shown(see [37] CR. VIII prop 6.2) that it is equivalent to say that χf factors through(OK/m′)∗. The conductor of a Grössencharakter χ is the smallest divisor f ofm such that χ is the restriction of a Grössencharakter mod f

The characters χf and χ∞ are uniquely determined by the Grössencharakterχ. Characters of the ideal class groups are determined by Grössencharakters inthe following way(e.g [37] CR. VIII 6.10):

Proposition 2.2.1 The characters of the ideal class Cl(K),i.e the group ho-momorphisms χ : JK → S1 which are trivial on principal ideals, are preciselythe Grössencharakter χ mod 1 satisfying χ∞ = 1

To χ or χf we can associate a Gauss sum:

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2.2. HECKE GRÖSSENCHRAKTERS 19

Denition 2.2.2 Gauss SumsLet χf be a character of (OK/m)∗, let d be the dierent of K/Q andy ∈ m−1d−1. Then we might associate a Gauss sum of χf as be

τm(χf , y) =∑

x mod mgcd(x,m) = 1

χf (x)e2πıTr(xy)

We dene naturally L-functions with Grössencharakters χ as

LK(s, χ) =∑

06=a∈I

χ(a)N(a)s

=∏p

11− χ(p)N(p)−s

.

Hecke generalized theta functions to arbitrary number elds and along thesame lines as in the proof of the functional equations (e.g. [37]) of Dedekindzeta functions he proved the functional equations for L-functions with Grössen-charakters.

First of all he dened partial L-functions. For a ideal class R de Pie(K) wedene:

L(R, χ, s) =∑

a ∈ Ra integral

χ(a)N(a)s

To obtain the functional equation theta functions over number elds can beused, I refer to [37] for a general presentation. There is something else which isneeded: higher-dimensional Gamma functions which play the role of factors atinnity.

Higher-dimensional Γ-Function and characters on local elds In [37]a general presentation of Γ- functions is made which associate to any Gal(C/R)-set, X, a complex function ΓX . We only present the case of interest for us whichis whenX = Hom(K,C). So we deneX = Hom(K,C) = σ1, . . . , σr1 , σr1+1, . . . ,σr1+r2 , σr1+1, . . . , σr1+r2 where σi is real for 1 ≤ i ≤ r1 and complex otherwise.The fact that two conjugate complex embeddings dene the same place whereasthere are only one embedding in each conjugacy class has some consequences.Actually we can't deal them on the same footing.

For this reason we consider the space

C =∏σ∈X

C

and we dene homomorphisms N : C→ C and Tr : C→ C by

N(z) =∏σ∈X

zσ Tr(z) =∑σ∈X

zσ.

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20CHAPTER 2. DEDEKIND ZETA FUNCTIONS ANDHECKE'S L-FUNCTIONS

We dene the conjugation on C of an element z = (zσ) by (z)σ = zσ and wedene an involution ∗ by z∗σ = zσ. The space of interest is then:

R+∗ = z = (zσ) ∈ C|z = z, z = z∗ and ∀σ, zσ > 0.

ActuallyR+∗ =

∏p

R+∗p ,

where p = σ, σ is a conjugacy class of embeddings (a prime or a place) andR+∗

p is dened by

R+∗p =

R+∗ if p is real∆(R+∗ × R+∗) = (x, x)|x ∈ R+∗ if p is complex

.

There is a natural isomorphism R+∗p∼= R+∗ which sends x to x if p is real and

(x, x) to x2 if p is complex. Thus we obtain

R+∗−→∏p

R+∗.

Via this isomorphism we transport the product of the Haar measures dtt of the

right-hand side to a Haar measure dyy on R+∗.

Denition 2.2.3 Higher-dimensional Γ−functionsWith the preceding notations, for s = (sσ) ∈ C such that <(sσ) > 0 we denethe Γ−function for X = Hom(K,C) by

ΓX(s) =∫R

+∗N(e−yys)

dy

y.

The convergence reduce to the convergence of ordinary gamma functions.Then we dene the L-function of X (here K) by

LX(s) = N(π−s/2)ΓX(s/2)

The following lemma ([37]) is important for the purpose of the functionalequation of L-functions with Grössencharakters.

Lemma 2.2.1 If χ : JmK → S1 is a Grössencharakter with associated characters

χf and χ∞, there exist unique p ∈∏σ Z and

q ∈ z ∈ C|z = z and z = z∗ such that

χ∞(x) = N(xp|x|−p+ıq

)where xp stands for

∏σ(σx)pσ and same thing for |x|−p+ıq.

For such p and q, χ is called of type (p, q) and p− ıq is called its exponent.For χ of type (p, q) we dene

L∞(χ, s) = LX(s1 + p− ıq)

We are now ready to state the theorem of Hecke.

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2.2. HECKE GRÖSSENCHRAKTERS 21

Theorem 2.2.1 Functional equation of L-functions with Grössencharakters (1920)Let K be a number eld with n = [K : Q] and d the dierent of K. Let χ be aprimitive Grössencharackter of conductor f, then the function:

Λ(R, χ, s) = (|dK |N(m))s/2 L∞(χ, s)L(R, χ, s), <(s) ≥ 1.

has a meromorphic continuation to the complex plane C and satises the func-tional equation:

Λ(R, χ, s)) = W (χ)Λ(R′, χ, 1− s)

where R′ is the ideal class dened by R · R′ = [md] and the constant factor isgiven by

W (χ) =(ıTr(p)N

((md|md|)p

))−1 τ(χf )√N(m)

Where m and d are such that (m) = m and (d) = d.Furthermore |W (χ)| = 1 and Λ(R, χ, s) is holomorphic except for poles of

order at most one at s = Tr(−p+iq)/n and s = 1+tr(p+iq)/n and holomorphiceverywhere in the cases m 6= 1 or p 6= 0

The proof goes along the same lines as for the Dedekind zeta function andneeds the use of general theta functions.

Corollary 2.2.1 With the notations of the previous proposition, then the com-pleted Hecke L-function

Λ(χ, s) = (|dK |N(a))s/2L∞(χ, s)L(χ, s) =∑R

Λ(R, χ, s)

admits a holomorphic continuation to

C \ Tr(−p+ ıq)/n, 1 + tr(p+ ıq)/n

and satises the functional equation

Λ(χ, s) = W (χ)Λ(χ, 1− s).

Furthermore it is holomorphic on C if m 6= 1 or p 6= 0

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22CHAPTER 2. DEDEKIND ZETA FUNCTIONS ANDHECKE'S L-FUNCTIONS

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Chapter 3

From classical to

contemporary number theory

There is a natural generalization of this in more modern terms. It was conceivedby Hecke and concretized by Iwasawa and principally by Tate in his thesis [54].The principal idea is to consider idèles groups instead of ideals and characters onidèle class group. It turns out that, correctly set, it overlaps the previous theoryand oers the good theoretical framework. The result being a more involvedtheoretical point of view while technicalities are in some sense simpler. Thework of Tate has opened a new area of mathematics which is the basis of thecurrent treatment of the subject.

Independently, Artin considered L-functions associated to Galois represen-tations and Weil built a framework that include Artin L-functions and HeckeL-functions as special cases.

3.1 Tate's Thesis

Tate thesis [54], Fourier analysis in number elds and Hecke's zeta function hasbeen a major breakthrough in the theory.

3.1.1 From ideals to idèles

For a general presentation of restricted products, idèles and adèles see [29] or[44]. Here we show that we can dene characters on idèles, called Hecke charac-ters, and their associated Hecke L-functions which generalize L-functions withGössencharakters. This is the basis for an idelic treatment of the subject whichwas dealt with by Tate. First of all, remind the denition of the idèle groupof K. It is the restricted direct product of the multiplicative groups K∗ν of the

23

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24CHAPTER 3. FROMCLASSICAL TO CONTEMPORARYNUMBER THEORY

completions of K with respect to the local unit groups O×ν :

IK =

(xν) ∈

∏ν

K∗ν |xν ∈ O×ν for all but nitely many places ν of K

.

There is a natural algebraic embedding

K∗ → IKx 7→ (x, x, x, . . .) .

We dene the idèle class group to be the quotient IK/K∗ with respect to thisembedding. Then we can dene the characters of interest:

Denition 3.1.1 An idèle class quasi-character is a continuous homomorphismfrom IK to C∗ that is trivial on the image of K∗ in IK under the preceding di-agonal embedding.

A Hecke character, χ is a continuous homomorphism χ : IK → S1 such thatχ(K∗) = 1 (under the canonical embedding).

Before presenting Tate's thesis, I want to build the bridge betweenGrössencharakters and Hecke characters. It is only with this that we see thatTate's work generalizes and enriches what was done by Hecke. For this purpose,let us remind the basis of local elds theory. The most general denition of alocal eld is given in the book of Weil [57], it is a non-discrete locally compacttopological eld. On the additive group of a local eld there is a Haar measureµ that denes a discrete valuation. For this reason a local eld is sometimesdenes as a eld which is complete with respect to a discrete valuation and with(nite) perfect residue eld or as a completion of a global eld.

Let K be a number eld and m =∏

p pnp an integral ideal of K which isprime to ∞ (np = 0 if p|∞). Remind that for every prime p there is a basis ofneighborhoods of 1 in the multiplicative group K∗p given by

O×p = U(0)p ⊇ U (1)

p ⊇ U (2)p ⊇ · · ·

where U (n)p = 1 + πnpOp for πp a uniformizing parameter of the local ring Op.

Then we dene

Im

K = Imf × I∞ for Im

f =∏p-∞

U(np)p , and I∞ = (K ⊗Q R)∗ =

∏p|∞

K∗p

Denition 3.1.2 Module of denition of a Hecke character.With the preceding notations, m is called a module of denition of the Heckecharacter χ if

χ(ImK) = 1

Now we can build the correspondence between Hecke characters with moduleof denition m and Grössencharkters (mod m). Recall that we noted Jm

K thegroup of ideals relatively prime to m. We dene C(m) by

C(m) = IK/Imf K∗.

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3.1. TATE'S THESIS 25

For every prime p -∞ we choose a prime element πp (i.e a uniformizing param-eter of Op) of Kp. Then we dene a homomorphism

JmK −→ C(m)

p - m 7−→ (. . . , 1, 1, πp, 1, 1 . . .).

It doesn't depend of the choice of πp since the idèles (. . . , 1, 1, up, 1, 1 . . .), up ∈O∗p, for p - m, lie in Im

f . Then the composition

JmK → C(m)→ S1

dene a 1-1 correspondence between Hecke characters modulo m and Grössen-charackters modulo m. For an explanation of these facts see [37], this idelicinterpretation is due to Chevalley in [8].

Tate's thesis

Tate's thesis itself [54] is a good account of the theory, there is also the morerecent book [44] which is very pedagogical.

Let K be a number eld and consider its completion Kp at a prime p , Fqthe residue eld and a corresponding uniformizing parameter πp if p is non-archimedean. Then Kp is a local eld with absolute value | · |p. The additivegroup possesses a Haar measure µp normalized so that µ(Op) = 1. We then havean associated Haar measure µ×p on the multiplicative group K∗p normalized sothat µ×p (O∗p) = 1 given by

dµ×p (x) =

(1−Np−1)−1 dµp(x)

|x|p if p -∞dµp(x)|x|p if p | ∞

.

Generally, K∗p = Up ×Gp where Up is the subgroup of elements of unit absolutevalues and Gp = y ∈ R+∗ | ∃x ∈ K∗p , y = |x|p is the valuation group.

Characters of Gp are of the form t 7→ ts for some s ∈ C, whereas grouphomomorphisms (quasi-characters) Up → C∗ take their values in S1. Hence wehave:

Lemma 3.1.1 With the preceding decomposition, a quasi-character of K∗p , i.ea group homomorphism χ : K∗p → C∗ can be written

χ = ω| |sp

where ω is a character of Up (with values in S1) and s is a complex number.The number s is not uniquely determined by this decomposition but its real partσ = <(s) is. σ is called the real part or exponent of χ and noted σ = <(χ).

Denition 3.1.3 A (quasi-)character χ on a local eld F is said to be unram-ied if its restriction to the local units UF is trivial, χ|UF = 1

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26CHAPTER 3. FROMCLASSICAL TO CONTEMPORARYNUMBER THEORY

The basis of Tate's thesis rest on the following simple computation whichallow to transform the L-functions dened as a sum as an integral.

Let p - ∞ and ξp : K∗p → C an unramied quasi-character. We have thefollowing decompositions

K∗p =⋃n∈Z

πnpO×p Op \ 0 =⋃n≥0

πnpO×p .

Then ∫πnpO

×p

ξp(x)dµ×p (x) = ξp(πnp )

(∫O×p

χp(ε)dµ×p (ε)

)= ξp(πp)n,

the rst equality being due to the property of the Haar measure. This gives∫Op\0

ξp(x)dµ×p (x) =∑n≥0

∫πnpO

×p

ξp(x)dµ×p (x) = (1− ξp(πp))−1.

Now, let ψ be a Hecke character corresponding to the Grössencharkter χ bythe preceding correspondence. Then by the properties of the restricted directproducts ψ decomposes into a product ψ(x) =

∏p ψp(xp) for

x = (xp) ∈ IK such that ψp : K∗p → C∗ is a quasi-character, unramied foralmost all p, ψp . Then if ξp = ψp| |s which is also a quasi-character,ξp(πp) = χ(p)Np−s. So, in this case, we nd the Euler factor of the L-functionswith Grössencharakters.

With this in mind we would like to dene the L-functions associated toquasi-characters of the idèle class group, called Hecke L-functions. If χ is sucha quasi-character, we may dene L(χ) =

∏p L(χp), where for a quasi-character

ξp : K∗p → C at a nite place we dene the local L-factor ,L(ξp) = (1−ξp(πp))−1

if ξp is unramied and 1 otherwise. It remains to dene the factors at innity.There are dened as follow. If Kp = C, Up = S1 and ξp takes the form

χs,n : reıθ 7→ rseınθ, for some s ∈ C, n ∈ Z

and we dene the local factor

L(χs,n) = (2π)−(s+|n|2 Γ(s+

|n|2

).

If Kp is non-archimedean, χ is a quasi-character of K∗p and πp is a uniformiz-ing parameter of Op we put:

L(χ) =

1 If χ is ramied1

1−χ(π) If χ is unramied

If Kp = R, Up = −1, 1 and ξp is of the form µ| · |s where µ is either thetrivial character or the character sgn : x 7→ x/|x|. We dene the correspondinglocal L-factors are dened by

L(µ| · |s) =π−s/2Γ(s/2) if µ = 1π−(s+1)/2Γ( s+1

2 ) if µ = sgn

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3.1. TATE'S THESIS 27

Denition 3.1.4 Let ψ be an idèle class quasi-character, it can be written

ψ =∏p

ψp.

We dene

L(ψ) =∏p

L(ψp)

where the local factors L(χp) are dene according to what precedes. With thisdenition, we further dene the Hecke L-function corresponding to χ as

L(s, ψ) = L(ψ| · |s)

Actually, Tate dened more general class of function called zeta integralsand used Fourier analysis to deduce the analytic continuation and functionalequation for them.

As before, we consider a Hecke character (mod m), ψ : JK/K∗ → S1 associ-ated to a Grössencharakter χ : IK → S1 (mod m).

If p is a nite place (prime) c : K∗p → C is a quasi-character andf ∈ L1(Kp) is such that fc ∈ L1(K×p ) for <(c) > 0, we dene the local zetaintegral by

ζp(f, c) =∫Kp

f(x)c(x)dµ×p (x).

The motivation for this denition is that if f = δOp\0 is the characteristicfunction of Op \ 0 then

ζp(f, ψp| · |s) =1

1− χ(p)Np−s.

For archimedean places we nd that the corresponding local zeta integralequals the local factors we computed previously (e.g. [7]).

We can work it out globally. For this purpose we consider a function f(x) =∏p fp(xp) on the additive group of adèles AK such that fp ∈ L1(K∗p) and

fp = δOp\0 for non-archimedean places prime to m, i.e for p - m and p - ∞.We dene S(m) := q : q - m and q -∞. Then

ζ(f, ψ| · |s) = L(s, χ)∏

p∈S(m)

ζp(fp, ψp| · |s).

Actually, the product term in the last equation correspond exactly to themissing factors for the functional equation of the L-functions with Grössen-charakters. The key theoretical tool of Tate's thesis is Fourier theory on adèlegroup AK that we present next.

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28CHAPTER 3. FROMCLASSICAL TO CONTEMPORARYNUMBER THEORY

3.1.2 Some Fourier theory

Let K be a number eld, AK its adèle group and if p is a prime of K, let Kp

the corresponding local eld.The following lemma is simple but important.

Lemma 3.1.2 ([7] p. 308)Let k be a local eld, k+ its additive group. If a 7→ χ(a) is a non-trivial characterof k+. The correspondence η ↔ (a 7→ χ(ηa)) is both a topological and analgebraic isomorphism between, k+ and its character group.

Now, we dene a particular character of AK/K by specifying a particularcharacter on each completion Kp.

Denition 3.1.5 Let p a prime of OK that we identify with a place of K andlet Kp be the corresponding completion. We dene:

λp(xp) =

exp(−2πıxp) if Kp

∼= R,exp(−2πı<(xp)) if Kp

∼= Cexp(−2πı(TrKp/Qpxp)) if [Kp : Qp] <∞.

and a global character

λ : AK/K → C∗x 7→

∏p λp(xp).

λp denes an isomorphism Kp∼= Kp (the character group) by the preceding

lemma and we have similarly an isomorphism AK ∼= AK dened by x 7→ (y 7→λ(yx)).

If p is non-archimedean, the local dierent dp ⊂ Op is dened by:

d−1p = x ∈ Kp|λp(xy) ∈ Z,∀y ∈ Op.

One has dp = dKOp and NdK = |dK | (the absolute discriminant of K).We then dene normalized measures on the completion of K.

Denition 3.1.6 (Self-dual measures)Let K be a number eld and p a place of K. We dene the measure

µp =

N(dp)−1/2µp if p is non-archimedeanLebesque measure dx if Kp

∼= R2dxdy for z = x+ iy ∈ Kp

∼= C.

The measure µ on AK is then (well) dened by µ =∏

p µp and is a Tama-gawa measure, in the sense that µ(AK/K) = 1.

We are ready to dene Fourier transforms:

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3.2. ARTIN L-FUNCTIONS 29

Denition 3.1.7 Let f ∈ L1(AK) and fp ∈ L1(Kp), the spaces L1 being denedby the self-dual measures. Their Fourier transform are

fp(x) =∫Kp

fp(y)λp(xpy)dµp(y), f(x) =∫

AKf(y)λ(xy)dµ(y).

The following inversion formula holds

ˆf(−x) = f(x), ˆ

fp(−xp) = f(xp).

One of the great feature of Tate's thesis is to replace the complex compu-tations of Hecke by quite simple Fourier analysis. In this process, the use ofgeneralized theta series is replaced by a Poisson summation formula.

Proposition 3.1.1 (Poisson summation formula)Let f be a continuous function on AK such that both |f | and |f | are summableover K ⊂ AK and the series

∑α∈K f(x + α) converges uniformly on every

compact subset of AK . Then∑α∈K

f(α) =∑α∈K

f(α).

Theorem 3.1.1 Let ψ a Hecke quasi-character and let f a function which satis-es the hypothesis of the Poisson summation formula and such that for all σ > 0,|f(x)|‖x‖σ is integrable over the group of idèles JK . Then for <(ψ| · |s) > 1 thefollowing integral

ζ(f, ψ| · |s) =∫

JK

f(x)ψ(x)|x|sdµ×(x)

is well dened.ζ(f, ψ| · |s) admits an analytic continuation to the entire complex plane and

satises the following functional equation

ζ(f, ψ| · |s) = ζ(f , ψ−1| · |1−s).

As one can see the result of Tate is much more general than the result ofHecke without been much more complex but theoretically more evolved andcertainly very elegant.

3.2 Artin L-functions

Artin around 1920 constructed a family of functions associated to a represen-tation of Galois groups Gal(L/K) of extensions of number elds L/K. Tate'sthesis and Artin L-functions are the origins of the current treatment of thesubject of L-functions, an open eld with many far reaching conjectures.

