LMS Symposia 2011

University of Durham

18 – 28 July 2011

ANABELIAN GEOMETRY

— A short survey —

Florian Pop, University of Pennsylvania

LECTURE 1: GT versus I/OM

I) Motivation/Introduction

Non-tautological description of

GalQ = Aut(Q)

The toy example: GalR via πtop1

- Var R category of R-varieties X.

- GalR ∼= {±1} acts on Xan := X(C) and πtop1 (Xan)

• Get a representation:

ρX : GalR → Out(πtop

1 (X))

- Consider πtop1 : Var R → Groups (outer hom’s)...

• (ρX)X gives rise to repr ıR : GalR → Aut(πtop1 )

FACT: ıR is an isomorphism.

Comment: Got non-tautological description of Gal R...

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•What about other subfields of C ?

- Λ ⊆ C base field, e.g. Q, Q(x1, . . . , xn), R, Qp

- VarΛ geometrically integral Λ-varieties X.

Comment/Note: In general, Gal doesn’t act on πtop1

∗ How to do it: Replace πtop1 by

• πalg1 := (profinite compl. of πtop

1 ) is a functor

πalg1 : VarΛ → prof.Groups (outer hom’s)

- There exists canonical exact sequence:

1→ πalg1 (X, ∗)→ πet

1 (X, ∗)→ GalΛ → 1

- Represent ρX : GalΛ → Out(πalg

1 (X))

• Finally (ρX)X gives rise to a morphism:

ıΛ : GalΛ → Aut(πalg1 )

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II) Studying GalQ via ıQ

Question/Problem:

1) Find categories V for which Aut(πalgV ) has

“nice” topological/combinatorial description.

2) Find such categories V for which

ıV : GalK → Aut(πalgV ) is an isomorphism.

• This would give a new description of GalΛ!!!

Example: Teichmuller modular tower

- Mg,n moduli space of curves

(genus g, with n marked pts.)

- “Connecting” morphisms

(“boundary” embeddings, gluings, etc.)

- T = {Mg,n | g, n} category of varieties

over Λ = Q, the Teichmuller modular tower.

- Note: M0,4∼= P1

Q \ {0, 1,∞} = P01∞.

M0,n∼= (M0,4)n−3 \ {fat diagonal}

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FACTS (to Question/Problem 1):

- Harbater-Schneps: Let V0 := {M0,4,M0,5}.Then Aut(πalg

V0) = GT is the famous

Grothendieck–Teichmuller group.

- [MANY]: The variants IGT , IIGT , IVGT , newGT

are all of the form VGT := Aut(πalgV )...

• Conclusion: Question/Problem 1 has quite

satisfactory answer(s) for V ⊆ T .

FACTS (to Question/Problem 2):

Injectivity: Λ number field. Then

ıV : GalK → Aut(πalgV ) injective, provided:

- Drinfel’d (using Belyi Thm): P01∞ ∈ V

- Voevodsky: C ∈ V, C ⊂ E hyperbolic curve...

- Matsumoto: C ∈ V, C affine hyperbolic curve

- Hoshi-Mochizuki: C ∈ V, C hyperbolic curve

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Surjectivity:

• Ihara/Oda–Matsumoto Conj (I/OM).

ıQ : GalQ → Aut(πalgVarQ

) is isomorphism.

FACTS:

- I/OM has positive answer.

- Actually much stronger assertions hold, e.g...

• Given Λ ⊂ C and X ∈ VarΛ, dim(X) > 1, set:

- VX = {Ui ⊂ X}i ∪ {P01∞}, with morphisms:

inclusions Uj ↪→ Ui, projections Ui → P01∞.

- (P): ıVX: GalQ → Aut(πalg

VX) is isomorphism.

Note: With X = P2Q, this gives (in principle)

a pure topol./combin. construction of GalQ!

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III) Variants

Replacing πalg1 by variants

(compatible with the Galois action of GalΛ)

A) The pro-` variant

∗ Replace πalg1 by its pro-` quotient, hence:

- πalg1 (X) by its pro-` quotient πalg,p

1 (X)

- πV by the corresponding π`V- Aut(πalg

V ) by Aut(π`V)...

Questions:

1) Describe the image ı`V(GalΛ)...

2) Consider the pro-` I/OM...

- Lot of intensive research here concerning 1),

e.g., by Matsumoto, Hain–M, Nakamura, etc.

Comment: This generalizes Serre’s question/result about/on...

- The pro-` I/OM is true in stronger form...

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B) The pro-` abelian-by-central I/OM

• Bogomolov’s Program

Comment:

- The “Yoga” of Grothendieck’s anabelian geometry...

Bogomolov (1990). Consider:

- K|k function field, td.deg(K|k) > 1, k = k.

- ΠcK → ΠK abelian-by-central/abelian

pro-` quotients of GalK , ` 6= char.

