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Atkin and Swinnerton-Dyer Congruences onNoncongruence Modular Forms
ICERMApril 18, 2013
Winnie Li
Pennsylvania State University, U.S.A.and
National Center for Theoretical Sciences, Taiwan1
Noncongruence subgroups
• Bass-Lazard-Serre: All finite index subgroups of SLn(Z) forn ≥ 3 are congruence subgroups.
• SL2(Z) contains far more noncongruence subgroups than con-gruence subgroups.
• Let Γ be a finite index subgroup of SL2(Z). The orbit spaceΓ\H∗ is a Riemann surface, called the modular curve XΓ forΓ. It has a model defined over a number field.
• The modular curves for congruence subgroups are defined overQ or cyclotomic fields Q(ζN ).
• Belyi: Every smooth projective irreducible curve defined over anumber field is isomorphic to a modular curve XΓ (for infinitelymany finite-index subgroups Γ of SL2(Z)).
2
Modular forms for congruence subgroups
Let g =∑n≥1 an(g)qn, where q = e2πiz, be a normalized
(a1(g) = 1) newform of weight k ≥ 2 level N and characterχ.
I. Hecke theory
• It is an eigenfunction of the Hecke operators Tp with eigenvalueap(g) for all primes p - N , i.e., for all n ≥ 1,
anp(g)− ap(g)an(g) + χ(p)pk−1an/p(g) = 0.
• The space of weight k cusp forms for a congruence subgroupcontains a basis of forms with algebraically integral Fourier co-efficients. An algebraic cusp form has bounded denominators.
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II. Galois representations
• (Eichler-Shimura, Deligne) There exists a compatible family ofdegree two l-adic rep’ns ρg,l of GQ := Gal(Q̄/Q) such that atprimes p - lN , the char. poly.
Hp(T ) = T 2 − ApT + Bp = T 2 − ap(g)T + χ(p)pk−1
of ρg,l(Frobp) is indep. of l, and
anp(g)− Ap an(g) + Bp an/p(g) = 0
for n ≥ 1 and primes p - lN .
• Ramanujan-Petersson conjecture holds for newforms. That is,|ap(g)| ≤ 2p(k−1)/2 for all primes p - N .
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Modular forms for noncongruence subgroups
Γ : a noncongruence subgroup of SL2(Z) with finite index
Sk(Γ) : space of cusp forms of weight k ≥ 2 for Γ of dim d
A cusp form has an expansion in powers of q1/µ.
Assume the modular curve XΓ is defined over Q and the cuspat infinity is Q-rational.
Atkin and Swinnerton-Dyer: there exists a positive integer Msuch that Sk(Γ) has a basis consisting of forms with coeffs. integraloutside M (called M -integral) :
f (z) =∑n≥1
an(f )qn/µ.
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No efficient Hecke operators on noncongruence forms
• Let Γc be the smallest congruence subgroup containing Γ.
Naturally, Sk(Γc) ⊂ Sk(Γ).
• TrΓcΓ : Sk(Γ)→ Sk(Γc) such that Sk(Γ) = Sk(Γc)⊕ker(TrΓc
Γ ).
• ker(TrΓcΓ ) consists of genuinely noncongruence forms in Sk(Γ).
Conjecture (Atkin). The Hecke operators on Sk(Γ) for p -M
defined using double cosets Γ
(1 00 p
)Γ as for congruence forms is
zero on genuinely noncongruence forms in Sk(Γ).
This was proved by Serre, Berger.So the progress has been led by computational data.
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Atkin and Swinnerton-Dyer congruences for ellipticcurves
Let E be an elliptic curve defined over Q with conductor M . ByBelyi, E ' XΓ for a finite index subgroup Γ of SL2(Z).Ex. E : x3 + y3 = z3, Γ is an index-9 noncong. subgp of Γ(2).
Atkin and Swinnerton-Dyer: The normalized holomorphic differ-
ential 1-form f dqq =∑n≥1 anq
ndqq on E satisfies the congruence
relation
anp − [p + 1−#E(Fp)]an + pan/p ≡ 0 mod p1+ordpn (1)
for all primes p -M and all n ≥ 1.
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Sketch of a proof using formal group laws
• The formal power series `(x) :=∑n≥1 anx
n is the formal log ofa formal group law G, which is isomorphic to the group law onthe elliptic curve E in a neighborhood of the identity element.