In the particular case of Dirichlet L-functions, the correspondence takes thefollowing form. Let χ be a modular character modulo m for m an integer and

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30CHAPTER 3. FROMCLASSICAL TO CONTEMPORARYNUMBER THEORY

let Gm = Gal(Q(µm)/Q) be the Galois group of the cyclotomic eld Q(µm)generated by m-th roots of unity. There is a classical isomorphism

(Z/mZ)∗ → Gmfor pmod[m] 7→ (φp : ζ 7→ ζp) ,

where we take only the prime number p, prime to m.φp being the Frobenius au-tomorphism. Thus we can interpret χ as a character ofGm, i.e a one-dimensionalrepresentation of Gm. Then L(s, χ) rewrites

L(χ, s) =∏p-m

11− χ(φp)p−1

.

Artin extended these notion to arbitrary complex representation of the Ga-lois group G := Gal(L/K) of an extension L/K. Let p be a prime ideal of OKand P a prime ideal of OL above p. We know by classical algebraic numbertheory that G acts transitively on prime ideals of OL lying above p. Thus wecan dene DP = τ ∈ G|τP = P, called the decomposition group of P andIP = τ ∈ G|τ(x) ≡ xmod P, ∀x ∈ OL, called the inertia group of P.

Let L(P) and K(p) denote the residue elds OL/P and OK/p. There is ([37]Ch. I (9.4)) a surjective homomorphism DP → Gal(L(P)/K(p)) with kernel IP ,hence an isomorphism DP/IP ∼= Gal(L(P)/K(p)). As L(P)/K(p) is an exten-sion of nite elds, DP/IP is cyclic with a generator, φp, called naturally theFrobenius automorphism and which is characterized by

∀x ∈ OE , φP(x) ≡ xN(p) mod P.

The action of the Galois group on primes above p being transitive, if P1 and P2

are above p there is an element of G which sends P1 to P2, DP1 to DP2 and IP1

to IP2 . This implies that if ρ : G→ AutC(V ) is a representation:

det(1− ρ(φP1)Np−s|V IP1 ) = det(1− ρ(φP2)Np−s|V IP2 ),

where V IP is the invariant subspace of V according to the action of IP . Fur-thermore two equivalent representations ρ and ρ′ dene the same factor det(1−ρ(φP)Np−s|V IP ) whereas two representations are equivalent if and only if theyhave the same character, so this factor depends only on χ.

Then, the following function is well dened.

Denition 3.2.1 Artin L-functionsLet L/K be a Galois extension of number elds with Galois group G and foreach prime ideal p of OK , P an arbitrary prime ideal of OL above p. Let IP bethe inertia group of P and (ρ, V ) a complex representation of G of character χ,the Artin L-function attached to (ρ, V ) and G is

L(s, ρ, L/K) =∏p

det(1− ρ(φP)Np−s|V IP ).

The series is convergent for <(s) > 1.

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3.2. ARTIN L-FUNCTIONS 31

3.2.1 Some usual facts about representations

In this section a representation mean a complex representation. A representation(ρ, V ) of a nite group G in a complex vector space V is a group homomorphismρ : G→ GLC(V ), equivalently it is a complex vector space with an action of G,i.e a G-module. When no ambiguity occurs we will denote the representation(ρ, V ) by ρ or even by V .

The degree of a representation V is the dimension of V . Two representations(ρ, V ) and (ρ′, V ′) are called equivalent if V and V ′ are isomorphic as G-modules,i.e. if there is an isomorphism T : V → V ′ compatible with the operation of G:

∀g ∈ G,∀v ∈ V, ρ′(g)T (v) = T (ρ(g)v).

The character χ of (ρ, V ) is the map χ : G→ C dened by χ(g) = Trace(ρ(g)).χ(1) is equal to the degree of ρ, i.e. to the dimension of V. Furthermore, χ isobviously, a class function, that is to say that χ is constant on each conjugacyclass of G.

The character of the direct sum of two representations is the sum of theircharacters and the character of the tensor product of two representation is theproduct of their characters.

A representation is called irreducible if V doesn't admit any G-invariantsubspace other than V itself, a character is called irreducible if it comes froman irreducible representation. Every representation V factors into a direct sumV = n1V1 ⊕ n2V2 ⊕ · · · ⊕ nrVr. Actually the multiplicities ni can be computedusing characters. We dene an hermitian inner product on the set of all classfunctions by

< χ,χ′ >=1

Card(G)

∑g∈G

χ(g)(χ′(g)).

Then ni =< χ,χi > where χi is the character of the irreducible representationVi. Thus, a representation is determined by its character.

There are two usual representations of a nite group G. The trivial repre-sentation, ρ : G → GL(V ) where dimV = 1 and ρ ≡ 1 which we denoted by1G. The regular representation of G, denoted RG, is given by the G-moduleV = C[G] =

∑g∈G xgg|xg ∈ C on which G acts by multiplication.

If H is a subgroup of G, there is a natural (in fact universal) way of as-sociating to a representation ρ of H, a representation of G called the inducedrepresentation. Let (ρ, V ) be a representation H the induced representation(Ind(ρ), IndGH(V ) is dened by the induced G-module

IndGH(V ) = f : G→ V |∀h ∈ H, f(hx) = hf(x)

on which g ∈ G acts by (gf)(x) = f(gx). If χ is the character of ρ the inducedcharacter IndHG (χ) of Ind(ρ) is:

IndHG (χ)(g) =∑

τ∈G/H

χ(τgτ−1),

where τ varies over a system of representatives on the right of G/H and we putχ(τgτ−1) = 0 if τgτ−1 doesn't belong to H.

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32CHAPTER 3. FROMCLASSICAL TO CONTEMPORARYNUMBER THEORY

3.2.2 Basic properties of Artin L-functions

Actually the Artin L-function L(s, ρ, L/K) doesn't depend essentially on ρ buton its equivalence class so we may write L(s, χ, L/K) for L(s, ρ, L/K) where χis the character of ρ.

The following theorems, due to Artin, can be found in [7] and [37], see also[39].

Theorem 3.2.1 Let G be the Galois group of a Galois extension L/K of num-ber elds. Let H and N be subgroups of G with N normal. Let χ and χ′ becharacters of G. Let ψ be a character of H and let IndHG (χ) the induced char-acter on G. Let η be a character of G/N and let η the pull back of η to G.Then

1. L(s, χ1 ⊕ χ2, L/K) = L(s, χ1, L/K)L(s, χ2, L/K)

2. L(s, ψ, L/LH) = L(s, IndHG (ψ), L/K)

3. L(s, η, LN/K) = L(s, η, L/K)

A rst link with zeta functions is made in the following theorem:

Theorem 3.2.2 Let G be the Galois group of a Galois extension L/K of num-ber elds. Let 1G and RG be the trivial character and the regular characterrespectively.

1. L(s,1G, L/K) = ζK(s)

2. L(s,RG, L/K) = ζL(s)

3. ζL(s) =∏χ∈G L(s, χ, L/K)χ(1)

3.2.3 Artin vs Hecke

In this section we begin by recalling local class eld theory in order to denethe norm residue symbol that will be used in the second part of this text. Nextwe recall global class eld theory. This is motivated by the denition of Weilgroups (see below) which involves reciprocity and by its use in the article [14]that we want to study. Finally we make the link between Artin L-functions andHecke L-functions following [37].

Some local class eld theory

Class eld theory describe abelian Galois extension. This section as two purpose: stating Artin reciprocity and dening the local norm residue symbol, also calledthe local Artin symbol. First of all we state the existence theorem of local classeld theory.

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3.2. ARTIN L-FUNCTIONS 33

Theorem 3.2.3 (Existence theorem, [37] ch. V) Let K be a local eld. Themap

L→ NL := NL/KL∗

, where NL/K is the usual norm, gives a one to one correspondence betweenthe nite abelian extension of K and the open subgroup of nite index in K∗.Furthermore

• L1 ⊆ L2 ⇔ NL1 ⊆ NL2

• NL1L2 = NL1 ∩NL2

• NL1∪L2 = NL1NL2

Theorem 3.2.4 (Local reciprocity law) For every nite Galois extension L/Kof local elds we have an isomorphism, called Artin reciprocity map,

rL/K : Gal(L/K)ab→K∗/NL/KL∗,

where Gal(L/K)ab is the abelianization of Gal(L/K) dene as its quotient bythe commutator group.

We are ready to dene the local norm residue symbol

Denition 3.2.2 The local norm residue symbol is dened by inverting the localreciprocity map, it gives

( , L/K) : K∗ → Gal(L/K)ab

which as kernel NL/KL∗.

Some global class eld theory

We denote by CK the idèle class group JK/K of a number eld K. For globalelds, the existence theorem takes the form

Theorem 3.2.5 (Existence Theorem)There is a 1:1 correspondence between nite abelian extensions L/K and closedsubgroups of nite index of CK given by

L 7→ NL = NL/KCL,

where NL/K is the usual norm. Moreover

1. L1 ⊆ L2 ⇔ N1 ⊇ N2

2. NL1L2 = NL1 ∩NL2

3. NL1∩L2 = NL1NL2

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34CHAPTER 3. FROMCLASSICAL TO CONTEMPORARYNUMBER THEORY

The eld L/K corresponding to a subgroup NL of CK is called the class eld ofNL and satises

Gal(L/K) ∼= CK/N

Theorem 3.2.6 (Global reciprocity law) For every Galois extension L/K ofnite algebraic number elds we have a canonical isomorphism

rL/K : Gal(L/K)ab→CK/NL/KCL.

We dene the global norm residue symbol by way of this isomorphism

Denition 3.2.3 We dene the global norm residue symbol ( , L/K) as theinverse of rL/K , it gives

( , L/K) : CK → Gal(L/K)ab

with kernel NL/KCL.

The link between local and global norm residue symbol is contained in thefollowing proposition.

Proposition 3.2.1 (Local-global relationship, product formula, [37] ch. VI) IfL/K is an abelian extension, p is a place of K and Lp a completion of L abovep, then

• The following diagram is commutative

K∗p( ,Lp/Kp)//

Gal(Lp/Kp)

CK

( ,L/K) // Gal(L/K)

. Where the left arrow is the canonical embedding.

• (Product formula) For a ∈ K∗∏p

(a, Lp/Kp) = 1.

and, more generally, for α = (αp) ∈ IK ,

(α,L/K) =∏p

(αp, Lp/Kp).

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3.2. ARTIN L-FUNCTIONS 35

Artin symbol and Artin L-functions

If n is an integer, we dened in (1.3.1) the higher unit groupsU (n) = 1 + πnpOp of the completion Kp of a number eld K.

Denition 3.2.4 (congruence group, ray class group)We dene a nite cycle as a formal product

m =∏p-∞

pnp , np ≥ 0.

The congruence subgroup modulo m is dened by

ImK =

∏p

U(np)p

and the ray class group modulo m is ClK := CK/CmK where

CmK = Im

KK∗/K∗.

Denition 3.2.5 (ray class elds)The class eld associated to Cm

K by the existence theorem is denoted Km/K andcalled the ray class eld modulo m.

The Galois group of a ray class eld is canonically isomorphic to the rayclass group modulo m, i.e Gal(Km/K) ∼= CK/C

mK and every abelian extension

L/K is contained in a ray class eld. This motivates the following denition.

Denition 3.2.6 Let L/K be a nite abelian extension and let NL = NL/KCL.The conductor f of L/K is the greatest common divisor of all nite cycles msuch that L ⊆ Km.

To an extension L/K and primes p (resp. P) of K and L respectively, weassociated a Frobenius automorphism φP . We dene[

L/K

p

]= φP .

We are going to extend this denition.Let m be a nite cycle of K such that L lies in the ray class eld modulo

m, then each p - m is unramied in L because there is an equivalence betweenbeing ramied in L and dividing the conductor of L/K. We then dene a grouphomomorphism, called the Artin symbol:[

L/K]

: JmK → Gal(L/K)

from the group of all ideals of K relatively prime to m by putting :[L/K

a

]=∏p

[L/K

p

]νp

,

for an ideal a =∏

p pνp .

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36CHAPTER 3. FROMCLASSICAL TO CONTEMPORARYNUMBER THEORY

Proposition 3.2.2 [37]Let m be a nite cycle of K. We dene Pm

K as the group of principal ideal (a) ofK such that a ≡ 1 mod m and a is totally positive.

The Artin symbol[L/K

a

]only depends on the class amod Pm

K and dene an

isomorphism [L/K

]: Jm

K/Hm→Gal(L/K),

where Hm = (NL/KJmL )Pm

K .

Hence, let K/L be an abelian extension and f be the conductor of L/K. The

Artin symbol[L/K

a

]gives a surjective homomorphism

J f/P f → Gal(K/L)a mod Pm 7→

[L/K

a

].

Let χ be an irreducible character of the abelian group Gal(L/K), compos-ing it with the Artin symbol we get a character of the ray class group J f/P f

which induced a character on J f that we denote χ. It appears that χ is aGrössencharakter modulo f of type (p, 0).

Finally, a link between Hecke L-functions with Grössencharakters and ArtinL-functions is given in the following theorem ([37]):

Theorem 3.2.7 Let L/K be an abelian extension, let f be the conductor ofL/K and χ 6= 1 be an irreducible character of Gal(L/K) and χ the associatedGrössencharaker modulo f.

Then the Artin L-function for the character χ and the Hecke L-function forthe Grössencharakter χ are related by

L(s, χ, L/K) = L(χ, s)∏p∈S

11− χ(φP)N−s

,

where S = p | f|χ(IP) = 1.

3.2.4 The functional equation

Let χ be an irreducible character of Gal(L/K) we dene the Artin conductorof χ as the ideal f(χ) given by:

f(χ) =∏p-∞

pnp(χ),

with

np(χ) =∑i≥0

Card(Gi)Card(G0)

codimV (V Gi),

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3.2. ARTIN L-FUNCTIONS 37

where Gi is the i-th ramication group of L:

Gi = σ ∈ Gal(L/K)|∀a ∈ OL, vL(σa− a) ≥ s+ 1vL a normalized discret valuation of L

.

It is a theorem of Artin that np(χ) is a rational integer.If we want to obtain a functional equation for Artin L-function we need to

complete the L-function

L(s, χ, L/K) =∏p-∞

1det(1− ρ(φP)N(p)−s|V IP )

,

with factor at ∞.For every innite place p of K we put:

Lp(s, χ, L/K) =

ΓC(s)χ(1) if p is complexΓR(s)n

+ΓR(s+ 1)n

−if p is real

where n+ = χ(1)+χ(φP)2 , n− = χ(1)−χ(φP)

2 , φP is the Frobenius automorphismintroduced previously and the Γ−factors are dened by

ΓR(s) = π−s/2Γ(s/2), ΓC(s) = 2(2π)−sΓ(s).

Denition 3.2.7 (Completed Artin L-function)The completed Artin L-function for a character χ of Gal(L/K) is dened to be

Λ(L/K) = A(χ)L∞(s, χ, L/K)L(s, χ, L/K),

where

A(χ) = |dK |χ(1)N(f(χ)), and L∞(s, χ, L/K) =∏p|∞

Lp(s, χ, L/K).

Proposition 3.2.3 ([37]) If χ is a character of degree 1 of Gal(L/K) and χis its associated Grössencharakter then the completed Artin L-function and thecompleted Hecke L-function coincide:

Λ(s, χ, L/K) = Λ(s, χ)

This proposition together with a theorem of Brauer according which anycharacter decomposes into a sum of characters induced by characters of degree1 on subgroups of Gal(L/K) permits to prove ([37]) the functional equationfor Artin L-function provided that we know the functional equation of HeckeL-functions with Grössencharakters.

Theorem 3.2.8 (Functional equation of Artin L-functions)The Artin L-function Λ(s, χ, L/K) admits a meromorphic continuation to Cand satises the functional equation

Λ(s, χ, L/K) = W (χ)Λ(1− s, χ, L/K)

with W (χ) ∈ C (called the root number) of absolute value 1.

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38CHAPTER 3. FROMCLASSICAL TO CONTEMPORARYNUMBER THEORY

3.3 Weil L-functions

Let us call Weil L-functions functions associated to representations of somegroups, called Weil groups, related to Galois groups. These L-functions includeas special cases the L-functions of Hecke and the L-functions of Artin. Webegin by presenting Weil groups. A good synthetic reference for what followsis [55], [54] and [58] are also good references. Remark that the term Weil L-function is not used in the literature, one prefers to use the general term ofArtin L-functions as does Langlands in [30] or simply L-functions.

3.3.1 Generalities

Every eld K is equipped with a distinguish Galois extension, the separableclosure K/K, its Galois group GK = Gal(K/K) is called the absolute Galoisgroup of K. This group is often of innite degree but it collects all informationabout the Galois nite extension ofK. Hence, it is an important object of study.However the main theorem of Galois theory needs to be adapted, it is false forinnite extension, even in simple cases as K = Fp.

There are two main ingredients to overcome this diculty. First of all for anyGalois extension (nite or innite) Gal(L/K) carries a natural topology calledthe Krull topology, secondly even if L/K is innite Gal(L/K) it is a pronitegroup, that is to say that Gal(L/K) is the projective limit of nite groups, eachgiven the discrete topology.

The Krull topology is dened as follows: for every σ ∈ Gal(L/K) and forL/K ranging over the nite subextensions of L/K we take the cosets σGal(L/L))as a basis of neighborhoods of σ. As a consequence the Krull topology is com-pact Hausdor.

The main theorem of Galois theory is restated in this context by:

Theorem 3.3.1 (Main theorem of Galois theory)Let L/K be any Galois extension. Then there is a 1:1 correspondence betweenthe subextensions L/K of L/K and the closed subgroups of Gal(L/K) such thatthe open subgroups of Gal(L/K) are given by the nite subextensions of L/K.This correspondence is given by

L 7→ Gal(L/L).

If L/K is a Galois extension, the Galois group Gal(L/K) with the Krulltopology is the projective limit of the Galois groups Gal(Lf/K) of the niteGalois extensions Lf/K which are contained in L/K.

Let us remind that we denoted CK = IK/K the idèle class group of a numbereld K. We then state the global reciprocity law of class eld theory that wewill need later.

If G is a topological group we denote by Gab = G/[G,G] the maximal abelianHausdor quotient of G, where [G,G] is the closure of the commutator subgroupof G.

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3.3. WEIL L-FUNCTIONS 39

Theorem 3.3.2 (Artin reciprocity law)For every Galois extension L/K of nite algebraic number elds we have acanonical isomorphism

rL/K : Gal(L/K)ab−→CK/NL/KCL.

The inverse map of rL/K yields a surjective homomorphism

( , L/K) : CK −→ Gal(L/K)ab,

called the global norm residue symbol.

If H is a subgroup of nite index of a topological group there is a transferhomomorphism (cf. [47] VII.3 propositions 7-8)

t : Gab −→ Hab,

dened as follows: if s : H \G→ G is any section, then for g ∈ G,

t(g[G,G]) =∏

x∈ H\G

hg,x mod [H,H],

where hg,x ∈ H is dened by s(x)g = hg,xs(xg).

3.3.2 Weil Groups, according to Tate in [55]

Here, K will denote a local or a global eld and K its separable closure. By Ewe will mean a nite extension of K in K/K, and GE will denote the Galoisgroup Gal(K/E).

Denition 3.3.1 A Weil group for (K/K) is a triple (WK , φ, rE) where

• WK is a topological group.

• φ is a continuous homomorphism φ : WK → GK with dense image.

We dene WE := φ−1(GE) for each nite extension E of K in K.

• For each E, rE is an isomorphism of topological groups rE : CE→W abE

where

CE =

The multiplicative group E∗ in the local caseThe idèle class group A∗E/E∗ in the global case.

This triplet is subject to the conditions (W1) to (W4) below.

• (W1)For each E, the composed map

CErE // W ab

E

induced by φ // GabE

is the reciprocity law of class eld theory (Theorem 1.3.8).

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40CHAPTER 3. FROMCLASSICAL TO CONTEMPORARYNUMBER THEORY

• (W2)Let w ∈ WK and σ = φ(w) ∈ GK . For each E the following diagram iscommutative

CErE //

induced by σ

W abE

conjugaison by w

CEσ

rEσ // W abEσ

• (W3)For E′ ⊂ E the following diagram is commutative

CE′rE′ //

induced byinclusionE′ ⊂ E

W abE′

transfer

CE

rE // W abE

• (W4)If WE/K denotes WK/W

cE where W c

E = [WE : WE ], the following map isan isomorphism of topological groups

WK −→ proj limEWE/K.