Conjecture (Bogomolov’s Program, 1990):K|k can be recovered from Πc

K functorially.

Comments:- tr.deg(K|k) > 1 is necessary, because...

- This goes far beyond Grothendieck’s anabelian idea...

FACT (State of the Art):Bogomolov’s Progr OK for tr.deg > dim(k) + 1.

Comments:- dim(k) = 0: tr.deg > dim(k) + 1 is equiv to...

- The case tr.deg > dim(k) + 1 follows by...

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Back to: Pro-` abelian-by-central I/OM:

∗ Replace the profinite groups by the theirpro-` abelian-by-central quotient, hence:

- πalg1 (X) by its quotient Πc(X)→ Π(X)

- πV by the corresponding ΠcV → ΠV

- Aut(πalgV ) by Autc(ΠV) := im

(Aut(Πc

V)→ Aut(ΠV))...

• Get the pro-` abelian-by-central I/OM...

FACT (P): ıcVX: GalΛ → Autc(ΠVX

) is isom.

Comments:- Actually one proves the “birational variant” which is stronger.

- This is a special case of Bogomolov’s Program.

- Bogomolov’s Program would imply stronger assertions, like:

∗ Let dim(X) > dim(Λ) + 1 be geom rigid...

- Note: If Λ ⊂ Q, then dim(X) > 2 is enough.

- Consider UX = {Ui}i subcategory of VX .

FACTS (P): ıcUX: GalΛ → Autc(ΠUX

) and

ıUX: GalΛ → Aut(πUX

) are isomorphisms.

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C) The tempered GT and I/OM (Andre)

∗ Replace C by Cp and πalg1 by the the corresp.

tempered (alg.) fundam. group πtemp1 hence:

- πV by the corresponding πtempV

- Aut(πalgV ) by Aut(πtemp

V )...

• Get the tempered variant GTtemp

of GT .

• Get the tempered I/OM for Λ|Qp finite.

FACTS (Andre):

1) ıtempΛ : GalΛ → Aut(πtemp

VarΛ) is isom.

2) ıQ(GalQ) ∩ GTtemp

= ıtempQp

(GalQp).

Comments:- Actually πtemp

1 (X) ⊂ πalg1 (X) for all X .

- There are “tempered variants” of other

aspects involving fundamental groups too.

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IV) Final Question/Comments

• Arithmetical I/OM (question by David Burns)

• Pro-linear/pro-unipotent I/OM (question by Minhyong Kim)

• (Generalized) Drinfel’d upper half plane

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Supplement 1: On GT

• Exact sequence:

1→ F2 → π1(P01∞)→ GalQ → 1,

where P01∞ := P1Q\{0, 1,∞} tripod...

and F2 = 〈τ0, τ1, τ∞ | τ0τ1τ∞ = 1〉 = Fτ0,τ1 ...

• GT = {(λ, f) | λ ∈ Z×, f ∈ [F2, F2], rel.I,II,III}the famous Grothendieck–Teichmuller group.

• ∃ can embedd GalQ ↪→ Aut(F2).

- Intensively studied by Drinfel’d, Ihara, Deligne,

Schneps, Sch.–Lochak, Sch.–Nakamura,

Sch.–Harbater, I–Matsumoto, Furusho, etc...

- Several variants IGT , IIGT , IVGT , etc. of GT .

- Actually: rel. I, II, III, are not independent

(Schneps, Sch.–Lochak; Furusho: III suffices)

- Boggi–Lochak (to be thoroughly checked):

Some variant newGT equals Aut(πalgT ).

Supplement 2): Belyi’s Theorem

- Recall RET: There exists equiv of categories

Topology&Geometry: Compl. algebraic curves:

compact Riemann projective smoothsurfaces X complex curves X

Function fields:

function fields F inone variable over C

X ←→ M(C) = F = C(X) ←→ X

• Basic Question (Grothendieck):

Which X , hence which X, hence which F ,

are defined over Q ⊂ C, hence number fields?

Theorem (Grothendieck/Belyi).

X is defined over Q ⇔ ∃ X → P01∞ etale.

Proof: “⇒” by Belyi (nice and tricky!)

“⇐” by Grothendieck (etale fundam. groups)

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Comments:- This is the origin of Grothendieck’s “Designs d’enfants”.

- A cover X → P1C as in Theorem is a Belyi map.

- Study the action of GalQ on the space of “Designs”

(many many people: Malle, Kluners–M., Schneps,

Lochak–Sch., Zapponi, math-physicists, etc. etc. etc...)

Interesting open Question/Problem:

Higher dim extensions of Theorem above.

Two possible ways:a) Describe all X → P1

C with at most n branch points.

b) Replace curves by higher dim varieties, e.g., surfaces (?!?).