• The Hasse-Weil L-function of E is
L(s, E) =∏p-M
1
1− bpp−s + p1−2s
∏p|M
1
1− bpp−s=∑n≥1
bnn−s,
in which bp = p + 1−#E(Fp) for p -M .
• Honda: ˜̀(x) :=∑n≥1 bnx
n is the formal log of a formal grouplaw which is strictly isomorphic to G.
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• Hence for p -M , the sequences {an} and {bn} satisfy the samecongruence relation of ASD type:
cnp−Apcn +Bp,1cn/p +Bp,2cn/p2 + · · · ≡ 0 mod p1+ordpn.
• The bn’s satisfy the three term recursion
bnp − bpbn + pbn/p = 0 for all p -M and n ≥ 1.
So Ap = bp, Bp,1 = p, and other Bp,i = 0. This proves (1).
• For p - M , Hp(T ) = T 2 − bpT + p is the characteristic poly.of the Frobp acting on the Tate module Tl(E) for l 6= p.
• Taniyama-Shimura modularity theorem: g =∑n≥1 bnq
n is anormalized congruence newform. Note that f ∈ S2(Γ). Thus(1) gives congruence relations between f and g.
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ASD congruences in general
Back to general case where XΓ has a model over Q, and thed-dim’l space Sk(Γ) has a basis of M -integral forms.
ASD congruences (1971): for each prime p - M , Sk(Γ,Zp)has a p-adic basis {hj}1≤j≤d such that the Fourier coefficients ofhj satisfy a three-term congruence relation
anp(hj)− Ap(j)an(hj) + Bp(j)an/p(hj) ≡ 0 mod p(k−1)(1+ordpn)
for all n ≥ 1. Here
• Ap(j) is an algebraic integer with |Ap(j)| ≤ 2p(k−1)/2, and
• Bp(j) is equal to pk−1 times a root of unity.
This is proved to hold for k = 2 and d = 1 by ASD.
The basis varies with p in general.
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Galois representations attached to Sk(Γ) and Schollcongruences
Theorem[Scholl] Suppose that the modular curve XΓ has amodel over Q such that the cusp at infinity is Q-rational. At-tached to Sk(Γ) is a compatible family of 2d-dim’l l-adic rep’nsρl of GQ unramified outside lM such that for primes p > k+1not dividing Ml, the following hold.
(i) The char. polynomial
Hp(T ) = T 2d + C1(p)T 2d−1 + · · · + C2d−1(p)T + C2d(p)
of ρl(Frobp) lies in Z[T ], is indep. of l, and its roots are alge-
braic integers with complex absolute value p(k−1)/2;
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(ii) For any form f in Sk(Γ) integral outside M , its Fouriercoeffs satisfy the (2d + 1)-term congruence relation
anpd(f ) + C1(p)anpd−1(f ) + · · ·++ C2d−1(p)an/pd−1(f ) + C2d(p)an/pd(f )
≡ 0 mod p(k−1)(1+ordpn)
for n ≥ 1.
Theorem If Sk(Γ) is 1-dimensional, then
• the ASD congruences hold for almost all p;
• the degree two l-adic Scholl rep’ns of GQ are modular.
The 2nd statement follows from Kahre-Wintenberger’s work onSerre’s conjecture on modular rep’ns, and various modularity lift-ing theorems.
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Application: Characterizing noncongruence modularforms
The following conjecture, supported by all known examples,gives a simple characterization for noncongruence forms. If true,it has wide applications.
Conjecture. A modular form in Sk(Γ) with algebraic Fouriercoefficients has bounded denominators if and only if it is a con-gruence modular form, i.e., lies in Sk(Γc).
Theorem[L-Long] The conjecture holds when Sk(Γ) is 1-dim’l,containing a basis with Fourier coefficients in Q.
The proof uses ASD congruences, modularity of the Scholl rep-resentations, and the Selberg bound on Fourier coefficients of a wtk cusp form f : an(f ) = O(nk/2−1/5).
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From Scholl congruences to ASD congruences
Ideally one hopes to factor
Hp(T ) =∏
1≤j≤d(T 2 − Ap(j)T + Bp(j))
and find a p-adic basis {hj}1≤j≤d, depending on p, for Sk(Γ,Zp)such that each hj satisfies the three-term ASD congruence rela-tions given by Ap(j) and Bp(j).
