LMS Symposia University of Durham July 2011 ANABELIAN GEOMETRY. A short survey. Florian Pop, University of Pennsylvania

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1 LMS Symposia 2011 University of Durham July 2011 ANABELIAN GEOMETRY A short survey Florian Pop, University of Pennsylvania

2 LECTURE 1: ĜT versus I/OM I) Motivation/Introduction Non-tautological description of Gal Q = Aut ( Q ) The toy example: Gal R via π top 1 - Var R category of R-varieties X. - Gal R = {±1} acts on X an := X(C) and π top 1 (X an ) Get a representation: ρ X : Gal R Out ( π top 1 (X) ) - Consider π top 1 : Var R Groups (outer hom s)... (ρ X ) X gives rise to repr ı R : Gal R Aut(π top 1 ) FACT: ı R is an isomorphism. Comment: Got non-tautological description of Gal R... 1

3 What about other subfields of C? - Λ C base field, e.g. Q, Q(x 1,..., x n ), R, Q p - Var Λ geometrically integral Λ-varieties X. Comment/Note: In general, Gal doesn t act on π top 1 How to do it: Replace π top 1 by π alg 1 := (profinite compl. of π top 1 ) is a functor π alg 1 : Var Λ prof.groups (outer hom s) - There exists canonical exact sequence: 1 π alg 1 (X, ) πet 1 (X, ) Gal Λ 1 - Represent ρ X : Gal Λ Out ( π alg 1 (X)) Finally (ρ X ) X gives rise to a morphism: ı Λ : Gal Λ Aut(π alg 1 ) 2

4 II) Studying Gal Q via ı Q Question/Problem: 1) Find categories V for which Aut(π alg V ) has nice topological/combinatorial description. 2) Find such categories V for which ı V : Gal K Aut(π alg V ) is an isomorphism. This would give a new description of Gal Λ!!! Example: Teichmüller modular tower - M g,n moduli space of curves (genus g, with n marked pts.) - Connecting morphisms ( boundary embeddings, gluings, etc.) - T = {M g,n g, n} category of varieties over Λ = Q, the Teichmüller modular tower. - Note: M 0,4 = P 1 Q \ {0, 1, } = P 01. M 0,n = (M 0,4 ) n 3 \ {fat diagonal} 3

5 FACTS (to Question/Problem 1): - Harbater-Schneps: Let V 0 := {M 0,4, M 0,5 }. Then Aut(π alg V 0 ) = ĜT is the famous Grothendieck Teichmüller group. - [MANY]: The variants I GT, II GT, IV GT, new GT are all of the form V ĜT := Aut(π alg V )... Conclusion: Question/Problem 1 has quite satisfactory answer(s) for V T. FACTS (to Question/Problem 2): Injectivity: Λ number field. Then ı V : Gal K Aut(π alg V ) injective, provided: - Drinfel d (using Belyi Thm): P 01 V - Voevodsky: C V, C E hyperbolic curve... - Matsumoto: C V, C affine hyperbolic curve - Hoshi-Mochizuki: C V, C hyperbolic curve 4

6 Surjectivity: Ihara/Oda Matsumoto Conj (I/OM). ı Q : Gal Q Aut(π alg Var Q ) is isomorphism. FACTS: - I/OM has positive answer. - Actually much stronger assertions hold, e.g... Given Λ C and X Var Λ, dim(x) > 1, set: - V X = {U i X} i {P 01 }, with morphisms: inclusions U j U i, projections U i P (P): ı VX : Gal Q Aut(π alg V X ) is isomorphism. Note: With X = P 2 Q, this gives (in principle) a pure topol./combin. construction of Gal Q! 5

7 III) Variants Replacing π alg 1 by variants (compatible with the Galois action of Gal Λ ) A) The pro-l variant Replace π alg 1 by its pro-l quotient, hence: - π alg 1 (X) by its pro-l quotient πalg,p 1 (X) - π V by the corresponding π l V - Aut(π alg V ) by Aut(πl V )... Questions: 1) Describe the image ı l V (Gal Λ)... 2) Consider the pro-l 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-l I/OM is true in stronger form... 6

8 B) The pro-l 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. - Π c K Π K abelian-by-central/abelian pro-l quotients of Gal K, l char. Conjecture (Bogomolov s Program, 1990): K k can be recovered from Π c 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... 7