- Several partial results to b), but...

Theorem (Ronkine 2004; unpublished).

The birat. class of a complex proj. surfaceX0 of general type is defined over Q iff

∃ smooth fibration X0→→P1C\{0, 1,∞}.

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Supplement 3) Bogomolov’s Program

Given: ΠcK → ΠK . Reconstruct K|k functorially.

Strategy of proof (P):

Main Idea: Consider P(K,+) := K×/k×

the “projectivization” of the k-v.s. (K,+).

• (K,+, ·) can be recovered from

P(K,+) endowed with its collineations,

via Artin’s Fundam. Thm. Proj. Geometries

NOW:

- Kummer Theory: K× = Homcont(ΠK ,Z`).

- And P(K,+) = K×/k× ↪→ K×.

Hence to do list: Given ΠcK→→ΠK ,

1) Recover K×/k× ↪→ K×.

2) Recover the collineations inside K×/k×.

3) Check compatibility with Galois Theory.

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PLAN:

• Local Theory, i.e., recover:

- primes of K|k; divisorial sets DX of primes.

• Global Theory, i.e., recover:

- Div(X), then K×/k×, then collineations;

and finally check Galois compatibility.

THE GENERAL “NONSENSE”

Case k = Fp:- Pic0(X) is torsion group

- Valuations of k are trivial

Case tr.deg(K|k) > dim k:

- Specialization (Deuring, Roquette, Mumford)

- 1-motives techniques

- Reduce to the case k = Fp- Recover the “nature” of k

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Local Theory (few words):

- primes of K|k: DVR Rv with k ⊂ Rv ⊂ Ksuch that tr.deg(Kv |k) = tr.deg(K|k)− 1.

- D = {vi}i geometric, if ∃ normal model X → k

such thatD = DX := {v |Weil prime div. of X}.

- Quasi prime divisors

• Recovering the (quasi) primes:

- 1st Method: Use B.-Tsch. “commuting pairs”...

- 2nd Method: Use techniques developed byWare, Koenigsmann, Mıac et al, Topaz...

Comment: This is very very technical stuff...

• Recovering (quasi) decomposition graphs

• Recovering rational quotients

• · · ·

LECTURE 2: Section conjectures

I) Motivation/Introduction

Effective version of the Mordell Cojecture...

(Grothendieck: Letter to Faltings, June 1983)

• Evidence: Minhyong Kim’s work...

Setting:

- k arbitrary base field; ki, ksep ⊂ k- X0 geom. integral k-variety, d = dim(X0)

- X ⊂ X0 open k-subvariety

• Canonical exact sequence:

1→ πalg1 (X, ∗)→ πet

1 (X, ∗)→ Galk → 1

• For x ∈ X0 regular ∃ vx on k(X) with:

vx(K×) = Zd and κ(vx) = κ(x).

- vx|vx prolong to k(X), get split exact seq

1→ Tvx → Zvx → Galκ(x) → 1.

Comments (about the splitting; tangential...)

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Conclude: κ(x) ⊂ kins ⇒ Galκ(x) = Galk andprk(X) : Galk(X) → Galk has sections svx .

- X → X pro-etale universal cover

- X0 → X0 normaliz of X0 in k(X) ↪→ k(X)

- Galk(X) → πet1 (X) can projection

- x 7→ x centers of vx on X0 → X0

- Functoriality: Zvx → Zx and Tvx → Tx.

Conclude: κ(x) ⊂ kins ⇒ Galκ(x) = Galk andprX : π1(X)→ Galk has sections sx.

Cases:

a) x ∈ X: Then Tx = {1} and Zx = Galκ(x)

• x ∈ X with κ(x) ⊂ kins defines conj class of sx.

b) x ∈ X0\X: Then in general one has

Tx 6= {1} and Zx 6= Galκ(x).

• x ∈ X0\X with κ(x) ⊂ kins defines a

“bouquet” of sections ≈ H1cont(Galk, Tx).

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Special case: X ⊆ X0 are smooth curves. Then:

- x ∈ X0\X iff Tx 6= 1.

- char(k) = 0 ⇒ Tx ∼= Z(1) as Gk-modules.Get H1

cont(Galk, Tx) ∼= k×...

Curve SC (Section Conjecture / Grothendieck)

k fin gen infinite, X → k hyperbolic non-isotrivial.Then all sections of prX are of the form sx.

Birational SC: k, X as above.

Then sections of prk(X) are of the form svx .

• Variants...

- p-adic Curve/Birat SC: Replace k fromCurve/Birat SC by a finite extension k|Qp.

- Geom pro-C Curve/Birat SC: Replace πalg1

from Curve/Birat SC by its pro-C completion...

- Etc. e.g., [ p-adic] (pro-C) Curve/Birat SC

•Note: ...Curve SC’s⇒ ...Birat SC’s [Comments]

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II) Evidence/Facts SC:

A) “No sections” results...