For a congruence subgroup Γ, this is achieved by using Heckeoperators to further break the l-adic space and Sk(Γ) into pieces.For a noncongruence Γ, no such tools are available.
Theorem[Scholl] If ρl(Frobp) is diagonalizable and ordinary(i.e. half of the eigenvalues are p-adic units), then ASD con-gruences at p hold.
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Examples of ASD congruences
The group Γ1(5) has genus 0, no elliptic elements, and 4 cusps.The wt 3 Eisenstein series
E1(z) = 1− 2q1/5 − 6q2/5 + 7q3/5 + 26q4/5 + · · · ,E2(z) = q1/5 − 7q2/5 + 19q3/5 − 23q4/5 + · · · .
have simple zeros at all cusps except ∞ and −2, resp., and non-
vanishing elsewhere. So XΓ1(5) is defined over Q with t = E2E1
as
a Hauptmodul, and tn = n√t is a Hauptmodul of a smooth irred.
modular curve XΓn over Q.
Let ρn,l be the l-adic Scholl representation attached to S3(Γn).
Ex 1. When n = 2, S3(Γ2) =< E1t2 > is 1-dim’l, ASD congru-ences hold for odd p, and ρ2,l are isom. to ρη(4z)6,l.
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Ex 2. (L-Long-Yang)
(1) When n = 3, the space S3(Γ3) =< E1t3, E1t23 > has a basis
f±(z) = q1/15 ± iq2/15 − 11
3q4/15 ∓ i16
3q5/15 −
−4
9q7/15 ± i71
9q8/15 +
932
81q10/15 + · · · .
(2) (Modularity) There are two cuspidal newforms of weight 3 level27 and quadratic character χ−3 given by
g±(z) = q ∓ 3iq2 − 5q4 ± 3iq5 + 5q7 ± 3iq8 +
+9q10 ± 15iq11 − 10q13 ∓ 15iq14 − · · ·such that ρ3,l = ρg+,l ⊕ ρg−,l over Ql(
√−1).
(3) f± satisfy the 3-term ASD congruences with Ap = ap(g±) and
Bp = χ−3(p)p2 for all primes p ≥ 5.
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Ex 3. (Atkin, Li, Long) S3(Γ4) =< h1, h2, h3 >, where hi = E1ti4
and h2 = E1t2 satisfies ASD congruences for all odd p.
The space < h1, h3 > has a basis satisfying the ASD congruencedepending on the residue of odd p mod 8:
1. If p ≡ 1 mod 8, then both h1 and h3 satisfy ASD with Ap =
sgn(p)a1(p) and Bp = p2, where sgn(p) = ±1 ≡ 2(p−1)/4
mod p;
2. If p ≡ 5 mod 8, then h1 (resp. h3) satisfies ASD with Ap =
4ia5(p) (resp. −4ia5(p)) and Bp = −p2;
3. If p ≡ 3 mod 8, then h1±h3 satisfy ASD withAp = ±2√−2a3(p)
and Bp = −p2;
4. If p ≡ 7 mod 8, then h1±ih3 satisfy ASD withAp = ∓8√−2a7(p)
and Bp = −p2.
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Here a1(p), a3(p), a5(p), a7(p) are the Fourier coefficients of thewt 3 congruence forms f1, f3, f5, f7 given below:
f1(z) =η(2z)12
η(z)η(4z)5= q1/8(1 + q − 10q2 + · · · ) =
∑n≥1
a1(n)qn/8,
f3(z) = η(z)5η(4z) = q3/8(1− 5q + 5q2 + · · · ) =∑n≥1
a3(n)qn/8,
f5(z) =η(2z)12
η(z)5η(4z)= q5/8(1 + 5q + 8q2 + · · · ) =
∑n≥1
a5(n)qn/8,
f7(z) = η(z)η(4z)5 = q7/8(1− q − q2 + · · · ) =∑n≥1
a7(n)qn/8.
For the last two examples, both l-adic space and S3(Γn) admitquaternion multiplications, which are used to break the spaces.