9 Back to: Pro-l abelian-by-central I/OM: Replace the profinite groups by the their pro-l abelian-by-central quotient, hence: - π alg 1 (X) by its quotient Πc (X) Π(X) - π V by the corresponding Π c V Π V - Aut(π alg V ) by Autc (Π V ) := im ( Aut(Π c V ) Aut(Π V) )... Get the pro-l abelian-by-central I/OM... FACT (P): ı c V X : Gal Λ Aut c (Π VX ) 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 U X = {U i } i subcategory of V X. FACTS (P): ı c U X : Gal Λ Aut c (Π UX ) and : Gal Λ Aut(π UX ) are isomorphisms. ı UX 8

10 C) The tempered ĜT and I/OM (André) Replace C by C p and π alg 1 by the the corresp. tempered (alg.) fundam. group π temp 1 hence: - π V by the corresponding π temp V - Aut(π alg V ) by Aut(πtemp V )... Get the tempered variant ĜT temp of ĜT. Get the tempered I/OM for Λ Q p finite. FACTS (André): 1) ı temp Λ : Gal Λ Aut(π temp Var Λ ) is isom. 2) ı Q (Gal Q ) ĜT temp = ı temp Q p (Gal Qp ). Comments: - Actually π temp 1 (X) π alg 1 (X) for all X. - There are tempered variants of other aspects involving fundamental groups too. 9

11 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

12 Supplement 1: On ĜT Exact sequence: 1 F 2 π 1 (P 01 ) Gal Q 1, where P 01 := P 1 Q \{0, 1, } tripod... and F 2 = τ 0, τ 1, τ τ 0 τ 1 τ = 1 = F τ0,τ 1... ĜT = {(λ, f) λ Ẑ, f [ F 2, F 2 ], rel.i,ii,iii} the famous Grothendieck Teichmüller group. can embedd Gal Q Aut( F 2 ). - Intensively studied by Drinfel d, Ihara, Deligne, Schneps, Sch. Lochak, Sch. Nakamura, Sch. Harbater, I Matsumoto, Furusho, etc... - Several variants I GT, II GT, IV GT, 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 new GT equals Aut(π alg T ).

13 Supplement 2): Belyi s Theorem - Recall RET: There exists equiv of categories Topology&Geometry: compact Riemann surfaces X Compl. algebraic curves: projective smooth 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 P 01 étale. Proof: by Belyi (nice and tricky!) by Grothendieck (étale fundam. groups) 1

14 Comments: - This is the origin of Grothendieck s Designs d enfants. - A cover X P 1 C as in Theorem is a Belyi map. - Study the action of Gal Q 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 P 1 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. surface X 0 of general type is defined over Q iff smooth fibration X 0 P 1 C \{0, 1, }. 2

15 Supplement 3) Bogomolov s Program Given: Π c K Π K. 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 = Hom cont (Π K, Z l ). - And P(K, +) = K /k K. Hence to do list: Given Π c K Π K, 1) Recover K /k K. 2) Recover the collineations inside K /k. 3) Check compatibility with Galois Theory. 1

16 PLAN: Local Theory, i.e., recover: - primes of K k; divisorial sets D X of primes. Global Theory, i.e., recover: - Div(X), then K /k, then collineations; and finally check Galois compatibility. THE GENERAL NONSENSE Case k = F p : - Pic 0 (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 = F p - Recover the nature of k 2

17 Local Theory (few words): - primes of K k: DVR R v with k R v K such that tr.deg(kv k) = tr.deg(k k) 1. - D = {v i } i geometric, if normal model X k such that D = D X := {v Weil prime div. of X}. - Quasi prime divisors Recovering the (quasi) primes: - 1 st Method: Use B.-Tsch. commuting pairs nd Method: Use techniques developed by Ware, Koenigsmann, Míac et al, Topaz... Comment: This is very very technical stuff... Recovering (quasi) decomposition graphs Recovering rational quotients

18 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; k i, k sep k - X 0 geom. integral k-variety, d = dim(x 0 ) - X X 0 open k-subvariety Canonical exact sequence: 1 π alg 1 (X, ) πet 1 (X, ) Gal k 1 For x X 0 regular v x on k(x) with: v x (K ) = Z d and κ(v x ) = κ(x). - v x v x prolong to k(x), get split exact seq 1 T vx Z vx Gal κ(x) 1. Comments (about the splitting; tangential...) 1