Proposition 3.3.1 (Properties of Weil groups [54])

1. Weil groups exist and are unique up to isomorphism: If WK and W ′K aretwo Weil groups for K/K, there exists an isomorphism θ : WK→W ′K suchthat the following diagrams are commutative

WK

φ

""EEE

EEEE

E

θ

GK

W ′K

φ′

==zzzzzzzz

W abE

induced by θ

CE

rE;;wwwwwwwww

rE′ ##GGG

GGGG

GG

(W ′E)ab

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3.3. WEIL L-FUNCTIONS 41

2.

WK/WE

∼= induced by φ // GF /GE

3. If (WK , φ, rE) is a Weil group for K/K and if E is a nite Galoissubextension, (WE , φ|WE , rE′) is a Weil group for K/E.

4. If WK is a Weil group, then for F ⊂ E′ ⊂ E the following diagram iscommutative

CErE //

normNE′/E

W abE

induced byWE ⊂WE′

CE′

r′E // W abE′

So the construction of Weil groups is functorial and satises a local-globalrelationship given by

Proposition 3.3.2 (prop. 1.6 of [55], local-global relationship) Let K be aglobal eld, ν a place of K and Kν the corresponding completion. Let K(resp.Kν) the separable closure of F(resp. Kν and WK(resp. WKν ) the correspondingWeil group.

Let iν : K → Kν be an F-homomorphism. For each nite extension E ofK in K, let Eν = i(E)Kν the induced completion of E. There is a continuoushomomorphism θν : WKν → WK such that the following diagrams are commu-tative.

WKν//

θν

GKν

induced byiν

WK

// GK

E∗ν∼ //

W abKν

induced byiν

CE

∼ // W abE

where nν maps a ∈ E∗ν to the class of the idèle whose ν-component is a andwhose other components are 1

3.3.3 Special cases

Here we discuss the particular cases when K is a local eld, a global functioneld or a global number eld.

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42CHAPTER 3. FROMCLASSICAL TO CONTEMPORARYNUMBER THEORY

• if k is a nite eld with q elements, the Weil group Wk is the subgroup ofGal(k/k) generated by ( topological generator) the Frobenius endomor-phism φ : x 7→ xq in k. So Wk

∼= Z. We make Wk into a topological groupby giving it the discrete topology.

Denition 3.3.2 (Geometric Frobenius)If φ ∈Wk is the Frobenius endomorphism, we dene the geometric Frobe-nius as F = φ−1 ∈Wk.

• K local non-archimedeanFor each E we consider, kE the residue eld and qE = Card(kE). We putk =

⋃E kE . WK is the dense subgroup of GK consisting of the elements

σ ∈ GK which induce on k the map x→ xqnK for some n ∈ Z. The inertia

group IK is then contained in WK and the topology of WK is such thatIF gets the pronite topology induced from GK and is open in WK . φ isthe inclusion and rE is the reciprocity law.

There is an exact sequence

1 // IK // WKπ // Wk

// 1 ,

andWK =

⋃n∈Z

ΦnIK ,

for any Φ ∈ Gal(K/K) such that π(Φ) = F , where F is a geometricFrobenius.

• K local archimedeanIf K ∼= C we take WK = K∗, φ trivial and rK the identity.If K ∼= R we take WK = K

∗ ∪ jK∗ with the relations j2 = −1 andjcj−1 = c for c the non trivial element of GK . φ maps K

∗to 1 and jK

to c. rK is the identity and rK is characterized by

rK(−1) = j[WK : WK ]rK(x) =

√x[WK : WK ], for x > 0

• K a global function eldThe construction is similar to that of a local non-archimedean eld withthe constant eld in place of residue eld and geometric Galois groupGal(K/Kk) in place of inertia group.

• K a global number eldThere is no simple known construction in this case, only a cohomologicalconstruction due to Weil in [58]

3.3.4 L-functions and ε-factors

Here we bring together the idea of Hecke, Tate, Artin and others to build atheory of L-functions associated to representations of Weil groups.

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3.3. WEIL L-FUNCTIONS 43

Local case

Let K be a local eld, OK its integer ring, π a uniformizing parameter, k itsresidue eld, p the characteristic of k, q its the cardinal and I the inertia groupof K/K. We have q = pd where d = [k : Fp]. Moreover, let K be a separableclosure of K, OK the integral closure of OK in OK and k the correspondingresidue eld which is a separable closure of k.

On note ωs : x 7→ ‖x‖s the (quasi-character) group homomorphism K∗ →C, where ‖x‖ = q−v(x) is the normalized absolute value corresponding to thevaluation v.

If χ is a quasi-character χ : K∗ → R∗ we dene the L-factor L(χ) by (1.3.1p. 20).

Proposition 3.3.3 (Tate)Let K be a local eld, ψ an additive character of K, dx an additive Haar measureon K, χ : K∗ → C∗ a quasi-character and f a smooth function with compactsupport on K.

We dened the Fourier transform of f by

f(y) :=∫K

f(x)ψ(xy)dx.

There exists a constant ε(χ, ψ, dx) ∈ C∗ independent of f which satises Tate'slocal functional equation:∫

K∗f(x)ω1χ

−1(x)d∗xL(ω1χ−1)

= ε(χ, ψ, dx)

∫K∗

f(x)χ(x)d∗xL(χ)

.

Corollary 3.3.1 (Basic properties of ε-factors)

1. ∀ r > 0, ε(χ, ψ, rdx) = rε(χ, ψ, dx)

2. ∀ a ∈ K∗, ε(χ, ψ(ax), dx) = χ(a)‖a‖−1ε(χ, ψ, dx)

The construction of L-functions associated to representations of Weil groupsneeds the notion of virtual representations to be fully understood. It would betoo long to present it here, I send the reader back to the article of Deligne [9]which is a complete exposition of the subject. I present only what is necessaryfor understanding the subject.

Artin showed that there is a L-functions of representations of Weil groupsof local elds which is additive on exact sequences and such that L(V ) = L(χ)for a representation V of degree 1 associated to the quasi-character χ. As L isadditive, it can be dened by giving its values on irreducible representations.

K ∼= C In this case WK = K∗ is abelian and the only irreducible characters ofV are quasi-characters for which L as been dened previously.

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44CHAPTER 3. FROMCLASSICAL TO CONTEMPORARYNUMBER THEORY

K ∼= R In this case the only irreducible representations V of degree dierentfrom 1 are of the form V = IndFFχ for χ a quasi-character of F

∗= WF . We

then dene L(V ) = L(χ).

K non-archimedean With I the inertia group and F the geometric Frobeniuswe put

L(V ) = det(1− φ|V I)−1

The following proposition is proved in [9]

Proposition 3.3.4 1. For each exact sequence 0 → V ′ → V → V ′′ of rep-resentations of W (K/K):

L(V ) = L(V ′)L(V ′′)

2. Let E be a nite subextension of K/K and VE a complex representation ofW (K/E). Let VK be the induced representation of VL on W (K/K) then

L(VK) = L(VL).

The following theorem, rstly due to Langlands [30] has been proved another way by Deligne and can be found in [9].

Theorem 3.3.3 (Existence and uniqueness of local constants)There exists a unique function ε associated to a (K,K,ψ, dx, V, ρ) composedof a local eld K, an algebraic closure K, an additive character ψ on K, anadditive Haar measure dx, a complex nite dimensional vector space V and arepresentation ρ : W (K/K)→ GL(V ), such that

1. For each exact sequence of representations V ′ −→ V −→ V ′′

ε(V, ψ, dx) = ε(V ′, ψ, dx)ε(V ′′, ψ, dx)

2.

∀ a > 0, ε(V, ψ, adx) = adim(V )ε(V, ψ, dx)

3. If E is a nite separable subextension of K/K and VK is a representationof W (K/K) induced by a representation VL of W (K/L)

ε(VK , ψ) = ε(VL, ψ TrL/K)

4. If dim(V ) = 1, ρ corresponding to a quasi-character χ, we have

ε(V, ψ, dx) = ε(χ, ψ, dx)

I nish this section with some properties of ε-factors that will be needed laterin this text.

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3.3. WEIL L-FUNCTIONS 45

Denition 3.3.3 Let K be a non-archimedean local eld, OK its integer ring,π a uniformizing parameter. In accordance with the preceding notation we denethe numbers n(ψ) and a(V ) as:

n(ψ) is the largest integer n such that ψ(π−nOK) = 1.a(χ) is the exponent of the conductor of χ. It equals 0 if χ is unramied

and is the smallest integer m such that χ is trivial on U (m) = 1 + πnOK if χ isramied.

Proposition 3.3.5 (Properties of ε-factors)We keep the notations in use.•

∀ a > 0, ε(V, ψ(a·), dx) = (det(V ))(a)‖a‖−dim(V )ε(V, ψ, dx)

ε(V ⊗ ωs, ψ, dx) = (f(V ))−sδ(ψ)−sdim(V )ε(V, ψ, dx),

where δ(ψ) = qn(ψ) in the non-archimedean case and 1 in the archimedeancase, f(V ) the absolute norm of the Artin conductor of V which is denedadditively from the Artin conductors of characters (cf. paragraph "Artinvs Hecke in 1.3.1). f(V ) = qa(V ) in the non-archimedean cases and 1 inothers.

• If K is non archimedean and W unramied (inertia invariant)

ε(V ⊗W,ψ, dx) = ε(V, ψ, dx)dim(W ) det(W )(πa(V )+n(ψ)dim(V ))

Global case Let K be a global eld, for any place p let Kp be the completionof K at p. Let AK be the adèle group of K , dx a Tamagawa measure on AK (i.e.dx(AK/K) = 1 and dene the norm ‖ ‖ on AK by d(ax) = ‖a‖dx. Moreoverdene the quasi-character ωs : A∗K → C by ωs(a) = ‖a‖s.

Let ψ be an additive character on AK and denote its local component at aplace p by ψp. Similarly let χ be a quasi-character on A∗K = IK and denote itslocal components by χp.

We dene the global L-function and global factor of χ and ψ by

L(χ) =∏p

L(χp), and ε(χ) =∏p

ε(χp, ψp, dxp).

Denition 3.3.4 With the previous notation we dene L(s, V ) and ε(s, V ) by

L(s, V ) = L(V ⊗ ωs), and ε(V, s) = ε(χ, ωs ⊗ ψ, dx).

The central theorem of this theory is then

Theorem 3.3.4 (Functional equation)The previous products converge for s in some right half plane and dene afunction L(V, s) which is meromorphic in the whole s-plane and satises thefunctional equation

L(V, s) = ε(V, s)L(V ∗, 1− s),where V ∗ is the contragredient representation.

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46CHAPTER 3. FROMCLASSICAL TO CONTEMPORARYNUMBER THEORY

Actually, for "classical" Artin L-function, the ε-factors can be computedbecause they come from Hecke L-functions with Grössencharakters for whichthey are known. For Weil L-functions these factors come from representationsof Weil groups, what explains why they are uncomputable in general.

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Chapter 4

The geometric objects case

4.1 The general picture

Let X be a scheme of nite type over Spec Z, the residue eld at a closed pointx ∈ X is nite, let N(x) denote its order. The zeta function of X is dened tobe the Euler product

ζ(X, s) =∏x∈X

11− N(x)−s

,

where X is the set of closed point of X, the product is absolutely convergentfor <(s) > dim X and we have in particular :

ζ(X, s) =

ζ(s) if X = Spec Zζ(s− n) if X = Spec Z[T1, . . . , Tn]ζK(s) if X = Spec(OK)

.

4.1.1 Étale and l-adique cohomology

The good settings to study these zeta functions is by cohomology. Étale co-homology has been constructed in order to prove the Weil conjectures in thegeneral case and appears to be the correct basis for any further theory. For ageneral presentation of the subject I refer to Milne's notes [32]. In this sectionwe introduce what is necessary to dene l-adic cohomology.

If X and Y are non-singular varieties over an algebraically closed eld amorphism X → Y is étale at a point if the corresponding map of tangent spaceis an isomorphism. In the case of schemes the denition is more involved. Theterm étale means local isomorphism, in the case of schemes an étale morphismis a at and unramied morphism:

Denition 4.1.1 (atness)A homomorphism of rings A → B is at if the corresponding functor: M →M ⊗AM from A−modules to B −modules is exact.

47

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48 CHAPTER 4. THE GEOMETRIC OBJECTS CASE

A morphism of schemes φ : X → Y is at if, for all x ∈ X, the correspondinglocal homomorphism OY,φ(x) → OX,x is at.

Denition 4.1.2 (ramication)A local homomorphism of local rings f : A→ B is unramied if B/f(mA)B isa nite separable extension of A/mA.A morphism ψ : X → Y is unramied if it is of nite type (for the denitionsee [20] p. 84) and if, for all, x ∈ X the map OY,φ(x) → OX,x is unramied.

Denition 4.1.3 (Étale morphisms)A morphism of schemes φ : X → Y is étale if it is at and unramied (henceof nite type).

For the properties of étale morphism see [32] chap. 2.In order to prove the Weil conjecture, one may hope to dene a cohomol-

ogy theory on schemes for which a Lefschetz xed point formula holds as wasremarked by Weil. Grothendieck in a paper of 1957 constructed a general topol-ogy theory for categories. The ambient space is a category and the topology onis dened by a site.

Denition 4.1.4 (sites)Let C be a category. A site on C is dened by giving for every U ∈ Ob(C) aset of families of arrows, (Ui → U)i∈U , called a coverings of U , subject to thefollowing compatibility conditions:

1. for any covering (Ui → U)i∈I and any arrow V → U of C the berproducts Ui ×U V exist and (Ui ×U V → V )i∈I) is a covering of V;

2. if (Ui → U)i∈I) is a covering of U and if for each i ∈ I, (Vij → Ui)j∈Ji ,is a covering of Ui, then the composed family (Vij → U)i,j is a coveringof U;

3. for any U ∈ Ob((C)) the family ( Uid // U ) is a covering of U .

A compatible system of coverings on C is called a (Grothendieck) topology anda category C together with such a topology is called a site

If X is a scheme or a variety, we dene the étale site on X, denoted Xet

as the category Et/X whose objects are étales morphisms U → X and whosearrows are commutative triangles

U //

@@@

@@@@

V

~~~~~~

~~~

X

.

The coverings of Xet being the families of étales morphism (Ui → U)i∈I .

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4.1. THE GENERAL PICTURE 49

A sheaf of sets on Xet (resp. groups, rings, A-modules...) is a contravariantfunctor F : Et/X → Sets (resp.:Ab,Rg,A-mod, . . .) such that the followingsheaf axiom is satised:

For any (U → X) ∈ Et/X (étale) and for any étale covering (Ui → U)i∈I ,the following sequence is exact:

(sheaf axiom) F(U)→∏i∈IF(Ui)⇒

∏i,j

F(Ui ×U Uj)

With some technicalities, we can dene a category, Sh(Xet),whose objectsare sheaves of abelian groups on Et/X and whose arrows are natural transfor-mations. It can be shown (cf. [32]) that this category is abelian and possessenough injectives. The functor

Sh(Xet) −→ Ab

F 7−→ Γ(X,F)

is left exact. So we can dene classically the cohomology groups, Hr(Xet, ·), bytaking the rth right derived functor (cf. [23] for the corresponding theory).

There is a natural way to associate to Hr(Xet, ·) a vector space with goodproperties ; the rst step is to dene

Hr(Xet,Zl) = proj limnHr(Xet,Z/lnZ),

where l is prime and Z/lnZ stands for the constant sheaf. Actually, there is ageneral construction for what are called l-adic sheafs but we don't need it. Thenwe dene the vector spaces of l-adic cohomology by:

∀r ≥ 0, Hr(Xet,Ql) := Hr(Xet,Zl)⊗Ql,

this space is also denoted Hret(X,Ql).

4.1.2 Zeta functions for schemes over nite elds

Let X be a scheme over a nite eld Fq (q = pf ). For all closed point x ∈ X(the set of closed points), the residue eld k(x) at x is a nite extension of Fq,hence its cardinal is of the form N(x) = qdeg(x) for deg(x) = [k(x) : Fp]. Wedene the big" zeta function of X by

Z(X, t) =∏x∈X

1

1− tdeg(x),

whereas the classical zeta function is

ζ(X, s) = Z(X, q−s) =∏x∈X

11− N(x)−s

.

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50 CHAPTER 4. THE GEOMETRIC OBJECTS CASE

It can be shown (cf. [31]) by taking the developpement of log Z that:

Z(X, t) = exp

( ∞∑l=1

CardX(Fql)tl

l

),

where X(Fql) is the number of point of X in Fql .The particular properties of Zeta functions over nite eld are the subject

of the celebrated Weil's conjectures of Weil (1949) whose proofs were completedby Deligne in 1973. They rest on the fact that a Lefschetz xed-point formulaholds for endomorphisms acting on étale cohomology groups.

More precisely

Denition 4.1.5 The Frobenius morphism F : X → X of a scheme X over Fqis dened on every ane subscheme Spec A ⊂ X by the ring homomorphisma 7→ aq; on the topological space, X, F acts as the identity.

The Lefschetz-xed point formula implies that

CardX(Fql) =∑r

(−1)rTr(F l|Hret(X,Ql)),

where Tr(·) is the trace. This can be used to prove the rationality of Z(X, s).But the l-adic cohomology space are fundamental for other reasons.One of them is that they furnish the correct point of view for Zeta and

L-functions of varieties and schemes.Let X be a non-singular projective variety over a nite eld Fq, Fq the

algebraic closure of Fq and Xex = X ×Fq Fq the variety obtained by extensionsof scalars. For l 6= p and for m ≥ 0, we can dene the l-adic cohomologyspace Hm

et (Xex,Ql) on which the Frobenius morphism acts by functoriality as

an endomorphism Fl,m. Then we can dene

Pl,m(X,T ) = det(1− TFl,m|Hmet (X

ex,Ql)).

Then the following functions

Z(X,T ) =2 dim X∏m=0

Pl,m(T )(−1)m+1

are taken as the basic zeta functions and equal the preceding denition in spe-cialized cases.

4.1.3 The global case

In order to simplify the exposition, we consider only varieties in this section.The general case of scheme has been dealt with principally by Grothendieck andDeligne and can be found in the SGAs (Séminaire de Géométries Algébriques).

For details I refer to the text of Serre [46], it is to his merit to expose theprincipal ideas and conjectures of the subject.

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4.1. THE GENERAL PICTURE 51

Let K be a local eld, we denote by MK and M∞K the sets of place and ofnite places respectively. If p ∈ MK we denote by Kp the local eld obtainedby completion of K at p and by Op, k(p) and p the ring of integers, the residueeld and the residue characteristic at p respectively. For simplicity we supposethat k(p) is nite. The construction depends essentially on the nite eld case.

Let X be a non-singular projective variety over K. Let S ⊂ MK be a nitesubset such that X has good reduction apart from S. This means that, for allp ∈MK \ S, there exists a smooth projective Spec(Op)−scheme, Xp, such thatXp ×Op Kp

∼= X ×K Kp. For such an Xp, we denote X(p) = Xp ×Op k(p) itsreduction modulo p.

As X(p) veries the Weil conjecture, we can associate a polynomial Pm,p ∈Z[T ] to it, as done in the previous section, its degree Bm is the m − th Bettinumber of X.

Then we can dene

ζS(s) =∏

p∈MK\S

1Pm,p(N(p)−s)

which is absolutely convergent in the right half plane <(s) > 1 + m/2. Itimplies that ζS(s) is holomorphic in this right half plane and that is the sum ofa Dirichlet series with integral coecients.

The question of analytic continuation and of functional equation is dicult.We know that, for abelian varieties of C.M. type, we there is an analytic con-tinuation to the left of <(s) > m + 1 and for certain cases of automorphy ormodularity as for elliptic curves over Q with help of Wiles' big theorem.

In all these cases, their exist an rational number A > 0, a polynomial Pm,pfor each p ∈ S, and gamma factors, Γp(s) for p ∈M∞K such that if we put

ζ(s) = ζS(s)∏p∈S

1Pm,p(N(p)−s)

, and then ξ(s) = As/2ζ(s)∏

p∈M∞K

Γp(s)

then the following functional equation holds:

(∗) ξ(s) = w ξ(m+ 1− s), w = ±1.

Jean-Pierre Serre in [46] gives precise conjectures about the possible de-nition of the previous factors A and Pm,p, (p ∈ S). The previous ones seemto depend on the l-adic cohomology of the varieties X ×K Kp whereas the Γp,(p ∈ M∞K ), depend on the Hodge decomposition of the complex cohomology ofX ×K Kp.

