     # ANABELIAN GEOMETRY - University of Pennsylvania III) Variants Replacing ث‡alg 1 by variants (compatible

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• LMS Symposia 2011

University of Durham

18 – 28 July 2011

ANABELIAN GEOMETRY

— A short survey —

Florian Pop, University of Pennsylvania

• LECTURE 1: ĜT 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 ( πtop1 (X)

) - Consider πtop1 : Var R → Groups (outer hom’s)... • (ρX)X gives rise to repr ıR : GalR → Aut(π

top 1 )

FACT: ıR is an isomorphism.

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

1

• •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 π top 1 ) is a functor

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

- There exists canonical exact sequence:

1→ πalg1 (X, ∗)→ πet1 (X, ∗)→ GalΛ → 1

- Represent ρX : GalΛ → Out ( πalg1 (X)

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

ıΛ : GalΛ → Aut(π alg 1 )

2

• 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: Teichmüller 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 Teichmüller modular tower.

- Note: M0,4 ∼= P1Q \ {0, 1,∞} = P01∞. M0,n ∼= (M0,4)n−3 \ {fat diagonal}

3

• FACTS (to Question/Problem 1):

- Harbater-Schneps: Let V0 := {M0,4,M0,5}. Then Aut(πalgV0 ) = ĜT is the famous

Grothendieck–Teichmüller group.

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

are all of the form VĜT := 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(π alg V ) 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

4

• Surjectivity:

• Ihara/Oda–Matsumoto Conj (I/OM). ıQ : GalQ → Aut(π

alg VarQ

) is isomorphism.

FACTS:

- 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(π alg VX ) is isomorphism.

Note: With X = P2Q, this gives (in principle) a pure topol./combin. construction of GalQ!

5

• 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 π

alg,p 1 (X)

- πV by the corresponding π`V - Aut(πalgV ) 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...

6

• 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 ΠcK 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...

7

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

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

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

- πV by the corresponding ΠcV → ΠV - Aut(πalgV ) by Aut

c(ΠV) := im ( Aut(ΠcV)→ Aut(ΠV)

) ...

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

FACT (P): ıcVX : GalΛ → Aut c(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 → Aut c(ΠUX ) and

ıUX : GalΛ → Aut(πUX ) are isomorphisms.

8

• C) The tempered ĜT and I/OM (André)

∗ Replace C by Cp and πalg1 by the the corresp. tempered (alg.) fundam. group πtemp1 hence:

- πV by the corresponding π temp V

- Aut(πalgV ) by Aut(π temp V )...

• Get the tempered variant ĜT temp

of ĜT .

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

FACTS (André):

1) ıtempΛ : GalΛ → Aut(π temp VarΛ

) is isom.

2) ıQ(GalQ) ∩ ĜT temp

= ıtempQp (GalQp).

Comments: - Actually πtemp1 (X) ⊂ π

alg 1 (X) for all X .

- There are “tempered variants” of other

aspects involving fundamental groups too.

9

• Arithmetical I/OM (question by David Burns)

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

• (Generalized) Drinfel’d upper half plane

10

• Supplement 1: On ĜT

• Exact sequence: 1→ F̂2 → π1(P01∞)→ GalQ → 1,

where P01∞ := P1Q\{0, 1,∞} tripod... and F̂2 = 〈τ0, τ1, τ∞ | τ0τ1τ∞ = 1〉̂ = F̂τ0,τ1 ...

• ĜT = {(λ, f) | λ ∈ Ẑ×, f ∈ [F̂2, F̂2], rel.I,II,III} the famous Grothendieck–Teichmüller group.

• ∃ can embedd GalQ ↪→ Aut(F̂2). - 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 ĜT .

- 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 smooth surfaces X complex curves X

Function fields:

function fields F in one 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∞ étale.

Proof: “⇒” by Belyi (nice and tricky!) “⇐” by Grothendieck (étale fundam. groups)

1

• 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, Klüners–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 → P1C 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. surface X0 of general type is defined over Q iff ∃ smooth fibration X0→→P1C\{0, 1,∞}.

2

• 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: K̂× = Homcont(ΠK ,Z`).

- And P(K,+) = K×/k× ↪→ K̂×.

Hence to do list: Given ΠcK→→ΠK ,

1) Recover K×/k× ↪→ K̂×. 2) Recover the collineations inside K×/k×.

3) Check compatibility with Galois T

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