19 Conclude: κ(x) k ins Gal κ(x) = Gal k and pr k(x) : Gal k(x) Gal k has sections s vx. - X X pro-étale universal cover - X 0 X 0 normaliz of X 0 in k(x) k( X) - Gal k(x) π et 1 (X) can projection - x x centers of v x on X 0 X 0 - Functoriality: Z vx Z x and T vx T x. Conclude: κ(x) k ins Gal κ(x) = Gal k and pr X : π 1 (X) Gal k has sections s x. Cases: a) x X: Then T x = {1} and Z x = Gal κ(x) x X with κ(x) k ins defines conj class of s x. b) x X 0 \X: Then in general one has T x {1} and Z x Gal κ(x). x X 0 \X with κ(x) k ins defines a bouquet of sections H 1 cont(gal k, T x ). 2

20 Special case: X X 0 are smooth curves. Then: - x X 0 \X iff T x 1. - char(k) = 0 T x = Ẑ(1) as G k -modules. Get H 1 cont(gal k, T x ) = k... Curve SC (Section Conjecture / Grothendieck) k fin gen infinite, X k hyperbolic non-isotrivial. Then all sections of pr X are of the form s x. Birational SC: k, X as above. Then sections of pr k(x) are of the form s vx. Variants... - p-adic Curve/Birat SC: Replace k from Curve/Birat SC by a finite extension k Q p. - Geom pro-c Curve/Birat SC: Replace π alg 1 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] 3

21 II) Evidence/Facts SC: A) No sections results... - Stix, using the following local result: k Q p finite, X hyperbolic. Then existence of sections of pr X I(X) is a p-power. - Harari Szamuely (using the Manin-Brauer obstruction) - Hain: k = Q(M g ), X g gen curve of genus g. Curve SC true for X g for g 5... Geom pro-p Curve SC: - Hoshi:...does not hold over k = Q[ζ p ] for X p 0 + Xp 1 + 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 ( Jac(X) ) finite. If pr : Gal geom.ab k(x) has sections, then X has index 1. Gal k 4

22 B) Unconditional results: Koenigsmann: The p-adic Birat SC holds. Comment: Actually a much stronger fact holds: k Q p finite, and K k regular extension. Then sections of Gal K Gal k originate from k-rational places of K. - In particular, if K = k(x 0 ) with X 0 proj smooth curve, then the sections originate from X 0 (k)... Refinement: Minimalistic p-adic Birational SC Context: k Q p finite, µ p k, K = k(x 0 ). - 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 X 0 (k). Comments: The less Galois theory, the better... 5

23 C) Partial results: Tamagawa: - k finite field, X X 0 hyperbolic. - X X univ pro-étale cover, K = k( X). - Given s section of pr X, consider all Y X finite sub-cover of X X such that im(s) π et 1 (Y ). - Y 0 (k) = 2 i=0 ( 1)i Tr(ϕ k ) H i ( Y 0, Z l (1) ) A section s comes from a point x X 0 (k) iff Y 0 (k) non-empty for all Y X as above. Nakamura: - k number field, X X 0 affine hyperbolic. The points x X 0 \X are in bijection with conjugacy classes of max subgroups = Ẑ of π alg 1 (X) of pure weight 2. 6

24 For the next three results: - k Q p fin, X X 0 hyperbolic, X X univ cover - pr X : π et 1 (X) Gal k can projection Mochizuki: Suppose X 0 defined over Q. Then X 0 \X x maximal = Ẑ πalg 1 (X) on which Gal k acts via the cycl character. Comment:...the absolute form of the anab conj for curves. Saidi: Defines good sections of pr X... The good sections X 0 (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 pr X valuation w on k( X) such that im(s) Z w. Comments:...relation to Mochizuki s combinatorial SC - Equivalently: Every section comes from Berkovich points. 7

25 D) Section conjectures over R Here: k = R, X X 0 hyperbolic, etc... Wojtkowiak, Mochizuki, Wickelgren, etc... π et 1 (X) Gal R has sections iff X 0 (R) One cannot expect a Curve SC over R... Wickelgren: - The geometrically pro-2 curve version holds. - The pro-2 birational SC holds. 8

26 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 π alg 1 designs an algorithm (of p -adic nature) which under the conjectural properties of his Selmer varieties produces the rational points of the curve in discussion; and more impressively, the effectiveness of the algorithm is guaranteed by the validity of the Curve SC. This I would claim sheds the right light on the relation and the rôle of the Curve SC to an effective Mordell. 9

27 IV) Short list of open Problems: 1) Prove/disprove: ı V : Gal Q Aut(π V ) 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 ı V and linear represent of Gal Q, respectively Gal Qp 10