The bad reduction case: p ∈ S Let p be a nite place of K, let Kp andG = Gal(Kp/Kp). Let l 6= p = char(k(p)), to (V, ρ), an l-adic representation,i.e. V is a d-dimensional vector space over Ql and

ρ : G −→ Aut(V )

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52 CHAPTER 4. THE GEOMETRIC OBJECTS CASE

a continuous representation , Serre associates in [46] two non-negative integers εand δ which dene the "ramication of ρ in some sense. He denes f(p) = ε+δ.

Remind that we supposed that k(p) is nite. If I is the inertia group, G/Icomes with the Frobenius generator φ that induces an automorphism of V I

called the geometric Frobenius and denoted Fρ. Then we dene

Pρ(T ) = det(1− TFρ|V I).

If X is a non-singular projective variety over Kp , if m ≥ 0 is an integer andif l 6= p we dene

Vl = Hmet (X ×Kp Kp).

Then G = Gal(Kp/Kp) acts on Vl which denes an l-adic representation ρl.Serre's conjecture C8 in [46] permits to dene:

Pm,p = det(1− TFρl |V Il ), for p ∈ S.

The gamma factors:p|∞ The denition of Γ-factors depend on Hodge struc-tures on the complex cohomology space and on the theorem of Hodge givenbelow.

An integral complex Hodge structure of weight k ∈ Z on a complex vectorspace VC, is a decomposition

VC =⊕p+q=k

V p,q, such that V p,q = V q,p (the complex conjugate).

A real integral Hodge structure of weight k on a complex vector space VC is acomplex decomposition of wight k together with an automorphism J of VC suchthat J2 = 1 and J(V p,q) = V q,p for all p+ q = k.

Let X be a complex variety, we denote by AkC(X) the space of complexdierential forms of degree k on X. Elements of AkC(x) are C∞ sections of thevector bundle ΩkX of dierential forms of degree k. We have a natural (DeRham) complex given by the exterior dierential d

d : AkC −→ Ak+1C

such that d d = 0. This allows to dene the k-th complex cohomology groupby:

Hk(X,C) :=Ker(d : AkC(X)→ Ak+1

C (X))Im(d : Ak−1

C (X)→ AkC(X))

. The following due to Hodge is proved in [56] in which Claire Voisin developHodge theory and more.

Theorem 4.1.1 (Hodge)Hk(X,C) admits a Hodge decomposition

Hk(X,C) =⊕p+q=k

Hp,q(X) such thatHp,q(X) = Hq,p(X),

where Hp,q(X) is given by the set of closed forms of type (p, q) on each x ∈ X.

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4.1. THE GENERAL PICTURE 53

If X is a complex non-singular variety over a global eld K there are twocases of innite places

• If p is a complex place of K, then from what Hodge theorem we canconsider the group Hk(X(Kp),C) and the corresponding Hodge decom-position.

• If p is a real place of K, C is a quadratic extension of Kp and we can con-sider the group Hk(X(C),C) and its Hodge decomposition. Furthermore,complex conjugation on C induces a real Hodge structure on this space.

To clarify the exposition, we will suppose that the following spaces and theirdecomposition are the preceding ones without mentioning them.

Denition 4.1.6 (Gamma factor at complex places)If V k =

⊕V p,q is a complex vector spaces endowed with a Hodge structure we

deneΓV =

∏p+q=k

ΓC(s− inf(p, q))dim V p,q .

where ΓC is the gamma function dened in 1.3.2

If V =⊕

p+q=k Vp,q is a real Hodge decomposition with automorphism J ,

Serre denes a decomposition

V n,n = V n,+ ⊕ V n,−

from the invariance of V n,n under J, where:

V n,+ = x ∈ V n,n|J(x) = (−1)nxV n,− = x ∈ V n,n|J(x) = (−1)n+1x

Denition 4.1.7 (Gamma factors at real places). With the preceding nota-tions, if V k =

⊕V p,q is a real Hodge decomposition, we dene the gamma

factor

ΓV (s) =∏

2n=k

ΓR(s−n)dim V n,+ΓR(s−n+1)dim V n,−∏

p<q,p+q=k

ΓC(s−p)dim V p,q ,

where ΓC and ΓR are dened in 1.3.2 .

The constant A Remind that we consider a number f(p) attached to an l-adic representation of Gal(Kp/Kp) for p a place of bad reduction. This factoris completely dened in [46] and we associate to it a divisor

f =∑p∈S

pf(p)

and its corresponding norm

N(f) =∏p∈S

pf(p).

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54 CHAPTER 4. THE GEOMETRIC OBJECTS CASE

Moreover, we dene

D =|dK/Q|, if K is a number eldq2g−2, if K is a function eld of genus g over Fq

The constant A of the functional equation is then conjecturally equal to

A = N(f)DBm ,

where Bm is the m-th Betti number of X.

Conjecture 4.1.1 With the preceding denition, and with

ξ(s) = As/2ζ(s)∏p|∞

Γp(s)

the following functional equation holds :

ξ(s) = wξ(m+ 1− s), with w = ±1.

4.2 L-functions

Here we present L-function with the help of what we previously learned.

4.2.1 Weil groups again: The Weil-Deligne group

We saw that l-adic representations are of special importance in the theory ofL-functions. However some l-adic representations of usual Weil's group don'tprovide good objects because we can't associate them continuous complex rep-resentations which are natural in the theory of L-functions.

The reason for this is that, if K is a local non-archimedean eld, a complexrepresentation ρ : WK → Gln(C) is by denition a continuous group homomor-phism. In order to be continuous, it is necessary and sucient (cf. [42]) thatρ be trivial on a open subgroup of IK (the inertia group seen as a subgroup ofGal(K/K) with the Krull topology), whereas it is not true for important casesof l-adic representations. The way to bypass this is to consider representationsof a larger group: the Weil-Deligne group

Let K be an non-archimedean local eld. Remind from 1.3.3 Special cases,that we have an identication

WK =⋃n∈Z

ΦnIK ,

where Φ maps to the inverse of the Frobenius endomorphism of Gal(k/k) for kthe residue eld of K. There is a natural quasi-character ω of WK given by

ω : WK −→ C×IK 7−→ 1Φ 7−→ q−1

,

ω is unramied by denition.

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4.2. L-FUNCTIONS 55

Denition 4.2.1 (Weil-Deligne group)The Weil-Deligne group WK of K is the semidirect product

WK := WK n C

where the action of WK on C is given by

∀g ∈WK ,∀z ∈ C, gzg−1 = ω(g)z.

Actually the Weil-Deligne group may be seen as a group scheme over Q as in[55] (denition (4.1.1)) or [9] (8.3.6) but we don't need this.

If L/K is a nite separable extension of K and if ΦL is an inverse Frobeniusof L, i.e. if ΦL is mapped to the geometric Frobenius FL de Gal(k/l) then asan element of Gal(k/k), FL = F f(L/K), where F is the geometric Frobenius ofk/k and f(L/K) is the residue class degree. So ω(ΦL) = q−1

L and

ω|WL = ωL.

Hence, WL may be viewed as a subgroup of WK .A representation of WK over a complex vector space V is a continuous

homomorphismρ :WK → GL(V ).

This is equivalent to the following denition

Denition 4.2.2 A (resp. l-adic) representation ofWK over a (resp. Ql)complexvector space V (resp. Vl) is a pair ρ′ = (ρ,N) (resp. (σl, Nl)) consisting of

1. a homomorphism ρ :WK → GL(V ) (resp.Vl) which is trivial on an opensubgroup of IK .

2. a nilpotent endomorphism N of V (resp. Nl of Vl) such that ρ(g)Nρ(g)−1 =ω(g)N , for all g ∈WK . (resp. σl(g)Nlσl(g)−1 = ω(g)Nl)

One obtains a representation of WK by putting:

ρ′(gz) = ρ(g)exp(zN), for g ∈WK and z ∈ C,

so that

N =log σ′(z)

z

which as a sense because ([42], p.129) σ′(z) is unipotent.It is straightforward to verify that if σ′ = (σ,N) and τ ′ = (τ, P ) are two

representations of WK :

• σ′ ⊕ τ ′ = (σ ⊕ τ,N ⊕ P )

• σ′⊗τ ′ = (σ⊗τ,N⊗1⊕1⊗P ), 1 being the adequate identity automorphism.

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56 CHAPTER 4. THE GEOMETRIC OBJECTS CASE

What are of particular importance for us are l-adic representations σl ofGal(K/K), i.e. continuous group homomorphism

σl : Gal(K/K)→ Gl(Vl),

where Vl is a nite dimensional Ql-vector space. The reason for us to considerWeil-Deligne groups and their representations is that there is a simple method,due to Grothendieck and Deligne, for associating to an l-adic representation σlof Gal(K/k) a complex representation σ′l,ı of WK for an embedding ı : Ql → C.

Actually, if σl is an l-adic representation of WK , composing it with theextension of scalars Gl(Vl) → Gl(C⊗ı Vl gives

σl,ı :W → Gl(C⊗ı Vl).

Similarly, applying extension of scalars End(Vl) → End(C⊗ı Vl) gives Nl,ı.So representations of the Weil-Deligne group will be of use to us.

Denition 4.2.3 A representation σ′ = (σ,N) of WK is called Φ-semisimpleor admissible if σ is semisimple.

A representation σ′ is called indecomposable if it space can not be written asa direct sum of invariant spaces.

Actually, a representation σ′ is Φ-semisimple if σ(Φ) is semi-simple for someinverse Frobenius element Φ.

There is a particular representation which play a central role, they are calledspecial representations and dened by :

Denition 4.2.4 (Special representation)Let e0, e1, . . . , en−1 be the canonical basis for Cn. The special representation

of dimension n, denoted sp(n), is the representation σ′ = (σ,N) of WK , where:

• ∀g ∈ WK ,∀ 0 ≤ j ≤ n− 1, σ(g)ej = ω(g)jej

• ∀ 0 ≤ j ≤ n− 2, Nej = ej+1, andNen−1 = 0

The special representation is admissible, indecomposable, n-dimensional. (e.g.see [42] p. 132). Special representations appears in particular if we want tocharacterize admissible indecomposable representation which is the purpose ofthe following proposition.

Proposition 4.2.1 ([42] p. 133) Every admissible indecomposable representa-tion of WK is equivalent to a representation of the form π ⊗ sp(n), where π isan irreducible representation of WK and n is a positive integer.

It is a proposition of Deligne in Modular functions in one variable II, Formesmodulaires et representations of Gl(2)

Furthermore, admissible indecomposable representations of WK satisfy theequivalence of Schur's lemma for this case.

Rohrlich in [42] page 133 prove the following corollary by induction on virtualrepresentations.

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4.2. L-FUNCTIONS 57

Proposition 4.2.2 ([42] p. 133)Let σ′ be an admissible representation of WK . Then

σ′ =s⊕j=1

πj ⊗ sp(nj)

where πj is an irreducible representation of WK and nj is a positive integer.Furthermore, if

σ′ =t⊕

j=1

ρj ⊗ sp(mj)

then s = t, and after renumbering the summands nj = mj and πj ∼= ρj.

There is a natural Φ-semisimple representation associated to any represen-tation of WK .

Denition 4.2.5 ([55] (4.1.3) and [9] (8.5)) Remind that ω denote the un-ramied character of WK which takes the value q−1 on any Frobenius element.Thus there exists a map v : WK → Z such that ω(g) = q−v(g)

Let σ′ = (σ,N) be a representation of WK on a vector space V, there ex-ists a unique unipotent automorphism u of V such that u commutes with Nand σ(WK) and such that exp(aN)ρ(g)u−v(g) is a semisimple automorphismof V for all a ∈ K and all g ∈ WK \ I. Then σ′ss = (σu−v, N) is called a(Φ−)semisimplication of σ′ and σ′ is (Φ−)semisimple if and only if σ′ = σ′ss,which means that the Frobenius acts semisimply.

4.2.2 Conductors and ε-factors

As usual, let OK and πK be the ring of integers and a uniformizer of K. Ifσ′ = (σ,N), is a representation of the Weil-Deligne group WK , Deligne in [9]dene the conductor of σ′. It is an ideal of OK , denoted N(σ′), hence it takesthe form :

N(σ′) = πa(σ′)K OK .

The exponent of the conductor of σ′, i.e. a(σ′) is an integer which is related tothe exponent of the conductor, a(σ) of σ by :

a(σ′) = a(σ) + b(σ′),

where b(σ′) = Codim(V IN ) for V IN = V I ∩kerN and V I is the invariant subspaceof V under the inertia. In order to dene a(σ) we need more framework.

Remind that the inertia can be dened as I = Gal(K/Kur) where K isa separable closure of K and Kur is the maximal unramied subextension ofK/K. At present, let R be a nite Galois extension of Kur such that σ is trivialon the subgroup Gal(K/R), and put G = Gal(R/Kur). If vR is the valuationon R then we have a ltration G = G0 ⊃ G1 ⊃ G2 ⊃ · · · by higher ramicationgroups :

Gj = g ∈ G|vR(g(πR)− πR) ≥ j + 1,

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58 CHAPTER 4. THE GEOMETRIC OBJECTS CASE

where πR is a uniformizer on R. Then we dene :

a(σ) =∞∑j=0

Card(Gj)Card(G)

Codim(V Gj ).

This generalize the case of characters. This exponent satises the followingproperties (e.g. [42] p. 141)

Proposition 4.2.3 ([42] p. 141) Let σ′ and τ ′ be representations of WK , letL be a nite subextension of K/K, d(L/K) (resp. f(L/K)) be the exponent ofthe relative discriminant of L/K (resp. the residue class degree of L over K)and ρ′ a representation of L, then

1. a(σ′ ⊕ τ ′) = a(σ′) + a(τ ′)

2. a(IndL/Kρ′) = dim(ρ′)d(L/K) + f(L/K)a(ρ′)

In the case of admissible indecomposable representations, Rohrlich provedthe following proposition :

Proposition 4.2.4 ([42] p.141) Let σ′ = π ⊗ sp(n) be an admissible indecom-posable representation (proposition 4.2.1), where π is an irreducible representa-tion of WK and n is a positive integer. Then

a(σ′) =na(π) if π is ramiedn− 1 if π is unramied

.

ε-factors We dened previously ε-factors in the case where we are given atuple (K,K,ψ, dx, V, σ) (theorem 3.3.3) and we would like to dene a similarconstant for a representation σ′ = (σ,N) of WK . It was done by Deligne in [9]but the exposition of Rohrlich in [42] is straightforward.

Denition 4.2.6 (ε-factors) Let K,K,ψ and dx be as usual (theorem 3.3.3)and let now σ′ = (σ,N) be a representation of WK . Let ε(σ, ψ, dx) be theepsilon factor of theorem 3.3.3, then :

ε(σ′, ψ, dx) = ε(σ, ψ, dx)δ(σ′),

whereδ(σ′) = det(−σ(Φ)|V I/V IN ).

For this, we have properties given in 3.3.3 together with :

• δ(σ′ ⊕ τ ′) = δ(σ′)δ(τ ′)

• δ(IndL/Kρ′) = δ(ρ′)

with the preceding notations (e.g. proposition 4.2.3). The article of Rohrlich [42]contains the essential formulas, from which the following will be of importanceto us.

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4.2. L-FUNCTIONS 59

• ([42] Lemma p.144) If ψ is a character of K there exists a unique Haarmeasure dxψ on K such that the Fourier transform on the Schwartz spaceS(K) is an isometry of the L2 norm. We call such a measure a self-dualmeasure.

Let σ′

4.2.3 L-?

Denition 4.2.7 Let σ′ = (σ,N) be a representation of WK on a complexvector space V and I the inertia group of K/K. As usual V I = V σ(I) is theI−invariant subspace of V . Let V IN = V I ∩ ker N , as two inverse Frobeniuselement dier by an element of I, the action of σ(Φ) on V IN is independent ofthe inverse Frobenius Φ. Thus the following denition of the L-factor attachedto σ′ makes sense :

L(σ′, s) = det(1− q−sσ(Φ)|V IN )

This denition is similar to the denition of Artin L-functions we are know farfrom our Riemann-Dirichlet's origins. The similarity goes further :

Proposition 4.2.5 ([42] p. 137-138)Let σ′ and τ ′ be representation of WK , L/K a nite extension and ρ′ a repre-sentation of WL. Then :

• L(σ′ ⊗ τ ′, s) = L(σ′, s)L(τ ′, s)

• L(IndL/Kρ′, s) = L(ρ′, s)

If l is a prime dierent from the characteristic p and σ′l : Gal(K/K) →GL(Vl) is an l-adic representation they are two natural way to construct anL-factor.

• The rst one, in the continuation of what precede, is to consider an embed-ding ı : Ql → C and then to construct the representation σ′l,ı = (σl,ı, Nl,ı)of the Weil-Deligne group WK which gives us L(σ′l,ı, s)

• The other way is, given an embedding ı : Ql → C, to dene :

L(σ′l, ı, s) = ı(det(1− xσ′l(Φ)|V Il ))−1|x=q−s

Actually, this two denitions agree :

Proposition 4.2.6 ([42] p. 139)With the previous notations :

L(σ′l, ı, s) = L(σ′l,ı, s)

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60 CHAPTER 4. THE GEOMETRIC OBJECTS CASE

Furthermore, we dene the root number associated to an algebraically closedextension of local eld K/K, an Haar measure dx on K, a character ψ of Kand a representation σ′ of the Weil-Deligne group WK by

W (σ′, ψ) =ε(σ′, ψ, dx)|ε(σ′, ψ, dx)|

.

Usually, we take for dx a self-dual measure with respect to dx and denote theresulting factor by W (σ′).

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Part II

Parity conjecture for elliptic

curves

61

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Chapter 5

Elliptic Curves

For basic facts about algebraic geometry and elliptic curves see [20], [27], [52]and, [53], among others.

An elliptic curves is an abelian variety of dimension 1, i.e. it is a complete,non-singular, connected curve of genus one with a specied rational point de-noted O. What are of particular importance to us are elliptic curves over niteelds, over local elds and (of course) over number elds. It is worth notingthat elliptic curves have a long history going back to the computation of the arclength of ellipses. On C, an elliptic curve is up to homothety of the form C/Λwhere Λ is a lattice in C (i.e. a discrete subgroup which contains an R− basis),thus an elliptic curve over C looks like a torus (e.g. [52] ch. VI).

Let E/K be an elliptic curve over a eld K, then ([52] Prop. 3.1 p. 63)E can be embedded in the projective plane P2(K) as a cubic curve with thefollowing type of equation :

Weierstrass Equation

Y 2Z + a1XY Z + a3Y Z2 = X3 + a2X

2Z + a4XZ2 + a6Z

3

with a1, a2, · · · , a6 ∈ K and such that the basepoint O is represented by (0 : 1 :0).

This dening equation can be simplied (e.g. [52] p.46-47) if char(K) 6= 2and char(K) 6= 2, 3 as follows.

Weierstrass equation for char(K) 6= 2

Y 2Z = 4X3 + b2X2Z + 2b4XZ2 + b6Z

3.

63

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64 CHAPTER 5. ELLIPTIC CURVES

Weierstrass equation for char(K) 6= 2, 3

Y 2Z = X3 − 27c4XZ2 − 54c6Z3.

Where the coecients are given by

b2 = a21 + 4a2 c4 = b22 − 24b4

b4 = 2a4 + a1a3 c6 = −2b32 + 36b2b4 − 216b6b6 = a2

3 + 4a6

.

Furthermore let us dene to important quantity ∆ and j called respectively thediscriminant and the j−invariant of the elliptic curve by

∆(E) =c34 − c261728

j(E) =c34

∆(E).

A curve given by a Weierstrass equation has at most one singular point :

Proposition 5.0.7 (proposition 1.4 p.50 of [52]) The curve given by a Weier-strass equation can be classied as follows.

1. It is non-singular if and only if ∆ 6= 0.

2. It has a node if and only if ∆ = 0 and c4 6= 0.

3. It has a cusp if and only if ∆ = c4 = 0.

Moreover to elliptic curves over K are isomorphic over K is and only if theyhave the same j−invariant and every element of K can be the j− invariant ofan elliptic curve on an appropriate eld.

What is a particular characteristic of elliptic curves (and abelian varieties)is that there denition implies that they are endowed with an algebraic grouplaw. Namely, for this law, three points P1, P2, P3 sum to O if and only if theWeil divisor P1 + P2 + P3 is the intersection of E with a line.

The natural morphisms for elliptic curve are called an isogenies and aredened as follows :

Denition 5.0.8 (Isogenies) Let E and E′ be elliptic curves, an isogeny be-tween E and E′ is a morphism

φ : E → E′

such that φ(O) = O.