- Stix, using the following local result:

• k|Qp finite, X hyperbolic. Then existenceof sections of prX ⇒ I(X) is a p-power.

- Harari–Szamuely(using the Manin-Brauer obstruction)

- Hain: k = Q(Mg), Xg gen curve of genus g.Curve SC true for Xg for g ≥ 5...

• Geom pro-p Curve SC:

- Hoshi: ...does not hold over k = Q[ζp]for Xp

0 +Xp1 +Xp

2 ⊂ P2 and p regular prime.

•Finally: Curve/Birational SC are widely open.

• Conditional result on Birational SC: k # field.

- Esnault–Wittenberg: X proj hyperbolic, with

qqqq(Jac(X)

)finite. If pr : Galgeom.ab

k(X) → Galkhas sections, then X has index 1.

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B) Unconditional results:

• Koenigsmann: The p-adic Birat SC holds.

Comment: Actually a much stronger fact holds:

• k|Qp finite, and K|k regular extension. Then sections of

GalK → Galk originate from k-rational places of K .

- In particular, if K = k(X0) with X0 proj smooth curve,

then the sections originate from X0(k)...

Refinement: Minimalistic p-adic Birational SC

Context: k|Qp finite, µp ⊂ k, K = k(X0).

- k′′|k′ ↪→ K ′′|K ′ max Z/p metabelian ext’s.

- pr : Gal(K ′|K)→ Gal(k′|k) can projection.

- Liftable section s of pr is one coming from a

section of pr : Gal(K ′′|K)→ Gal(k′′|k)

• (P): “Bouquets” of liftable sections ↔ X0(k).

Comments: The less Galois theory, the better...

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C) Partial results:

Tamagawa:

- k finite field, X ⊆ X0 hyperbolic.

- X → X univ pro-etale cover, K = k(X).

- Given s section of prX , consider all

Y → X finite sub-cover of X → X

such that im(s) ⊂ πet1 (Y ).

- |Y0(k)| = ∑2i=0

(−1)iTr(ϕk)∣∣Hi(Y 0,Z`(1)

)• A section s comes from a point x ∈ X0(k) iff

Y0(k) non-empty for all Y → X as above.

Nakamura:

- k number field, X ⊂ X0 affine hyperbolic.

• The points x ∈ X0\X are in bijection with

conjugacy classes of max subgroups ∆ ∼= Zof πalg

1 (X) of pure weight −2.

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For the next three results:

- k|Qp fin, X ⊂ X0 hyperbolic, X → X univ cover

- prX : πet1 (X)→ Galk can projection

Mochizuki: Suppose X0 defined over Q.

• Then X0\X 3 x↔ maximal ∆ ∼= Z ⊂ πalg1 (X)

on which Galk acts via the cycl character.

Comment: ...the “absolute form” of the anab conj for curves.

Saidi: Defines “good sections” of prX ...

• The good sections ↔ X0(k).

Comment: Proof relies on:

- Mochizuki’s p-adic cuspidalization methods...

- Pop’s methods developed for the “minimalistic” result...

P–Stix:

• For every section s of prX ∃ valuation

w on k(X) such that im(s) ⊂ Zw.

Comments: ...relation to Mochizuki’s “combinatorial SC

- Equivalently: Every section comes from Berkovich points.

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D) Section conjectures over R

Here: k = R, X ⊆ X0 hyperbolic, etc...

Wojtkowiak, Mochizuki, Wickelgren, etc...

πet1 (X)→ GalR has sections iff X0(R) 6= ∅

• One cannot expect a Curve SC over R...

Wickelgren:

- The geometrically pro-2 curve version holds.

- The pro-2 birational SC holds.

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III) Final Comments

- Initial motivation of Grothendieck......Mordell Conjecture, now Faltings’ Theorem...)

- No relation between Curve SC and(an effective) Mordell Conjecture yet...

Minhyong Kim:

Using pro-unipotent completions of πalg1 designs

an algorithm (of p -adic nature) which —under

the conjectural properties of his “Selmer vari-

eties”— produces the rational points of the curve

in discussion; and more impressively, the effec-

tiveness of the algorithm is guaranteed by the va-

lidity of the Curve SC. This — I would claim —

sheds the right light on the relation and the role

of the Curve SC to an effective Mordell.

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IV) Short list of open Problems:

1) Prove/disprove:

ıV : GalQ → Aut(πV) is not onto for V ⊆ T

2) Clarify/prove the pro-unipotent I/OM

3) Clarify the relation between the global and

the p-adic Curve/Birational SC

4) Prove/disprove the global/p-adic (birational)

section Conjecture.

5) Relation between the representations ıV and

linear represent of GalQ, respectively GalQp

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