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An example of failure of ASD congruences
Ex 4. (Kibelbek) X : y2 = x5 + 1 is a genus 2 curve defined overQ. By Belyi, X ' XΓ for a finite index subgroup Γ of SL2(Z).Put
ω1 =dx
2y= f1
dq1/10
q1/10, ω2 = x
dx
2y= f2
dq1/10
q1/10.
Then S2(Γ) =< f1, f2 >, where
f1 = q1/10 − 8
5q6/10 − 108
52q11/10 +
768
53q16/10 +
3374
54q21/10 + · · · ,
f2 = q2/10 − 16
5q7/10 +
48
52q12/10 +
64
53q17/10 +
724
54q22/10 + · · · .
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The l-adic representations attached to S2(Γ) are the dual of theTate modules on the Jacobian of XΓ. They are strongly ordinaryat p ≡ ±1 mod 5, so ASD congruences hold for such p.
For primes p ≡ ±2 mod 5, Hp(T ) = T 4 + p2 (not ordinary).
S2(Γ) has no nonzero forms satisfying the ASD congruences forp ≡ ±2 mod 5.
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Scholl congruences
Back to the finite index subgroup Γ of SL2(Z); Sk(Γ) is d-dim’lwith an M -integral basis.
• The 2d-dim’l Scholl rep’ns ρl ofGQ are generalizations of Deligne’sconstruction to the noncongruence case.
• For p -M , Scholl constructed p-adic de Rham spaceDR(Γ, k,Zp)of rank 2d, which contains Sk(Γ,Zp).• Scholl: de Rham cohomology is isomorphic to the crystalline
cohomology. Thus the φ operator in the crystalline theory istransported to the de Rham space.
• The action of Frobp on l-adic side (l 6= p) and φ on the p-adicside have the same characteristic poly.
Hp(T ) = T 2d + C1(p)T 2d−1 + · · · + C2d(p).
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Scholl congruences for weakly holomorphic modularforms
Mwkk (Γ, R) ⊃Mwk−ex
k (Γ, R) ⊃ Swk−exk (Γ, R) ⊃ Dk−1Mwk2−k(Γ, R)
• A modular form is weakly holo. if it is holo. on H and mero.at cusps.
• f ∈ Mwkk (Γ, R) is called weakly exact if at each cusp c of Γ
the Fourier coefficients an(f, c) of f at c is divisible by nk−1
in R for each n < 0.
• f ∈Mwk−exk (Γ, R) is a cusp form if it has vanishing constant
term at all cusps.
•Dk−1 :∑n cnq
n/µ 7→∑n n
k−1cnqn/µ maps Mwk
2−k(Γ, R) to
Swk−exk (Γ, R).
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• Using geometric interpretation of weakly holo. modular forms,Kazalicki-Scholl proved
DR(Γ, k,Zp) =Swk−exk (Γ,Zp)
Dk−1(Mwk2−k(Γ,Zp))
.
∑n anq
n/µ and∑n bnq
n/µ in Swk−exk are equal inDR(Γ, k,Zp)⇔ an ≡ bn mod p(k−1)ordpn.
• φ :∑n anq
n/µ 7→ pk−1∑n anq
np/µ.
• Since Hp(φ) = 0 on DR(Γ, k,Zp), any f =∑n anq
n/µ in
Swk−exk (Γ,Zp) satisfies the congruence
anpd(f ) + C1(p)anpd−1(f ) + · · · + C2d(p)an/pd(f )
≡ 0 mod p(k−1)ordpn for all n ≥ 1.
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• Since φ(Sk(Γ,Zp)) ⊂ pk−1DR(Γ, k,Zp), for f ∈ Sk(Γ,Zp)the above congruence holds mod p(k−1)(1+ordpn).
Ex 5. S12(SL2(Z)) is 1-dim’l spanned by ∆(z) = η(z)24 =∑n≥1 τ (n)qn. The char. poly Hp(T ) = T 2 − τ (p)T + p11.
E4(z)6/∆(z)− 1464E4(z)3 = q−1 +
∞∑n=1
anqn
= q−1 − 142236q + 51123200q2 + 39826861650q3 + · · ·lies in DR(SL2(Z), k,Z). For every prime p ≥ 11 and integersn ≥ 1, its coefficients satisfy the congruence
anp − τ (p)an + p11an/p ≡ 0 (mod p11ordpn).
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