From the theory of curves (e.g. [52] ch. II), an isogeny is either trivial orsurjective. As an elliptic curve is an algebraic group, we have the followingnatural isogenies : let m be a positive integer we dene the isogeny [m], calledthe multiplication by m map, by

[m]P = P + P + · · ·+ P︸ ︷︷ ︸m times

,

for every P ∈ E.An interesting property of isogenies is that they are nearly invertible in

the sense of the following theorem

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65

Theorem 5.0.1 ([52] ch. III) Let φ : E → E′ be an isogeny of degree m.

• There exists a unique isogeny, called the dual isogeny,

φ : E′ → E

satisfyingφ φ = [m].

• This last isogeny being dened, it also satises

φ φ = [m] on E′.

If φ : E → E′ is an isogeny, let us dene E(φ) by E(K)[φ] = ker(φ : E(K) →E′(K) and E[φ] = E(K)[φ], in particular for the endomorphism [m] we willdenote it E[m]. This last group is of the following kinds

Proposition 5.0.8 ([52] Ch. III Cor. 6.4 p 89) Let E/K be an elliptic curveover a eld K and m be a nonnegative integer then

1. If char(K) = 0 and if m is prime to char(K), then

E[m] ∼= (Z/mZ)× (Z/mZ).

2. If char(K) = p, prime, then either :

E[pe] = O for all e = 1, 2, . . . ; orE[pe] = Z/peZ for all e = 1, 2, . . . .

When K is of characteristic p and E[pe] = O for all e, we say that E issupersingular otherwise we say that it is ordinary.

From the denition of elliptic curves we can deduce the notable fact thatit is an algebraic group, the second notable fact is that isogenies are actuallyalgebraic group homomorphisms (e.g. [52] Ch. III Th. 7.5).

From the groups E[ln] we can construct a Galois representation which isactually the Galois representation that come from l−adic cohomology groupsH1(e.g. ch.4) and so can be used to construct the L-functions.

Let E/K be an elliptic curve and K the algebraic closure of K. ThenGal(K/K) acts on E[m] because, for σ ∈ Gal(K/K) and P ∈ E[m], [m]Pσ =([m]P )σ = O. This gives a representation :

Gal(K/K)→ Aut(E[m]).

This is far from sucient for our purpose as we would like l−adic representation.The purpose of the following theorem is to dene a Zl−module which tensoredby Ql will be of use :

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66 CHAPTER 5. ELLIPTIC CURVES

Denition 5.0.9 Let E be an elliptic curve and l a prime. The l−adic Tatemodule of E is :

Tl(E) := proj limnE[ln],

the inverse limit coming from the natural maps

E[ln+1][l] // E[ln] .

From the previous proposition (prop. 5.0.2) we deduce immediately

Proposition 5.0.9 The Tate module of E/K is a Zl module and has the fol-lowing structure :

• Tl(E) ∼= Zl × Zl if l 6= char(K).

• Tp(E) = 0 or Zp if p = char(K) > 0

Furthermore, the action of Gal(K/K) on each E[ln] commutes with themultiplication by l map, so it acts on the Tate module Tl(E). Further as E[m]is nite, it is a discrete Gal(K/K)-module, so this action is continuous ([38] ch.I prop. (1.1.8)) and the resulting action on Tl(E) is also continuous. To obtainan action over an vector space we put :

Vl(E) := Tl(E)⊗Ql,

which gives the l−adic representation denoted ρE,l :

ρE,l : Gal(K/K)→ AutQl(Vl(E)).

5.1 Basic facts

The natural objects of study are elliptic E curves over number elds K. Thenotable fact is that the group E(K) is nitely generated, this is known as theMordell-Weil theorem. However we won't be concerned by this here but by themachinery of number elds, their completion to local elds and the reduction tonite elds. This goes as follows : LetMK be the set of places of the number eldK and p ∈MK , let Kp, Op, mp, kp = Op/mp be the corresponding completion,ring of integers, its maximal ideal and residue eld respectively.

Denition 5.1.1 With the preceding notations, one says that E/K has goodreduction at p if one can nd a coordinate system in P2(K) such that the corre-sponding equation of E has coecient in Op and its reduction modmp denesa non-singular cubic Ep, called the reduction of E at p.

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5.1. BASIC FACTS 67

5.1.1 Elliptic curves over local elds

If E/K is an elliptic curves over a local eld, we dene its reduction E(k)the corresponding reduction according to the preceding denition (k denote theresidue eld of the local eld K), the reduced curve can be singular in general.We denote K,OK ,mK , k = OK/mK , v : K → Z a local eld, its ring of integer,the corresponding maximal ideal, the residue eld and the valuation of K.

We need to dene the sets

E0(K) = P ∈ E(K)|P ∈ Ens(k),E1(K) = P ∈ E(K)|P = O,

where Ens(k) denote the set of non-singular point of the curve E/k. The fol-lowing sequence is exact (e.g. [52] ch. VII prop. 2.1)

0 // E1[K]injection // E0(K) reduction //// Ens(k) // 0 .

If K = Kv is the completion of a number eld at a nite place v ∈ M0K we

denotecv = (E(Kv) : E0(Kv))

and call it the local Tamagawa number at v.From all Weierstrass equations that dene an elliptic curve over a local eld

with a1, a2, . . . , a6 ∈ OK we can select the one for which the valuation of thediscriminant is minimal (see [52] ch. VII §1).

Denition 5.1.2 Let E/K be an elliptic curve the minimal Weierstrass equa-tion of E over K is such that v(∆(E)) is minimal with the condition a1, a2, . . . , a6 ∈OK .

The reduction type of E/K is of three kind that we resume in the followingtheorem.

Theorem 5.1.1 (Theorem and Denition [52] ch. VII)Let K be a local eld,E/K an elliptic curve over K and E/k the reduced curve for a minimal Weier-strass equation :

y2 + a1xy + a3y = x3 + a2x2 + a4x+ a6

• E has good reduction, one also says stable reduction, over K if E is non-singular. This is the case if and only if v(∆(E)) = 0.

• We say that E has multiplicative reduction, one says semi-stable reduction,over K if E has a node. This is the case if and only if v(∆(E)) > 0 andv(c4) = 0, if so :

Ens(k) ∼= Gm(k) ∼= k∗

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68 CHAPTER 5. ELLIPTIC CURVES

• E has additive reduction, one also says unstable reduction, over K if Ehas a cusp. This is the case if and only if v(∆(E)) > 0 and v(c4) > 0, ifso :

Ens(k) ∼= Ga(k) ∼= k+

In the cases of multiplicative and additive reduction we say that E has badreduction. The reduced curve is an elliptic curve if and only if E has goodreduction

Moreover, we can distinguish two types of multiplicative reduction :

Denition 5.1.3 Let E/K be an elliptic curve with multiplicative reduction.One says that E/K has split multiplicative reduction if the slopes of the tangentlines are in k. Otherwise we say that E/K has non-split multiplicative reduction

The meaning of stable, semi-stable and unstable is given just below.

Denition 5.1.4 Let E/K be an elliptic curve, one says that E has potentialgood reduction over K if it acquires good reduction over a nite extension K ′/K.One dene in the same way potential multiplicative reduction.

One can show that E/K has potential good reduction if and only if thej−invariant is integral, i.e. j(E) ∈ O∗K ([52] ch. VII).

Theorem 5.1.2 (semi-stable reduction theorem, [52] ch. VII) Let E/K be anelliptic curve.

• Let K ′/K be a nite unramied extension of K then E has the samereduction type over K and over K ′.

• Let K ′/K be any nite extension, if E has either good or multiplicativereduction over K, it has the same type of reduction over K ′.

• There exists a nite extension K ′/K so that E has either good or splitmultiplicative reduction over K ′

Recall that if Gal(K/K) acts on a set Σ, we say that Σ is unramied if it isinvariant under the inertia group I of K. There is a well known criterion for anelliptic curve to have good reduction :

Theorem 5.1.3 (Criterion of Néron-Ogg-Shavarevich) Let E/K be an ellipticcurve over a local eld K, the following are equivalent :

1. E ha good reduction over K

2. E[m] is unramied for all integers m ≥ 1 relatively prime to char(k)

3. The Tate module Tl(E) is unramied for some (all) primes l 6= char(k)

4. E[m] is unramied for innitely many integers m ≥ 1 relatively prime tochar(k).

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5.1. BASIC FACTS 69

5.1.2 Formal groups

We saw that an elliptic curve can be dened by Weierstrass equations and thatsuch a curve is equipped with a natural group law. The description of the grouplaw by the coordinates (functions) may be done. The resulting theory is equiv-alent to the theory of groups but with formal power series in the coordinates.It is known as formal groups. For some details on what is going to be said see[52] ch. IV.

Let E/K be an elliptic curve dened by a Weierstrass equation

E : y2 + a1xy + a3y = x3 + a2x2 + a4x+ a6.

We make the following change of variables :

z = −xy

w = −1y

so that the point O of E becomes the point (z, w) = (0, 0) and z is a localuniformizer at O. Furthermore the preceding equation becomes

w = f(z, w) := z3 + a1zw + a2z2w + a3w

2 + a4zw2 + a6w

3.

Proposition 5.1.1 ([52] ch. IV)

1. We can reiter the preceding equation (w = f(z, w) = f(z, f(z, w)) · · ·) thisdene a power series

w(z) = z3(1 +A1z +A2z2 + · · ·) ∈ Z[a1, . . . , a6][[z]].

2. w(z) is the unique power series satisfying

w(z) = f(z, w(z)).

3. If Z[a1, . . . , a6] is made into a graded ring by assigning weights wt(ai) = ithen, for all n, An is a homogeneous polynomial of weight n

We have by example

A1 = a1, A2 = a21 + a2, A3 = a3

1 + 2a1a2, A4 = a41 + 3a2

1a2 + 3a1a3 + a22 + a4 · · · .

From this we deduce the Laurent series for x(z) and y(z):

x(z) = zw(z) = 1

z −a1z − a2 − a3z − (a4 + a1a3)z2 − · · · ,

y(z) = − 1w(z) = − 1

z3 + a1z2 + a2

z + a3 + (a4 + a1a3)z + · · · .

This give a solution in the quotient eld of the ring of formal power series.Further, the inverse of the point with coordinates (z, w) can be easily denedas

i(z) =x(z)

y(z) + a1x(z) + a3=

z−2 − a1z−1 − · · ·

−z−3 + 2a1z−2 + · · ·∈ Z[a1, . . . , a6][[z]]

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70 CHAPTER 5. ELLIPTIC CURVES

and the z-coordinate of the sum of the points (z1, w1), (z2, w2) is a power seriesof the form (see [52]):

F (z1, z2) = z1 + z2 − a1z1z2 − a2(z21 + z1z

22)

−(a3z31z2 − (a1a2 − 3a3)z2

1z22 + 2a3z1z

32) + · · ·

∈ Z[a1, . . . , a6][[z1, z2]].

We have thus dened a formal group over the ring Z[a1, . . . , a6] as the basicdenition of a formal group is given by

Denition 5.1.5 Let R be a ring, a one-parameter commutative formal groupdened over R is a power series F (X,Y ) ∈ R[[X,Y ]] satisfying

1. F (X,Y ) = X + Y + (terms of higher degrees).

2. F (X,F (Y,Z)) = F (F (X,Y ), Z) (associativity).

3. F (X,Y ) = F (Y,X) (commutativity)

4. There is a unique power series i(T ) ∈ R[[T ]] such that F (T, i(T )) = 0.(inverse)

5. F (X, 0) = X and F (0, Y ) = Y .

We denote the formal group associated to the elliptic curve by E. Actually,if E/K is an elliptic curve over a local eld K with ring of integers O andmaximal ideal M, the power series x(z) and y(z) converge for z ∈ M. Thisgives the following injection : M→ E(K).

Denition 5.1.6 With the preceding notations, we denote by E(M), called thegroup associated to the formal group E, the group whose ambient set isM andwhose laws are given by the law of formal group. Similarly we dene the groupsE(Mn) for all non-negative integer n

Proposition 5.1.2 ([52] ch. IV)

1. For each n ≥ 1, the map

E(Mn)/E(Mn+1)→Mn/Mn+1

induced by the identity map on sets is an isomorphism.

2. Let p be the characteristic of the residue eld k. Then every torsion pointof E(M) has order a power of p.

Remind the denition of E1(K) = P ∈ E(K)|P = O, this group is in factisomorphic to the group associated to E:

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5.1. BASIC FACTS 71

Proposition 5.1.3 ([52] ch. VII prop. 2.2) Let E/K be an elliptic curve overa local eld given by a minimal Weierstrass equation, let E/OK be the associatedformal group and let w(z) ∈ R[[z]] be the previous power series. Then the map

E(M) → E1(K)z →

(z

w(z) ,−1

w(z)

)is an isomorphism.

For details about the particular structure of formal groups, the reader isinvited to consult the book of Frölich [17].

5.1.3 Elliptic curves over p-adic elds

Elliptic curves over local elds are principally of three type : elliptic curves overC, elliptic curves over R and elliptic curves over p-adic elds.

Elliptic curves over C are torus of the form C/Z + τZ which can be realizedvia the map z → exp(2πiz) as C∗/qZ, where qZ = qn|n ∈ Z with |q| ∈ C∗. Forelliptic curves over Qp, we can't dene a torus because Qp has no non-triviallattices because if 0 6= t ∈ Λ an hypothetical lattices, then limn p

nt = 0 so thatΛ can't be discrete. However a denition as Q∗p/qZ with |q|p < 1 denes anelliptic curve over Qp. Actually there are many similarities between the theoryof elliptic curves over this dierent local elds.

Basic facts about elliptic curves over p-adic elds, i.e. nite extensions ofQp, are stated and prove in the chapter V of [53]. The essential theorem being :

Theorem 5.1.4 (Tate, [53] ch. V 3) Let K be a p-adic eld with absolute value| · |, let q ∈ K∗ satisfy |q| < 1, and let

sk(q) =∑n≥1

nkqn

1− qn, a4(q) = −5s3(q), a6(q) = −5s3(q) + 7s5(q)

12.

• The series a4(q) and a6(q) converge in K. We dene the Tate curve, de-noted Eq, by the equation

Eq : y2 + xy = x3 + a4(q)x+ a6(q).

• The Tate curve is an elliptic curve over K with discriminant

∆ = q∏n≥1

(1− qn)24

and j−invariant

j(Eq) =1q

+ 744 + 196884q + · · · = 1q

+∑n≥0

c(n)qn,

where the c(n) are integers.

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72 CHAPTER 5. ELLIPTIC CURVES

• The seriesX(u, q) =

∑n∈Z

qnu(1−qnu)2 − 2s1(q),

Y (u, q) =∑n∈Z

(qnu)2

(1−qnu)3 + s1(q),

converge for all u ∈ K \ qZ and dene a surjective homomorphism

φ : K∗ → Eq(K)

u 7→

(X(u, q), Y (u, q)) if u /∈ qZ

O if u ∈ qZ.

The kernel of φ is qZ.

• The map φ is compatible with the action of the Galois group Gal(K/K)in the sense that

∀u ∈ K∗,∀σ ∈ Gal(K/K), φ(uσ) = φ(u)σ.

In particular, for any algebraic extension L/K there is an isomorphism

L∗/qZ ∼= Eq(L).

Actually |j(Eq)| = | 1q | > 1 and every element of Qp is the j−invariant of aTate curve ([53] ch. V lemma 5.1.).

The next theorem due also to Tate is called the p−adic uniformization the-orem, it build the bridge between elliptic curves over p−adic elds and theirparametrization by Tate curve. It is equivalent to the usual uniformization overC (see [52] ch. VI 5) and the parametrization by Weierstrass functions.

Theorem 5.1.5 (Tate, p−adic uniformization theorem, e.g. [53] ch. V 5) LetK be a p−adic eld and E/K be an elliptic curve with |j(E)| > 1, and dene

γ(E/K) = −c4c6∈ K∗/K∗2.

• γ(E/K) is well dened as an element of K∗/K∗2, i.e. independent of theWeierstrass equation for E/K.

• Let E′/K be an other elliptic curve with j(E′) 6= 0, 1728. Then E and E′

are isomorphic over K if and only if

j(E) = j(E′) and γ(E/K) = γ(E′/K).

• Let E/K and E′/K be elliptic curve with j(E) = j(E′) 6= 0, 1728 andsuppose that γ(E/K) 6= γ(E′/K) so that L(

√γ(E/K)/γ(E′/K)) is a

quadratic extension. Let χ : Gal(K/K) → Gal(L/K) → ±1 be thequadratic character associated to L/K. Then there exists an isomorphismψ : E → E′ with the property that

∀σ ∈ Gal(K/K),∀P ∈ E(K), ψ(σ(P )) = χ(σ)ψ(P ).

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5.2. L-FUNCTIONS OF ELLIPTIC CURVES 73

• There is a unique q ∈ K∗ with |q| < 1 such that E is isomorphic over Kto the Tate curve Eq. Further, q ∈ K.

• Let q be chosen by the previous item. Then the following conditions areequivalent

1. E is isomorphic to Eq over K

2. γ(E/K) = 1

3. E has split multiplicative reduction

5.2 L-functions of elliptic curves

We saw at the beginning of this chapter that there is a natural l-adic Galoisrepresentation associated to an elliptic curve E/K over a local eld K for ldierent from the residue characteristic. It is given by the action of Gal(K/K)on the l−adic Tate module Tl(E) and the associated Ql-vector space Vl(E) =Tl(E)⊗Zl Ql and reads

ρE/K,l : Gal(K/K)→ GL(Vl(E)).

However we saw in the rst part of this text that the natural way to deneL−factors associated to non-archimedean local elds is via étale cohomology(Part I ch. 4 sec. 4.1). It turns out that the l−adic representation comingfrom H1(E,Ql) is the contragredient representation to ρE/K,l. It's a good pointbecause the denition of ρE/K,l is clear. At present we put :

σ′E/K,l = ρ′E/K,l : Gal(K/K)→ GL(Vl(E)∗),

where Vl(E)∗ is the dual of Vl(E). Now we know (Part I ch.4 sec 4.2) that to anl−adic representation σ′E/K,l of Gal(K/K) and to an embedding ı : Ql → C wecan associate a representation σ′E/L,l;ı = (σE/K,l,ı, NE/K,l,ı of the Weil-Delignegroup WK .

In [42], it is proved that σE/K,l,ı is actually independent of l and ı. Theproof breaks in to case according to the semi-stable reduction theorem (Part. IIch.5 th. 5.1.2.): the case of potential good reduction and the case of potentialmultiplicative reduction.

The rst case depends on the criterion of Néron-Ogg-Shafarevich, on thesemi-simplicity of σE/K,l,ı and then on the reduction to its character which isknown to be independent by a theorem of Serre and Tate. The proposition ofRohrlich reads

Proposition 5.2.1 ([42] sec.14 p.148) Suppose E has potential good reduc-tion. Then σ′E/K,l,ı is independent of l and ı. Further, dropping the subscriptsNE/K = 0 and σE/K is semisimple and E has good reduction if and only if σE/Kis unramied.

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74 CHAPTER 5. ELLIPTIC CURVES

The second case follows by extending K to a quadratic extension where Eacquires split multiplicative reduction and as we saw on the section on ellipticcurves over p−adic elds, such a curve is isomorphic to a Tate curve. Aftersome work this gives the following proposition

Proposition 5.2.2 ([42] sec. 15 p. 150) Suppose that E has potential multi-plicative reduction, and let χ be a character of the Weil group WK such thatχ2 = 1 and the twist Eχ has split multiplicative reduction. Then σE/K,l,ı ∼=χω−1 ⊗ sp(2) is independent of l and ı. In particular, NE/K 6= 0, so that σ′E/K(subscripts deleted) is ramied. Furthermore, χ is trivial, unramied but non-trivial, or ramied according as E/K has split multiplicative reduction, nonsplitmultiplicative reduction, or additive reduction.

The L−factors for elliptic curves are known basically by dening the L−functionfrom the zeta function of the curve. Actually, what was presented on the use ofl−adic cohomology is theoretically motivated and nally the denition agree.

If E/K is an elliptic curve over a local eld K we dene its L−factors hasthe L−factor of the corresponding representation :

L(E/K, s) = L(σ′E/K , s).

Proposition 5.2.3 ([42] sec. 17 p.151)

1. If E has good reduction, put a = 1− |E(k)|+ q where q is the cardinal ofthe residue eld k. Then

L(E/K, s) =1

1− aq−s + q1−2s.

