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:

    - 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(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

  • 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

    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