2. If E has multiplicative reduction, then

L(E/K, s) =1

1− αq−s,

where α is 1 or −1 according as E has split or non-split multiplicativereduction.

3. If E has additive reduction, then L(E/K, s) = 1.

If E/K is an elliptic curve over a number eld, we dene the L-function ofE/K by means of its local factors at nite places:

L(E/K, s) =∏

v∈M0K

L(E/Kv, s),

whereM0K denote the set of nite places. This function is absolutely convergent

for <(s) > 3/2.

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5.3. SELMER AND SHAFAREVICH-TATE GROUPS 75

5.3 Selmer and Shafarevich-Tate groups

For more de tails about the contents of this section, the reader is invited toconsult [7], [38], [40], [50] or [52].

5.3.1 Basic group cohomology

Let G be a group and ModG the category of G-module, i.e. of Z[G]-modulesnow considered as a module in the usual sense. There exists a functor fromthe category of G−module to itself given by

ModG 3 A 7−→ AG ∈ModG,

where AG = a ∈ A|∀g ∈ G, ga = a is the set xed by G. This functoris left-exact and covariant, that is to say that if we have a exact sequence ofG−module

0→ A→ B → C → 0

then the functor gives rise to the exact sequence

0→ AG → BG → CG

but the last arrow is not surjective in general, i.e. the functor is not right-exact.It is the subject of group cohomology to complete this last sequence.

Theorem 5.3.1 (Existence and uniqueness of the cohomological extension) Thereexists one and only one cohomological extension of the functor A 7→ AG up tocanonical equivalence. This means that there is a sequence of left-exact functorsHi(G,−) for i ≥ 0 and natural transformations δ : Hi(G,−) → Hi+1(G,−)such that H0(G,A) = AG and

1. For all exact sequence of G−modules

0→ A→ B → C → 0

the following innite sequence

0 // AG // BG // CGδ // H1(G,A) // · · ·

· · · // Hn(G,A) // Hn(G,B) // Hn(G,C) δ // Hn+1(G,A) // · · ·

is exact. It is called the long cohomology sequence.

2. Furthermore, if we have a commutative diagram with exact rows

0 // A //

B //

C //

0

0 // A′ // B′ // C ′ // 0

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76 CHAPTER 5. ELLIPTIC CURVES

Then we have a commutative diagram

0 // AG //

BG //

CGδ //

H1(G,A) //

· · ·

0 // A′G // B′G // C ′Gδ // H1(G,A′) // · · ·

Usually, for the purpose of study of Galois groups actions, we usually study thisfor pronite groups G and discrete G−modules on which it acts continuously.Remind that a pronite group is the projective limit of discrete groups or, whatamounts to be the same, a compact totally disconnected topological group. TheHq(G,A) are called cohomology group of G with coecients in A.

This group cohomology can be described in terms of cochains, cocyclesand coboundaries as the usual cohomology theories. If A ∈ ModG we denoteCn(G,A) as the set of all continuous maps from Gn to A. The coboundary is amap

∂ : Cn(G,A)→ Cn+1(G,A)

dened by the formula

(∂f)(g1, . . . , gn+1) = g1 · f(g2, . . . , gn+1)+∑ni=1(−1)if(g1, . . . , gi−1, gigi+1, gi+2, . . . , gn+1)

+(−1)n+1f(g1, . . . , gn)

The groups C∗(G,A) form a complex (i.e. dd = 0) and the cohomology groupsHi(G,A) are the cohomology groups of this complex :

Hi(G,A) = i− cocycle/i− coboundaries,

where i-cocycles are the element of the image of ∂ and i-boundaries are theelement of the kernel of ∂.

The elements of H1(G,A) are called crossed homomorphism, they are thecontinuous functions x : G→ A such that

∀σ, τ ∈ G, x(στ) = x(σ) + σx(τ).

Restriction and Ination

If f : G→ G′ is a homomorphism of groups, it induces a homomorphism

f∗ : Hq(G,A)→ Hq(G′, A)

for any G−module A.(see e.g. [7] ch. IV or [50])In particular, taking G′ = H to be a subgroup of G and f to be the embed-

ding H → G, we have, on one hand, restriction homomorphisms :

Res : Hq(G,A)→ Hq(H,A).

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5.3. SELMER AND SHAFAREVICH-TATE GROUPS 77

On the other hand, if H is a normal subgroup of G we can consider G→ G/Hand for any G−module A we have the G/H−module AH , hence a homomor-phism Hq(G/H,AH) → Hq(G,AH). By composing it with the embeddingAH → A, we obtain the ination homomorphism :

Inf : Hq(G/H,A)→ Hq(G,A).

Proposition 5.3.1 (Restriction-Ination sequence) Let H be a normal sub-group of G, and let A be a G−module. Then the sequence

0 // H1(G/H,AH)Inf // H1(G,A) Res // H1(G,A)

is exact.

5.3.2 Selmer and Shafarevich-Tate groups

Let L/K be a Galois extension, then Gal(L/K) is the projective limit of thegroups Gal(Lf/K) where L is a nite Galois subextension of Gal(L/K), thus is apronite group. We are interested in the cohomology groups Hq(Gal(K/k), A),q ≥ 0. If k is a eld we denote Hq(k,A) := Hq(Gal(k/k), A), where k denotethe algebraic closure of k.

Of particular importance for us are the groups of K-points, A = E(K),or the torsion points E[m], of an elliptic curves on which Galois groups actsnaturally for a number eld K. Let E/K and E′/K be two elliptic curves overa number eld K and φ : E → E′ an isogeny dened over K (e.g. φ = [m], wehave a short exact sequence ofGK := Gal(K/K)−modules

0 // E[φ] // E(K)φ // E′(K) // 0 .

Galois cohomology yields the long exact sequence :

0 // E(K)[φ] // E(K)φ // E′(K)

δ // H1(GK , E[φ]) // H1(GK , E(K))φ // H1(GK , E′(K)).

We can cut this sequence in the third term and fth term and completing weobtain the following short exact sequence

0 // E′(K)φ(E(K))

δ // H1(GK , E[φ]) // H1(GK , E(K))[φ] // 0

The preceding sequence is often called Kummer sequence because Kummer rstextract such a sequence in the case of elds and their groups of roots of 1.

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78 CHAPTER 5. ELLIPTIC CURVES

Similarly, let MK be the set of places of K and v ∈ MK , we can identify Kwith the algebraic closure of K inside Kv which yields the embedding

Gv := Gal(Kv/K) → GK := Gal(K/K)σ 7→ σ|K

whose image is a decomposition group at v. As Gv acts on E(Kv) and E′(Kv)we can repeat the preceding procedure which yields

0 // E′(Kv)φ(E(Kv))

δ // H1(Kv, E[φ]) // H1(Kv, E(Kv))[φ] // 0 .

Furthermore, for each v ∈MK we have a composed morphism, denoted Resv

Resv : Hq(K,E(K))Res // Hq(Kv, E(K)) // Hq(Kv, E(Kv)) .

Taking together the dierent embedding yields the following diagram with exactrows and columns

0 // E′(K)φ(E(K))

δ //

H1(K,E[φ]) //

**TTTTTTTTTTTTTTTTTT H1(K,E(K))

// 0

0 // ∏v∈MK

E′(Kv)φ(E(Kv))

δ //∏v∈MK

H1(Kv, E[φ]) // ∏v∈MK

H1(Kv, E(Kv))[φ] // 0

.

Denition 5.3.1 (Selmer and Shafarevich-Tate group) Let φ : E/K → E′/Kbe a rational isogeny. The φ-Selmer group of E/K is the subgroup of H1(K,E[φ])dened by

S(φ)(E/K) = ker

H1(K,E[φ])→

∏v∈MK

H1(Kv, E(Kv))

The Shafarevich-Tate group of E/K is the subgroup of H1(K,E(K)) dened by

X(E/K) = ker

H1(K,E(K))→

∏v∈MK

H1(Kv, E(Kv))

.

More generally, we could dene Xq(E/K) by replacing H1 by Hq in the pre-ceding denition

Proposition 5.3.2 Let φ : E/K → E′/K be a rational isogeny, Sφ(E/K) andX(E/K) be the Selmer group and the Shafarevich-Tate group respectively, then

1. There is an exact sequence

0→ E′(K)/φ(E(K))→ Sφ(E/K)→X(E/K)[φ]→ 0

.

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5.3. SELMER AND SHAFAREVICH-TATE GROUPS 79

2. Sφ(E/K) is nite and as a consequence the m-torsion X(E/K)[m], isnite for every integer m

Actually, H1(K,E(K)) classify principal homogeneous spaces over K underE, i.e. varieties X over K equipped with a fully faithful action of E (a morphismE ×X → X for which the induced action of E(K) on X(K) such that for eachx1, x2 ∈ X(K) there is a unique P ∈ E(K) such that Px1 = x2). Principalhomogeneous spaces over K under E are also called K−torsors. A K−torsors iscalled locally trivial if it is in the kernel of every Resv, or equivalently if X(Kv)is not empty for every v. Thus X(E/K) is geometrically the set of locallytrivial K−torsors under E, thus elements of X(E/K) correspond to K−torsorthat violate the Hasse principle.

The classical and important conjecture being that X(E/K) is nite. It isknown that as GK is a pronite group, the cohomology groups Hq(K,E(K))are torsion groups for q ≥ 1 which means that every element has nite order.Hence X(E/K) ⊂ H1(K,E(K)) is a torsion group and we can write

X(E/K) =⊕p

Xp∞(E/K),

where Xp∞ is the p-primary part of X(E/K), i.e. the subgroup of elementskilled by a power of p. By abelian groups theory, we can write

Xp∞ = (Qp/Zp)np ×(

Zps1Z

× · · · × ZpslZ

)for some integers np, s1, . . . , sl. That's why if the Shafarevich-Tate group isnite, the Mordell-Weil rank is equal to the p−Selmer rank, dened as theMordell-Weil rank plus the nunmber of copies of Qp/Zp in X. This is used tomotivate the form of the parity conjecture (see section 5.4.3) proved by Timand Vladimir Dokchitser in their article.

Concerning the niteness of X we have the following expectation due toGoldfeld and Szpiro in [19]

Conjecture 5.3.1 (Goldfeld-Szpiro) Let K be a number eld, E/K an ellip-tic curve of conductor N(E/K) (see 4.2.2 and the next section) then for anyconstant ε > 0, there is a constant Cε(K) such that

CardX(E/K) ≤ Cε(K)NK/Q(N(E/K))1/2+ε.

5.3.3 The Cassels-Tate pairing

In [5], J.W.S constructed a pairing over the Shafarevich-Tate group with niceproperties :

Theorem 5.3.2 (Cassels [5]) Let E/K be an elliptic curve over an algebraicnumber eld and X be its Shafarevich-Tate group, then there exists a skew-symmetric form

< ·, · >: X×X −→ Q/Z.

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80 CHAPTER 5. ELLIPTIC CURVES

This pairing has the following property : Let m be a natural number and supposethat < ξ, η >= 0 for all ξ ∈X such that mξ = 0 then η = mα for some α ∈X.

Here the kernel of the Cassels-Tate pairing is Q =⊕

(Q/Zp)np , the set ofinnitely divisible, which is conjecturally trivial, elements of X so that it denesalternate, non-degenerate pairing on X/Q.

5.4 BSD and parity conjectures

5.4.1 L-functions again

Let E/K be an elliptic curve over a number eld K and denote MK and M0K

the sets of places and of nite places respectively. We know that the l−adicrepresentations associated to E are essentially unique. We denote by σ′E/K thecorresponding representation of the Weil-Deligne group WK . We dened insection 4.2.2 the conductor N(σ′) for a representation of the Weil-Deligne groupof a local eld. By proposition 3.2.2 a representation σ′ of the Weil-Delignegroup of a global eld dene representations of the local Weil-Deligne group.Hence, if K is a global eld we dene

N(E/K) =∏

v∈M0K

N(E/Kv)

seen as an ideal of the integer ring OK .Furthermore, for D the absolute value of the discriminant of K, we dene

A(E/K) =∏

v∈M0K

A(E/Kv) =∏

v∈M0K

D2vN(N(E/Kv)) = D2N(N(E/K)).

Like in 4.2.3 where we dened the local root numbers W (σ′) for representa-tions of local Weil-Deligne groups, which we denote byW (E/Kv) in our context,we dene the global root number of E/K as

W (E/K) =∏

v∈MK

W (E/Kv) = (−1)r1+r2∏

v∈M0K

W (E/Kv).

The completed L-function (see part I) then is given by

Λ(E/K, s) = A(E/K)s/2(2(2π)−sΓ(s))nL(E/K, s).

One conjectures that this function has an analytic continuation to an entirefunction and that it satises the following functional equation

Λ(E/K, s) = W (E/K)L(E/K, 2− s)

which implies thatW (E/K) = (−1)ords=1L(E/K,s).

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5.4. BSD AND PARITY CONJECTURES 81

5.4.2 The Birch and Swinnerton-Dyer conjecture

Remind the following formula of I.2.1

ress=1ζK(s) =2r1(2π)r2Reg(K)

d1/2K ω

.

The equivalent formula for L-functions of elliptic curves is conjectural and knownas the Birch and Swinnerton-Dyer conjecture. The weak Birch and Swinnerton-Dyer conjecture asserts that

L(E/K, s) ∼s=1 c(s− 1)−r + higher order terms,

wherer = rank E(K).

We can be more precise about the constant c, it is the subject of the completeconjecture which many believe to be true for empirical and numerical signs.

The equivalent of the regulator of the eld K is called the elliptic regulatorof the elliptic curve E/K for K a number eld. This regulator which we denoteReg(E/K) is dened in term of the Néron-Tate pairing of E/K, i.e. in terms ofthe canonical height of the elliptic curve. For the denition of the (canonical)Néron-Tate height and the corresponding pairing see [52].

If we denote by h the canonical height of E/K, the canonical pairing is thefollowing bilinear form

< , >: E(K)× E(K)→ R

dened by< P,Q >= h(P +Q)− h(P )− h(Q).

Denition 5.4.1 (Elliptic regulator) Let E/K be a elliptic curve over a num-ber eld K. Remind from Mordell-Weil theorem that the group of K−rationalpoints E(K) is of nite type. The elliptic regulator of E/K, denoted Reg(E/K)is the volume of a fundamental domain of E(K)/Etors(K), computed with re-spect to the Néron-Tate height. In other words, if P1, . . . , Pr ∈ E(K) generateE(K)/Etors(K) then

Reg(E/K) = det(< Pi, Pj >)1≤i,j≤r.

For v ∈ MK , let Kv be the corresponding completion and µv be the Haarmeasure onKv, normalized so that for each open U ⊂ Kv and x ∈ Kv :µv(xU) =|x|vµv(U).

The Birch and Swinnerton-Dyer conjecture takes the form

Conjecture 5.4.1 (Birch and Swinnerton-Dyer) Let E/K be an elliptic curveover a number eld K and L(E/K, s) its L-function then

lims→1

L(E/K, s)

(s− 1)rank E(K)= CE/K ·

Card(X(E/K))Reg(E/K)√|dK |(Card E(K)tors)2

,

where CE/K is a constant which is the product of local Tamagawa numbers andperiods and dK is the discriminant of K.

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82 CHAPTER 5. ELLIPTIC CURVES

5.4.3 The parity conjecture

From the two preceding sections we may conjecture that the parity of the an-alytic rank, that we dene (−1)ords=1L(E/K,s), may be equal to the parity of

Mordell-Rank, (−1)rank E(K), both being equal to the ± sign of the functionalequation. This is the basic parity conjecture, one can see it as a very weak formof the Birch and Swinnerton-Dyer conjecture.

Conjecture 5.4.2 (Basic parity conjecture) The parity of the analytic rank ofan elliptic curve over a number eld is equal to the parity of its Mordell-Weilrank:

ords=1 L(E/K, s) ≡ rank E(K) (mod 2).

As discussed in section 5.3.2, when X(E/K) is nite, the Mordell-Weil rankis equal, for every p, to the p−Selmer rank dened by

sp(E/K) = rank E(K) + rankZpX(E/K),

and the analytic rank by

rkan(E/K) = ords=1 L(E/K).

Then a rened version of the parity conjecture reads

Conjecture 5.4.3 (Parity conjecture for Selmer groups) For some prime p

sp(E/K) ≡ rkan(E/K) (mod 2).

Combining the dierent conjectures we can expect the following conjectureas propose by Tim and Vladimir Dokchitser in their paper [14].

Conjecture 5.4.4 (p−parity conjecture) For any (some) prime p, the rootnumber agrees with the parity of the p−Selmer rank :

W (E/K) = σ(E/K, p),

where σ(E/K, p) = (−1)sp(E/K).

This is this conjecture which has been dealt with recently, in particular,Tim and Vladimir Dokchister in [14] prove the following theorem (see the nextchapter for a discussion of the paper)

Theorem 5.4.1 ([14] theorem 2) If E/K has a rational isogeny of prime degreep ≥ 3, and E is semistable at all primes over p, then the p-parity conjectureholds for E/K and p. It also holds for p = 2 under the additional assumptionthat E is not supersingular at primes above 2.

In the case of elliptic curves over the rational numbers, Jan Nekovár ([33],[34], [35] and [36]), proved the following result

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5.5. NÉRON MODELS 83

Theorem 5.4.2 (Nekovár [34]) Let E be an elliptic curve over Q with goodordinary reduction at p. Then the parity conjecture for Selmer groups holds forE and p.

The proof of the preceding theorems are not of the same nature, as thetechniques are dierent over a number eld or over Q. Actually it is dicult tocompare this two theorem : they really live in dierent worlds.

5.5 Néron Models

An almost complete presentation of Néron models can be found in [3], in thecase of elliptic curves the books of Silverman [53] and of Liu [27] are sucientat least for our purposes.

Let E/K be an elliptic curve over a complete eld K with respect to adiscrete valuation v and with ring of integers R and residue eld k. R is adiscrete valuation ring and K is its fraction eld. For instance K can be thecompletion of a number eld at a non-archimedean place, then R = OK andk = Fq. A minimal equation of E/K over R dene a scheme over Spec(R)which can be singular. However if we resolve the singularities of this scheme weobtain a scheme C/Spec(R) whose generic ber is E/K and whose special beris a union of curves over k. If we choose C/Spec(R) minimal with respect toC → Spec(R) then it is unique up to unique homomorphism. The subschemeE ⊂ C obtained by discarding all the singular points of the special ber of Cis called the Néron minimal model of E/K, in what follows we will be moreprecise.

5.5.1 Algebro-geometric preliminaries

Remind that a integral domain is called normal if it is integrally closed in itseld of fractions. A scheme X is called normal at the point x if the correspondinglocal ring OX , x is normal, a scheme is normal if every local rings are normal.

Furthermore a scheme X with structural sheaf OX is called noetherian if itcan be covered by a nite number of open ane sets Xi such that the rings OXare noetherian (i.e. its ideals are nitely generated). A scheme is said locallynoetherian if every point has a Noetherian open neighborhood. Following Liu[27], we dene

Denition 5.5.1 A normal locally noetherian scheme of dimension 0 or 1 iscalled a Dedeking scheme.

Denition 5.5.2 (bered surfaces) Let S be a Dedekind scheme. A integral,noetherian, at S-scheme X → S is called a bered surface.

A regular bered surface X → S over a Dedekind scheme of dimension 1 iscalled an arithmetic surface, the denition of atness has been given in chapter4 denition 4.1.1..

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84 CHAPTER 5. ELLIPTIC CURVES

Remark: Integral means that for every open set U ⊆ X, OX(U) is an integraldomain. The denition of atness has been given in chapter 4.

As a consequence, the generic ber of a bered surface over a scheme ofdimension 1 is a curve over K(S) normal if X is normal and the special bersXs, for s ∈ S, is a projective curve over the residue eld k(s) ([27] ch. 8lemma 3.3). Furthermore the set of singular points of X is a nite set of closedpoints. It is called an arithmetic surface because it is a one dimensional familyof one-dimensional varieties.

As a particular case of what preceded we can take S = Spec(R) where Ris a Dedekind domain. The principal theorem of the theory been the theoremof existence and uniqueness of regular minimal models for curve of genus ≥ 1,more precisely

Theorem 5.5.1 (Existence and uniqueness of minimal regular models, [53] ch.IV, [27] ch. 9). Let R be a Dedekind domain with fraction eld K, and C/K anon-singular projective curve over K.

1. (Existence) There exists a regular arithmetic surface C/R, proper over R,whose generic ber is isomorphic to C/K. It is called a proper regularmodel for C/K.

2. (minimality) Suppose that the genus of C is ≥ 1 then there exists a properregular model Cm/R of C/K which is minimal with respect to the followingproperty :

If C′/R is another proper regular model, the R−birational map inducedfrom a xed isomorphism between their generic bers :

C 99K Cm

is an R−isomorphism.

Cm/R is called a proper regular model for C/K.

Cm/R is unique up to unique isomorphism.

Loosely speaking, a Néron model of an elliptic curve is a scheme-theoreticmodel of the elliptic curve which is universal in some sense, more precisely :

Denition 5.5.3 Let R be a Dedekind Domain with fraction eld K, and letE/K be an elliptic curve. A Néron model for E/K is a smooth group schemeE/R whose generic ber is E/K such that every K-rational map between curveφR : X/K → E/K extend, for every smooth R−scheme Ξ/R with generic berX/K, to a morphism φR : Ξ/R→ E/R.

This denition implies that if E/R is a Néron model for E/K then

E(R) ∼= E(K).

The following theorem and its generalization (e.g. [27]) is fundamental

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5.5. NÉRON MODELS 85

Theorem 5.5.2 (Existence and uniqueness of Néron models, [53] ch. IV)

1. (Existence) Let R be a Dedekind domain with function eld K and letE/K be an elliptic curve, let C/R be a minimal regular model for E/Kand let E/R be the largest subscheme of C/R which is smooth over R.Then E/R is a Néron model for E/K.

2. (Uniqueness) The Néron model is unique up to unique isomorphism.

Actually, if the elliptic curve E/K has good reduction,the subscheme W0 ob-tained as the largest subscheme of the scheme dened by a Weierstrass equation,is a Néron modél for E/K.

5.5.2 The bers of Néron models

The principal result of the theory of Néron models over a discrete valuation ringis that they can be classied according to their special ber.

Theorem 5.5.3 (Kodaira, Néron, see e.g. [53] ch. IV) Let R be a discretevaluation ring with maximal ideal p, fraction eld K, and algebraically closedresidue eld k. Let E/K be an elliptic curve and let C/R be a minimal properregular model for E/K. Further, put cp = (E(K) : E0(K)), where E0(K)is dened according to the section 5.1.1. and put υ(∆) the valuation of thediscriminant of E. Then the special ber Cp = C×Spec RSpec(k) and the numbercp are of the following forms :

• Type I0 Cp is a non-singular curve and cp = 1. Good reduction. υ(∆) =0

• Type I1 Cp is a rational curve with a node and cp = 1. Multiplicativereduction.

• Type In, n 6= 2 Cp consists of n non-singular rational curves arranged inthe shape of an n-gon and cp = n.Multiplicative reduction. υ(∆) = n

• Type II Cp is a rational curve with a cusp and cp = 1. Additive reduction.υ(∆) = 2

• Type III Cp consists of two non-singular rational curves which intersecttangentially at a single point and cp = 2. Additive reduction. υ(∆) = 3

• Type IV Cp consists of three non-singular rational curves which intersectat a single point and cp = 3. Additive reduction. υ(∆) = 4

• Type I∗0 Cp is a non-singular rational curve of multiplicity two with fournon-singular curves of multiplicity 1 attached and cp = 4. Additive reduc-tion. υ(∆) = 6

• Type I∗n Cp consists of a chain of n+ 1 non-singular rational curves ofmultiplicity two with two non-singular rational curves of multiplicity oneattached to either ends and cp = 4. Additive reduction.υ(∆) = 6 + n

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86 CHAPTER 5. ELLIPTIC CURVES

• Type IV ∗ For this more complex case see for instance the pictures inthe appendix C of [52] or the chapter IV of [53]. cp = 3. Additivereduction.υ(∆) = 8

• Type III∗ For this more complex case see for instance the pictures inthe appendix C of [52] or the chapter IV of [53].cp = 2. υ(∆) = 9

• Type II∗ For this more complex case see for instance the pictures in theappendix C of [52] or the chapter IV of [53].cp = 1. Additive reduction.υ(∆) = 10

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Chapter 6

Parity conjecture with a

cyclic isogeny

6.1 Presentation of the article of Tim and Vladimir

Dokchitser

Recall that the p-parity conjecture says that the sign of the global root num-ber W (E/K) shall be equal to σ(E/K, p) = (−1)sp(E/K) where sp(E/K) is thep−Selmer rank of E/K. The article has the following main directories: com-puting the root number by local root numbers, computing σ(E/K, p) by a localmethod due to Cassels and Fisher for p ≥ 3 (see the next section), making thecorresponding computation in the case of a 2−isogeny. This is what we presentin the remaining of the text.

First of all, the computation of the local roots number and of a kind of localp−Selmer rank leads to the following result

Theorem 6.1.1 ([14] theorem 3) Let K be a number eld and p be an oddprime. Let E/K be an elliptic with a rational p−isogeny φ, and assume that Ehas semistable reduction at all primes above p. Then for all place v of K

W (E/Kv) = (−1,Kv,φ/Kv)σφ(E/Kv).

Where W (E/Kv) are local root numbers, σφ(E/Kv) are local factors (denedbelow) whose product gives the parity of the p−Selmer rank and (−1,Kv,φ/K)is dened as a by-product of the norm residue symbol (dened in section 3.2.3).

Precisely, Kv,φ is the completion of the eld Kφ which is the smallest eldover which the point of ker(φ) are dened and (−1,Kv,φ) is the composition

K∗loc.rec. // Gal(Kφ/K) // (Z/pZ)∗ .

The last embedding coming from the faithful action of ker(φ), which has cardinalp, on Gal(Kφ/K). Thus (−1,Kφ/K) = 1 if −1 is a norm from Kφ/K and itequals −1 otherwise.

87

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88 CHAPTER 6. PARITY CONJECTURE WITH A CYCLIC ISOGENY

For p = 2, the existence of a 2−isogeny is equivalent to the existence of a2−torsion point. For if φ is a 2−isogeny and if φ is its dual, there exists a pointP such that φ(P ) = O which gives [2]P = φ φP = O. A similar argumentshows that the kernel of a p−isogeny is contained in E[p]. By using a translationwa can assume that the 2-torsion point is (0, 0), then there exists a model ofthe curve of the following form

E : y2 = x3 + ax2 + bx, a, b ∈ OK

and they use the following 2-isogeny

φ : E → E′

(x, y) 7→ (x+ ax+ bx−1, y − bx−2y)

where E′ is the curve with model

y2 = x3 − 2ax2 + δx, δ = a2 − 4b.

Then, by making a case by case computation they deduce the following result

Theorem 6.1.2 ([14] theorem 4) Suppose E/K has either ordinary or multi-plicative reduction at all prime above 2. Then for all places v of K,

W (E/Kv) = (a,−b)Kv (−2a, δ)Kvσφ(E/Kv),

where ( , )Kv denotes the Hilbert symbol (dene below).

Hence the product formulas, given in proposition 3.2.1 and in section 6.2below, imply that the parity conjecture holds in the conditions of the previoustwo theorems. I insist on the fact that almost all results of [14] remain oncase-by-case computations.

The most notable result of this article being the already stated

Theorem 6.1.3 ([14] theorem 2) If E is an elliptic curve over a number eldK which has a rational isogeny of prime degree p ≥ 3 and that E is semistableat all primes above p, then the p-parity conjecture (conjecture 5.4.4) holds forE/K and p. It also holds for p = 2 under the additional assumption that E isnot supersingular at primes above 2.

6.2 The p−Selmer rank

6.2.1 Basis for computation of parity of p−Selmer ranks

In the appendix of an article of Vladimir Dokchitser, [15], Tom Fisher givesresults which are essential in the computation of the p−Selmer rank. It allowsus to dene the parity of the p−Selmer rank σ(E/K, p) (see section 5.4.3) as aproduct of local factors which turn out to be computable. This is what makesthe existence of a p−isogeny (isogeny of prime degree p) essential in [14].

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6.2. THE P−SELMER RANK 89

Proposition 6.2.1 Let E/K and E′/K be two elliptic curves over K and φ :E → E′ be a rational isogeny of prime degree p. Denote by φ the dual isogeny.Then

T (E,E′, φ) :=|E′(K)[φ]| |Sφ(E/K)||E(K)[φ]| |Sφ(E′/K)|

= pe

for some integer e such that

e ≡ sp(E/K) (mod 2),

where sp(E/K) is the p−Selmer rank dened in 5.4.3..

Proof : First of all, from the exact sequence of proposition 5.3.2. the quotientis equal to

|E′(K)[φ]| |E′(K)/φ(E(K)||E(K)[φ]| |E(K)/φ(E′(K)|

|X(E/K)[φ]||X(E′/K)[φ]|

.

On one hand, as φφ = [p] and φφ = [p], on E′ and E respectively, hence wehave the inclusion

X(E/K)[φ] ⊂X(E/K)[p] ⊂X(E/K)[p∞]X(E′/K)[φ] ⊂X(E′/K)[p] ⊂X(E′/K)[p∞]

so the φ−torsion of the Shafarevech-Tate group has order a power of p as asubgroup of the p−torsion. Moreover we have the following exact sequence

0 //X(E/K)[φ] //X(E/K)[p]φ //X(E′/K)[φ] // 0 .

Hence|X(E/K)[φ]||X(E′/K)[φ]|

=|X(E/K)[φ]|2

|X(E/K)[p]|.

Moreover we know (section 5.3) that the p−primary part of X takes the form

X(E/K)[p∞] = (Qp/Zp)np × (Z/ps1Z)× · · · × (Z/pslZ).

Each factor Qp/Zp contains p elements of order p, so there are pnp . And eachZ/psZ contains also p elements of order p. But the Cassels-Tate pairing is non-degenerate and skew-symmetric on this nite part. As a nite group equippedwith a non-degenerate skew-symmetric form has square order, the order of thenite part of the p−torsion is of the form p2b for some natural integer b. Hence

|X(E/K)[φ]||X(E′/K)[φ]|

= pnp+2b, for np = corankZpX(E/K)[p].

On the other hand, we know from he Mordell-Weil theorem that E(K) ∼=(Z)g × Etors(K), where Etors is the nite abelian group of torsion points ofE(K). Similarly E′(K) has the same form with the same rank g because E

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90 CHAPTER 6. PARITY CONJECTURE WITH A CYCLIC ISOGENY

and E′ are isogenous. We denote by E∞(K) and E′∞(K) the innite parts ofE(K) and E′(K). We have a commutative diagram

0 // Etors(K) //

φ

E(K) //

φ

E∞(K) //

φ

0

0 // E′tors(K) // E′(K) // E′∞(K) // 0

Taking kernel and cokernel we obtain

|E′(K)/φ(E(K))||E(K)[φ]|

=|E′tors(K)/φ(Etors(K))| |E′∞(K)/φ(E∞(K))|

|Etors(K)[φ]| |E∞(K)[φ]|.

Moreover, as Etors(K) and E′tors(K) are nite groups we have

|E′tors(K)/φ(Etors(K))||Etors(K)[φ]|

=|E′tors(K)||Etors(K)|

and |E∞(K)[φ]| = 1 because E∞(K) is a free group. Using the dual isogeny weobtain similar results for φ : E′ → E, thus

|E′(K)[φ]| |E′(K)/φ(E(K)||E(K)[φ]| |E(K)/φ(E′(K)|

=|E′∞(K)/φ(E∞(K))||E∞(K)/φ(E′∞(K))|

|E′tors(K)|2

|Etors(K)|2.

Finally, as the E∞s are free of rank g

|E′∞(K)/φ(E∞(K))||E∞(K)/φ(E′∞(K))| = prank E(K).

The result follows.

If v ∈ MK is non-archimedean we denote by µv the additive Haar measurenormalized so that µv(Ov) = 1. If v is archimedean, µv = dx is the usualLebesgue measure if Kv

∼= R and µv = 2dxdy if Kv∼= C. If ω is an invariant

dierential on E/K, following Cassels we dene

µv(ω,E) =∫E

|ω|vµv.

If E/K and E′/K are two isogenous elliptic curves and if ω (resp. ω′) is aninvariant dierential on E (resp. E'), then from [28], for almost all v ∈MK

µv(ω,E(Kv)) = µv(ω′, E(Kv)) =NvN(v)

,

where Nv is the number of points in the reduced curve and N(v) is the norm ofv (norm of the corresponding prime ideal). Hence the following product is welldened : ∏

v∈MK

µv(ω,E(Kv))µv(ω′, E′(Kv))

.

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6.2. THE P−SELMER RANK 91

Proposition 6.2.2 With the preceding denitions

µv(φ∗ω′, E(Kv))µv(ω′, E′(Kv))

=|E(Kv)[φ]|

|E′(Kv)/φ(E(Kv))|

Proof :This relation can be rewritten as∫E(Kv)

|φ∗ω′|vµv =|ker(φ : E(Kv)→ E′(Kv))||coker(φ : E(Kv)→ E′(Kv))|

∫E′(Kv)

|ω′|vµv,

which is clear by considering the corresponding coverings.

Corollary 6.2.1 If we dene σ(E/K, p) = (−1)sp(E/K) then

σ(E/K) =∏

v∈MK

σφ(E/K),

where we dene σφ(E/K) = ±1 and equal to one if and only if the power of pin |E(Kv)[φ]||E′(Kv)/φ(E(Kv))| is even.

Proof This is a direct consequence of theorem 6.0.4 and proposition 6.0.1.

Theorem 6.2.1 (Cassels [5] theorem 1.1) Let φ : E/K → E′/K be a rationalisogeny of elliptic curves over a number eld and let φ then

T (E,E′, φ) =|E′(K)[φ]| |Sφ(E/K)||E(K)[φ]| |Sφ(E′/K)|

=∏

v∈MK

|E(Kv)[φ]||E′(Kv)/φ(E(Kv))|

.

What preceded is the basis of the computation of the parity of the p−Selmerrank in the article of Tim and Vladimir Dokchitser. They also use a result ofSchaefer, [51], which says that if we dene the number α by φ∗ω′ = αω for ωand ω′ invariant dierential on E and E′ respectively. It is always possible sinceelliptic curves are of dimension one. And if v ∈M0

K is a nite place of K then

|E(Kv)[φ]||E′(Kv)/φ(E(Kv))|

= |α|−1v

cv(E′)cv(E)

.

The the computation of the local Selmer factors reduces to the computationof the p-order of the quotients c(E′)/c(E) and of |α|v this is done mainly in thearticle [14] but it also rests on a result of Tom Fisher in the appendix of [15].We nish this section by presenting these ideas.

Proposition 6.2.3 (lemma 11 of [14]) Let E/K be an elliptic curve over anl-adic eld K. Suppose φ : E → E′ is a cyclic p−isogeny (the kernel is acyclic group of cardinality p) dened over K for a prime number p ≥ 3. Denote

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92 CHAPTER 6. PARITY CONJECTURE WITH A CYCLIC ISOGENY

c(E) = (E(K) : E0(K)) and c(E′) = (E′(K) : E′0(K)) the Tamagawa numbers,and let δ = ±1 such that δ = 1 unless p = 3, µ3 6⊂ K and E/K has Néron-Kodaira type IV or IV ∗ (e.g. section 5.5.2). Then

ordpc(E′)c(E)

=

0, if E has good or non-split multiplicative reduction,±1, if E has split multiplicative reduction,0, if E has additive reduction and δ = 1,±1, if E has additive reduction and δ = −1.

Proof : We distinguish the corresponding cases :

• E has good reduction In this case we have by denition c(E) = c(E′) = 1.

• E has non-split multiplicative reduction By Tate algorithm ([53] ch. IV 9step 2) E and E′ c(E), c(E′) = 1 or 2, hence the quotient is prime to p.

• E has split multiplicative reduction In this case we use the followinglemma:

Lemma 6.2.1 (Tom Fisher [15]) If E has split multiplicative reductionthe either

1. E[φ] ' Z/pZ over K, and

c(E′)c(E)

=1p, or

2. E[φ] ' µp over K andc(E′)c(E)

= p.

This justify the second case of the proposition.

• This last case is explain in [14]

Next we reformulate some results of [14]. Just remark that the number αthat we dened by φ∗ω′ = αω is also the leading coecient for the action of φon formal groups.

Proposition 6.2.4 Let K be an l−adic eld and E/K an elliptic curve witha cyclic p−isogeny, φ, for p a prime ≥ 3. Furthermore let α be dene as theleading coecient of the action of φ on formal groups.

1. For l 6= p, α is a unit.

2. for l = p, ordp|α|l is even if and only if (−1, Fφ/F ) = 1

Proof : The rst item is trivial since in this case, φ induce an isomorphism onformal groups.

The second item is not obvious and depends on three lemmas of Tim andVladimir Dokchitser.

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6.2. THE P−SELMER RANK 93

Lemma 6.2.2 ([14] lemma 12) Let Ql ⊂ K ⊂ K ′ be nite extension (p odd),with K ′/K cyclic Galois of degree dividing p− 1. Then (−1,K ′/K) = 1 if andonly if one of the following condition is satised :

1. The residue eld k of K is of even degree over F, or

2. p−1e(K′/K) is even, where e(K ′/K) denote the ramication degree of K ′/K.

Proof :[14] section 5 .For what follows we denote by K an l−adic eld and by OK the ring of

integers , mK = (π)K the maximal ideal and a uniformizer , υ the valuation andk = OK/mK its ring of integers. Let φ : E/K → E′/K be an cyclic isogeny ofprime degree p. Moreover we denote by E(mK) and E′(mK) the formal groupsand by f : E(mK)→ E(mK) the map induced by φ. Then by working directlyon formal groups they prove the following lemmas

Lemma 6.2.3 ([14] lemma 13) If α, (f(T ) := αT + · · ·), is a unit, then Kφ/Kis unramied.

Proof : See [14] section 6.

Lemma 6.2.4 ([14] lemma 14) Let f : E → E′be the reduction of f modulomK . If f is inseparable of degree p, then f has kernel of order p in the maximalunramied extension Kunr if and only if υ(α) is a multiple of p− 1.

Proof : See [14] section 6.End of the proof of proposition 6.3.4 As |α|l = p−[k:Fp]υ(α) we know

that if [k : Fp is even so is ordp(|α|) and (−1,Kφ/K) = 1 by the rst item oflemma 6.3.2. This prove a special case of the proposition.

So we suppose at present that [k : Fp] is odd. We denote by e the greatestexponent such that 2e|(p− 1). Then ordp |α| ≡ υ(α)mod 2. If υ(α) = 0 then bylemma 6.3.3, Kφ/K is unramied so (−1,Kφ/K) = 1 since all units are normsin this case.

On the other hand if υα > 0 the reduction f is an inseparable isogeny ofdegree p, so we may apply lemma 6.3.4. This gives that 2e|υKφ(α).

If υ(α) is odd, it follows that the ramication degree ofKφ/K is a multiple of2e because ([47] ch. II) vφ(α) = e(Kφ/K)υ(α). Hence in this last case p−1

e(Kφ/K)

is odd, so (−1,Kφ/K) = −1 by lemma 6.3.2.If υ(α) is even. As the valuation of α in the eld Kunr

((πK)p−1/2

)is

divisible by p − 1, lemma 6.3.4 show that Kφ ⊂ Kunr((πK)p−1/2

)hence the

ramication index of Kφ/K is not divisible by 22 the second case of lemma 6.3.2applies.

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94 CHAPTER 6. PARITY CONJECTURE WITH A CYCLIC ISOGENY

6.2.2 The computation of p−Selmer ranks

The purpose of this section is to state and explain the theorem 6 of [14]. Thistheorem and its consequences is one of the main result of this paper. It isoriginal, in the sense that it stay on the approach of Tim Fisher explainedearlier.

Theorem 6.2.2 ([14] theorem 6) For K = R or C, or [F : Ql] < ∞ and aprime p ≥ 3. Let E/K be an elliptic curve with a rational p−isogeny φ. Deneσφ(E/K) as in corollary 6.3.1, then

σφ(E/K) =

−(−1,Kφ/K), if K is archimedian(−1,Kφ/K), if E has good reduction,−(−1,Kφ/K), if E has split multiplicative reduction,(−1,Kφ/K), if E has non-split multiplicative reduction,δ if E has additive reduction and l 6= p

Proof:Here I reformulate the proof given in [14].

K archimedean

If K = Kφ = R or K = Kφ = C then |E′(K)/φ(E(K))| = 1 and |E(K)[φ]| = pso that, by corollary 6.3.1, σφ(E/K) = −1. If K = R and Kφ = C the generatorof ker(φ) is only dened over C so that |E(K)[φ]| = 1 while E′(K)/φ(E(K))| = 1so that σφ(E/K) = 1.

This gives σφ(E/K) = −(−1,Kφ/K).

K non-archimedean

Proposition 6.3.3 and 6.3.4 directly implies this case, it only remains to provethat (−1,Kφ/K) = 1 for places l 6= p of semistable reduction this is done byproving that Kφ/K is unramied and then applying lemma 6.3.1, see [14] p.4.

6.3 The root number

The purpose of this section is to present one of the main theorem of the article[14], the one dealing with local root numbers of elliptic curves.

Theorem 6.3.1 ([14] theorem 5) Assume K = R or C, or [K : Dl] < ∞, andlet p ≥ 3 be a prime. Let E/K be an elliptic curve with a rational p−isogeny φ.Then

W (E/K) =

−1 if K is archimedean,1 if E has good reduction,−1 if E has split multiplicative reduction,1 if E has non-split multiplicative reduction,δ · (−1,Kφ/K) if E has additive reduction and l 6= p.

Her δ = 1 unless p = 3, µ3 6⊂ K and E/K has reduction type IV or IV ∗, inwhich case δ = −1.

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6.3. THE ROOT NUMBER 95

Proof : The rst cases were known before the article [14], for example we cancite the following result of Rohrlich:

Proposition 6.3.1 (Rohrlich [42] section 19) Let E/K be an elliptic curve overa local eld K.

1. If E has good reduction over K, then W (E/K) = 1.

2. Suppose E has potential multiplicative reduction. If E has multiplicativereduction over K itself

W (E/K) =−1 if E has split multiplicative reduction,1 if E has non-split multiplicative reduction.

If E has additive reduction, let ξ be a character of the Weil group WK such thatthe quadratic twist Eξ has multiplicative reduction. Then

W (E/K) = ξ(−1).

This gives the four rst cases. The forme of the case of additive reduction whichinterest us is dealt with by Tim and Vladimir Dokchitser in [14]. They distin-guish three cases : potential multiplicative reduction, potential good reductionwith p ≥ 5, potential good reduction with p = 3. This gives rise to three lemma(lemma 8, 9, 10) which x this dierent cases.

6.3.1 Potential multiplicative reduction

We begin by proving some elements of the proof of lemma 8 which are notexplicit in the text.

Lemma 6.3.1 (Lemma 8 of [14]) Suppose that K is a l−adic eld and thatE/K is an elliptic curve with additive reduction of potential multiplicative reduc-tion type. Denote the Tate module, for p 6= l, of E by Tp(E) = proj limnE[pn]and put Vp(E) = Tp(E) ⊗ Qp. Then the action of the inertia subgroup of

Gal(K/K) over Vp(E) is of the form ±(

1 ∗0 1

), furthermore the action of

the full Weil group on Vp(E) is of the form(χ ∗0 χ−1 ‖

), for some ramied

quasi-character χ. The root number is given by W (E/K) = (−1,Kφ/K).

Proof: Suppose rst that E/K has split multiplicative reduction. In this case,by theorem 5.1.5, E is isomorphic to a Tate curve Eq for some q ∈ K∗. Thereis an isomorphism, the l−adic uniformization, given by

φ : K∗/qZ → Eq(K).

In this isomorphism a pn−torsion point correspond to a pn − th root of q. Letus x such a root, Q = q1/pn ∈ K. Then, naturally, if ζ is a pn − th root of onein K the l−adic uniformization (see the section on Tate curves) yields(

ζZ ·QZ) /qZ ∼= Eq[pn].

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96 CHAPTER 6. PARITY CONJECTURE WITH A CYCLIC ISOGENY

On the other hand if σ is an element of the inertia group IK/K , σ(Q) is a pn−throot of q. Hence there exists an integer 0 ≤ b ≤ pn − 1 such that σ(Q) = ζbQ.At present we put, P1 = φ(ζ) and P2 = φ(ζ).

It follows from the commutativity of the l−adic uniformization with respectto Galois action that σ(P1) = P1 (K(ζ) is totally ramied) and σ(P2) = bP1+P2.So inertia takes the form (

1 b0 1

).

Now, it is straightforward to verify that we can take the projective limit overn in what preceded. This yields, in the case of split multiplicative reduction,that the action of inertia on the p−adic Tate module is of the form(

1 b0 1

).

Moreover b can't be identically 0, for otherwise inertia would act trivially whichwould contradict the criterion of Néron-Ogg-Shafarevich.

Furthermore, we know from theorem 5.1.1 that if E/K has potential mul-tiplicative reduction it admits split multiplicative reduction over a quadraticextension, and that the corresponding curve are isomorphic over K. This iso-morphism commutes with Galois only by a twist by the quadratic characterassociated to this extension. This justify the sign ± in the lemma.

For the action of the full Weil group, recall that inertia is normal in the Weilgroup, so Frobenius preserve the invariant subspace of inertia. This gives anaction of the Weil group of the form(

χ ∗0 χ′

)but the determinant of the l-adic Galois representation of elliptic curves is equalto the cyclotomic character ‖·‖K we deduce that χχ′ = ‖·‖K and so, the actionof the full Weil group is of the form(

χ ∗0 χ−1‖ · ‖

).

The fact that χ is ramied comes from the form of inertia. The remaining ofthe proof is clearly explained in the article [14].

6.3.2 Potential good reduction and p ≥ 5

This case is dealt with by the following lemma :

Lemma 6.3.2 ([14] lemma 9) Suppose E has potential good reduction and p ≥

5. Then the action of the Weil group on Vp(E) is of the form(χ 00 χ−1‖ · ‖

)for some quasi-character χ. The root number is given byW (E/K) = (−1,Kφ/K).

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6.3. THE ROOT NUMBER 97

Proof : Here I just add some precisions to the original proof given in [14]. The

fact that inertia acts via

(1 ∗0 1

)on E[p] from the Galois invariance of the

Weil pairing. As E[p] ∼= (Z/pZ)⊕ker(φ) we want to show that ker(φ) is invariantunder inertia. Let ep( , ) denote the Weil paring on E′[p]. Let A ∈ ker(φ) and ian element of inertia. Then for all B ∈ E′[p]

ep(φ(iA), B) = ep(iA, φ(B)) = ep(A, i−1φ(B))i = ep(φ(A), i−1B)i = 0,

by the properties of the Weil pairing ([52] ch. III sec. 8) and the fact that theisogeny is dened over K. As it is true for all B we deduce that φ(iA) = Oand so ker(φ) is an invariant one dimensional subspace of E[p] which show that

inertia has the form

(1 ∗0 1

).

Furthermore in [50] and [48], it is shown that inertia acts via a nite subgroupof order dividing 24, the exact order depending on the Néron classication. Soinertia has no element of order p ≥ 5 (by classical group theory), this showthat E/Kφ has good reduction by the criteria of Néron-Ogg-Shafarevich. Theremaining of proof is clearly stated in the article.

6.3.3 Potential good reduction and p = 3

The purpose of this section is to add some clues to the proof of the last case oftheorem 6.4.1 (theorem 5 of [14]), namely the case of potentially good reductionwith p = 3. The result is stated as follow.

Lemma 6.3.3 ([14] lemma 10) Suppose E has potential good reduction andp = 3. Then W (E/F ) = δ(−1,Kφ/K).

Proof: Inertia of G = Gal(K(E[3])/K) is of the form

(∗ ∗0 ∗

)because it

as an invariant one-dimensional subspace, namely ker(φ). It is of determinant1 because the determinant of the action of Galois is the cyclotomic characterwhich gives 1 on inertia. Then an easy computation shows that inertia, I, is oneof the following cyclic group C2, C3 or C6. The fact that I = C3 if and only if Ehas reduction type IV or IV ∗ can be found in the article of Serre ( [50] section5.6.) in the case l 6= 2 and it is a result of Krauss ([25] theorem 2) in the casel = 2. In this case, G is either C3, C6 if µ3 ⊂ K and the permutation group S3

otherwise. The rst two case are explained clearly in the article, let me explainthe third case : G = S3. The remaining of the proof is quite technical, I referto the original article for it.

I owe the following explaination directly by Tim Dokchitser that I thank forit. We can use Schur's lemma to show that Frob2 acts as a scalar because V3(E)is an irreducible representation of of Gal(F /F ). But as Gal(F /F ) is generatedby the Frobenius and ny inertia, it suce to show that Frob2 commutes withinertia. On the other hand, inertia acts trough C3 and is normal in Gal(F /F ) sofor any element i of inertia ρ(Frob2)ρ(i)ρ(Frob2)−1) is an element of C3 which

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98 CHAPTER 6. PARITY CONJECTURE WITH A CYCLIC ISOGENY

is necessarilly trivial because it acts trivially on E[3] as on E[3] inertia acts byC3 and Frob acts via a transposition.

6.4 The case of a 2−isogenyThis case is treated by direct computations on the Weierstrass equation :

E : y2 = x3 + ax2 + bx, a, b ∈ OKE′ : y2 = x3 − 2ax2 + δx, δ = a2 − 4b,φ(x, y) = (x+ a+ bx−1, y − bx−2y).

On the other hand the discriminant of E is given by

∆(E) = 16δb2.

Here we present the proof of theorem 4 of [14] which states that if E/K haseither good ordinary or multiplicative reduction at all prime above 2. Then forall places v of K,

(?) W (E/Kv) = σφ(E/Kv)(a, b)Kv (−2a, δ)Kv .

The proof goes in several steps, namely: innite places, nite places such that[Kv : Ql] < ∞ and l 6= 2, and the case of nite places with l = 2. Actually,the purpose of this theorem is not to compute the local terms which has beendone here and in other article, but to correlate local root number and localSelmer rank in such a way that the product over all places will prove the parityconjecture.

Next I had a section on Hilbert symbol

6.4.1 Hilbert symbols

Let K be a nite extension of Qp with p either nite or innite. We dene theHilbert symbol for a, b ∈ K∗ by

(a, b)K =

1, if the from ax2 + by2 − z2 has a non-trivial zero inK−1, otherwise.

Equivalently (a, b)K = 1 if and only if a = z2 − by2 for some y, z ∈ K, whichmeans that a is a norm from the quadratic extension K(

√b)/K.

The Hilbert symbol satises the following elementary properties

1. (a, b)K = (b, a)K

2. (aa′, b)K = (a, b)K(a′, b)K and (a, bb′)K = (a, b)K(a, b′)K .

3. (a, b)K = 1 for all b if and only if a ∈ K∗2.

4. (a,−a)K = 1

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6.4. THE CASE OF A 2−ISOGENY 99

Furthermore, it satises the following product formula : Let K be a numbereld then ∏

p∈MK

(a, b)Kp = 1.

The following results is used twice in the article [14], st in the case of thelocal norm residue symbol, second in the lemma 15 which gives the computationof some Hilbert symbols.

Lemma 6.4.1 (see e.g. [29] ch.2 sec.2) Let K be a nite extension of Qp forsome nite prime p. Let L be a unramied extension of K. Then every unit ofK is a norm of some unit in L.

In [14] they prove the following lemma

Proposition 6.4.1 ([14] lemma 15) Let K/Qp be a nite extension (p nite).Then

1. If |x| < 1 and y ∈ K∗ then (1 + 4x, y) = 1.

2. If p = 2, |x| = 1 and y ∈ O∗K then (1 + 4x, y) = 1.

3. (−1,−2) = −1 if and only if p = 2 and [K : Qp] is odd

Proof : For odd p, 1) is a consequence of the fact that 1 + 4x is a unit, sothe preceding lemma (6.2.1) implies. Similarly, (−1,−2) = 1 for odd p.

The item 1) for p = 2 is proved in [53] : the series

(1 + 4x)1/2 =∞∑n=0

(−1/2n

)(4x)n

converge in K because |x| < 1 and(−1/2n

)4n = (−1)n

(2nn

)is an integer. This prove that 1 + 4x is a square in K∗, hence (1 + 4x, y) = 1.

For item 2) they also use the preceding lemma : it suces to show thatK(√

1 + 4x)/K is unramied, that is (1+4x) is a square in the unique quadraticunramifed extension L of K. For the remaining of the proof I refer to [14] whereit is clearly stated.

6.4.2 Discussion about the proof of theorem 6.1.2

Here we discuss the proof of the theorem 6.1.2, i.e. of equation (?). Here E isendowed with a 2-isogeny.

The case of innite places is clearly explained in the article (sec. 7.1) anddepends on the technic of Fisher and on the classication of the groups E(R)and E′(R) which I recall :

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100 CHAPTER 6. PARITY CONJECTURE WITH A CYCLIC ISOGENY

Proposition 6.4.2 ([53] corollary 2.3.1 ch. V) Let E(R) be an elliptic curve,and let ∆(E) be the discriminant of some Weierstrass equation for E/R. Thereis an isomorphism of real Lie groups

E(R) ∼=S1, if ∆(E) < 0,S1 × (Z/2Z), if ∆(E) > 0.

The case of nite places depends again on the technic of Fisher, presentedearlier : we need to compute the parity of the exponent of the power of 2 in|E(K)[φ]|/|(E′(K)/φ(E(K)))| = c(E′)

c(E) |α|−1.

The computation of the Tawagawa is mainly based upon Tate's algorithmwhich is exposed in [53] chapter IV section 9. In the case of good reductionand of characteristic l 6= 2 the global result is easy to obtain. ([14] sec. 7.3).Otherwise the reduction is either potentially good or potentially multiplicative.The cases of characteristic 2 and 6= 2 is treated separately.

For potential good reduction and l 6= 2 we know that E/F (E[4]) has goodreduction from a result of Serre and Tate, corollary 3 of [48]. As E/F (E[4])has good reduction the discriminant is a twelve power over that eld and asF (E[4])/F is a 2-extension 3|v(∆E). According to the classication of Néron-Kodaira the reduction type of E is either III, III∗, I∗0 or I∗n. With this and withthe Weierstrass equations of E and E', Tim and Vladimir Dokchitser obtainedthe required results. The computation of the root numbers is due mainly to aresult of Kobayashi in [26] who computed root numbers of elliptic curves denedby a Weierstrass equation. I recall the result.

Theorem 6.4.1 (Kobayashi [26] theorem 1.1) Let K be a local eld with residueeld k and odd characteristic p. Let E/K be an elliptic curve with potentialgood reduction. let y2 = x3 + ax2 + bx+ c be a Weierstrass equation and ∆ thediscriminant of this cubic polynomial. We denote the quadratic residue symbol

on k× by(k

)and the Hilbert symbol of K by ( , )K . We extend the residue

symbol by putting(

0k

)= 1

1. If the Néron-Kodaira type of E is I0 or I∗0 , then

W (E/K) =(−1k

) v(∆)2

.

2. If the Néron-Kodaira type of E is III or III∗, then

W (E/K) =(−2k

).

3. If the Néron-Kodaira of E is II, IV , IV ∗ or II∗ there exists a Weierstrassequation such that 3 - vK(c). For such an equation we have

W (E/K) = δ(∆, c)K

(vK(c)k

)(−1k

)v(∆)(v(∆)−1)/2

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6.4. THE CASE OF A 2−ISOGENY 101

where δ = ±1 and δ = 1 if and only if ∆12 ∈ K.

Note that if π is a uniformizer parameter of the l-adic eld K with residue

eld k, the Hilbert symbol (·, π) is the same thing as the residue symbol

(·k

)..

This explains the table given in section 7.4 of [14].The case of type III is clear from Néron-Kodaira classication and from

δ ≡ −4bmod π2 note also that υ(δ) = 1 implies that there is a unit u such thatδ = uπ so (−2, δ) = (−2, π)(−2, u) = (−2, π), nally

(a,−b)(−2a, δ) = (a,−bδ)(−2, δ) = (a+ 4b2 +O(π3))(−2, π) = (−2, π)

which agree with (?).The case III∗ is similar, for the type I∗0 they use Tate algorithm ([53] ch.

IV sec. 9 p 367). More precisely c(E) = 1 + |α ∈ k|P (α) = 0| whereP = T 3 + a

πT2 + b

π2T . The discriminant of the quadratic equation obtain being

w = a2−4bπ2 = δπ−2, c(E) = 4 if and only if w is a square otherwise c(E) = 2.

But w is a square if and only if (π, δ) = 1. And same things for E′. With someeasy computations (see [14] p. 12) this gives (?). The case of type I∗n goes alongthe same lines : using Tate's algorithm as explained in [53].

In the case of good ordinary reduction in residue characteristic 2,W (E/K) =1, c(E) = c(E′) = 1, the only question remaining is ord2|α|. The model of thecurves depend on the fact that the 2-torsion point reduce to O or not.

Tim and Vladimir Dokchitser indicate that if E and E′ are transformedinto their respective minimal model by standard substitutions (x, y) 7→ (w2x+. . . , w3y + . . .) and (x, y) 7→ (u2x+ . . . , u3y + . . .), then α = uw−1.

If the 2-torsion point reduce to O, they manipulate the model (see e.g. [52]Appendix A) of the curve to obtain w = 1 and u = 2 this gives ord2|α|K =ord2|2|K which is, almost by denition, even if and only if [K : Q2] is even whichby proposition 6.5.1 (lemma 15 of [14]) is equivalent to (−2,−1) = 1. On theother hand by manipulating the equation of E, they obtain (a,−b)(−2a, δ) =(−2,−1), proving (?) in this case.

If the 2-torsion point reduce to O by using appropriate model they showthat α = 1.

In the case of multiplicative reduction, they distinguish the case of splitmultiplicative reduction for which they use the theory of the Tate curve and ofnon-split reduction for which they reduce to split multiplicative reduction aftera twist by a quadratic character.

In the case of split multiplicative reduction, we know from the precedingpropositions (6.2.3 and theorem 6.3.1) that ord2(c(E′)/c(E)) = ±1 and thatW (E/K) = −1.They directly use the theory of the Tate curve, which furnishesthe 2-torsion points. Their article is clear about this subject, no need to re-produce it here. For non-split multiplicative reduction, they use the theorem ofTate to reduce to split-multiplicative reduction and Tate's algorithm as exposedin [53] ch IV sec. 9 to compute the Tamagawa numbers.

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102 CHAPTER 6. PARITY CONJECTURE WITH A CYCLIC ISOGENY

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Chapter 7

Conclusion

What can I conclude from this work ? First of all that it was helpful for me.Before the beginning I didn't realize that one must enter the vast territory ofmodern research by some way, for me it was the article of Tim and VladimirDokchitser and its related articles and subjects. I am happy to have gained someknowledge about the rich subject of L−functions as there is a gap between theelementary Dirichlet series and the theoretical framework of L−functions ofalgebraic varieties. This work has helped me to understand some basic butimportant facts that are essential in understanding the far reaching and activeeld of research linked to the L world.

Moreover I reinforced my knowledge of elliptic curves by working on thearticle [14]. I tried to make the text as self-contained as possible, at least bypresenting the denitions, theorems and ideas linked to this article. As everyoneworking in the eld of elliptic curves may be lled with wonder by this fruitfulsubject, I am satised to know some of its technics and results. I was evensurprised to observed that I can know read works about it without being toodisoriented as I now know some of the main research themes and guidelines.

However, having understood this article, one may be unsatised by the factthat there is no clear idea that link root numbers to Selmer ranks. Somethingimportant may be missing, not too surprisingly because the notions involvedare, up to now, mainly conjectural.

Nonetheless, the essential of my work is not contained in this text. In fact as Iwant to do research in Mathematics, I consider that the main consequence of mywork is that I have gained insight and familiarity with this working activity. Itwas necessary for me to take the plunge from school mathematics to professionalmathematics and I consider that this work was a good step. Actually, I thinkthat it is useful for every student to go from a passive school learning to anactive and personal point of view. This is what gives sense to studies and this

103

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104 CHAPTER 7. CONCLUSION

is what I enjoy having (partially) reached, the next step being my thesis.I am so grateful to the Professor Marc Hindry of Paris 7 University. He has

chosen the article [14] and I didn't realize immediately that It would help me somuch. The article is quite short (17 p.) but working on it I travelled throughimportant areas of number theory, from algebraic number theory, class eldtheory, zeta and L functions to the fertile land of elliptic curves. I discoveredfor myself various new things as suggested in the bibliography. I am also gratefulfor his two half-semester courses on elliptic curves and on abelian varieties whichme and the other students enjoyed so much.

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