Note on [Topics in Absolute Anabelian Geometry I, II] of Shinichi Mochizuki

Transcription

1 Note on [Topics in Absolute Anabelian Geometry I, II] of Shinichi Mochizuki by Fucheng Tan for talks at the Oxford workshop on IUT theory of Shinichi Mochizuki (Some materials in the slides will NOT be covered in the talks.) December 8th, 2015

2 Sub-p-adic fields and slim profinite groups NF=number field, MLF=mixed char. local field, FF=finite field. A sub-p-adic field is a field which is isomorphic to a subfield of a finitely generated extension field over Q p (for some prime p). For G a Hausdorff top. group and H a closed subgroup, we have the centralizer Z G (H) of H in G, the normalizer N G (H) of H in G, and the commensurator C G (H) = {g G ghg 1 H has finite index in H, ghg 1 } of H in G. We say H is normally terminal (resp. commensurably terminal) if N G (H) = H (resp. C G (H) = H). Let G be a profinite group. G is called slim if the centralizer (resp. center) of any of its open subgroup is trivial. If K is MLF then G K is slim. (For any open subgroup H G K, if g G K commutes with H then it induces trivial action on H ab. Via LCFT and varying H, we see g acts trivially on K, hence g = 1.) For F a number field and p a prime ideal in it, the slimness of the decomposition group G p together with the fact that C GF (G p ) = G p implies that G F is slim. For any sub-p-adic field, the absolute Galois group is slim. For X a hyperbolic curve over an algebraically closed field, π 1 (X ) is slim. (by, for instance, Lefschetz-Verdier) For X a hyperbolic curve over a sub-p-adic field, π 1 (X ) is slim.

3 Let G be a center-free profinite group. Get a short exact sequence 1 G Aut(G) Out(G) 1. If G is tfg (=topologically finitely generated) then the topology of G admits a basis of characteristic (i.e. invariant under Aut(G)) open subgroups, which determines a profinite topology on the groups Aut(G), Out(G). For X a hyperbolic curve over a field k, we have a morphism of extensions of profinite groups: 1 π 1 (X k ) π 1 (X ) G k 1 outer repn. 1 Inn(π 1 (X k )) Aut(π 1 (X k )) Out(π 1 (X k )) 1 For G as above and a continuous homomorphism ρ : H Out(G) of profinite groups, denote by G out H the pullback via ρ of the s.e.s. of profinite groups 1 G Aut(G) Out(G) 1. We then get an s.e.s. of profinite groups 1 G G out H H 1.

4 Set-up for absolute anabelian geometry k=perfect field, k/k solvably closed Galois, G k := Gal( k/k). Note k = k if k is FF or MLF. X /k an algebraic stack, geometrically connected, smooth, separated of finite type. Y /X a connected finite étale Galois covering which is a k-scheme s.t. Gal(Y /X ) acts freely on an open subscheme of Y. Y Y is an open immersion into a proper scheme Y /k, which is equipped with a log structure so that Y log is log-smooth over (Spec k, triv.). (Log purity theorem) A finite étale cover Z Y is tamely ramified over height one primes of Y the normalization Z of Y in Z gives a log étale cover Z log Y log. Thus, the finite étale covers of X whose pullbacks to Y are tamely ramified over height one primes of Y form a Galois category. The associated profinite group is written as π tame 1 (X, Y ) := π tame 1 (X ) (with Y X fixed). Have s.e.s. 1 π tame 1 (X k ) π tame 1 (X ) Gal(k/k) 1.

5 Σ= a set of primes. A profinite group is called pro-σ if the order of any of its finite quotient groups is a Σ-integer, i.e. a positive integer whose prime factors all lie in Σ. A profinite group is almost pro-σ if it admits an open subgroup which is pro-σ. Let π tame 1 (X k ) X be an almost pro-σ-maximal quotient. Suppose ker(π tame 1 (X k ) X ) is normal in π tame 1 (X ) so that it determines a quotient π tame 1 (X ) Π X. Assume the quotient π tame 1 (X ) Gal(Y /X ) factors thru Π X, and the kernel of the induced map X Gal(Y /X ) is pro-σ, hence can be identified with the maximal pro-σ quotient of ker(π tame 1 (X k ) Gal(Y /X )). s.e.s. 1 X Π X Gal(k/k) 1.

6 A profinite group isomorphic to X above is called a profinite group of GFG-type. Any profinite group of GFG-type is tfg. If moreover X is a hyperbolic orbicurve and Σ contains a prime invertible in k, is slim. An extension 1 Π G 1 of profinite groups which is isomorphic to the s.e.s. above is called an extension of AFG-type. For an AFG-type extension 1 Π G 1, set N := ker(g G k = Gal( k/k)). Assume is slim and the outer action of N on is trivial. Then N lifts to a closed normal subgroup N Π Π. An extension of profinite groups isomorphic to 1 Π /N Π G /N 1 is called of GSAFG-type.

7 For i = 1, 2, let 1 i Π i G i 1 be an extension of AFG-type or GSAFG-type. Let ϕ : Π 1 Π 2 be a continuous homomorphism of profinite groups. ϕ is called absolute if it has open image. ϕ is called semi-absolute if it is absolute and ϕ( 1 ) lies in 2. (There are absolute homomorphisms which are not semi-absolute.) One form of anabelian geometry is relative anabelian geometry, in which instead of starting from (various quotients of) the profinite group Π X, one starts from the profinite group Π X equipped with the natural augmentation Π X G k to the absolute Galois group of k. By contrast, absolute anabelian geometry refers to the study of properties of X as reflected solely in the profinite group Π X.

8 Some known results in anabelian geometry (Neukirch-Uchida) For two number fields F 1, F 2, we have Isom field (F 1, F 2 ) Out(G F1, G F2 ) := Isom top. gp. (G F1, G F2 )/Inn(G F2 ). (In particular, two number fields with the same absolute Galois group are isomorphic.) (Mochizuki) The version of Grothendieck Conjecture for p-adic local fields: Let k 1, k 2 be local fields over Q p. Then we have a natural bijection Isom Qp (k 1, k 2 ) Out Fil (G k1, G k2 ) where Fil on G ki is given by the higher ramification groups in the upper numbering. Jarden-Ritter: There exist nonarchimedean fields which are not isomorphic but have the same absolute Galois group.

9 Let K be a sub-p-adic field and X K, Y K two hyperbolic curves over K. In the notation above, set Σ = {p}. (Mochizuki) The étale fundamental group functor determines bijections between dominant morphisms of schemes and open outer homomorphisms between fundamental groups: Hom dom K (X K, Y K ) open, outer Hom G k (π 1 (X K ), π 1 (Y K )) open, outer HomG k (Π XK, Π YK ). (Mochizuki) The Galois Section map is injective: X K (K) Sec (G K, Π XK ) /Inn(Π XK ).

10 Hyperbolic curves Let g Z 0. A family of curves of genus g is a smooth proper geometrically connected morphism of schemes X S whose geometric fibers are curves of genus g. Let r 0 be another integer such that 2g 2 + r > 0. A family of hyperbolic curves of type (g, r) is a morphism X S which factors as X Y S with X Y an open immersion, the complement D Y of whose image is a relative divisor which is finite étale over S of relative degree r, and Y S a family of curves of genus g. X =proper smooth connected algebraic curve over Q. A Belyi map is a dominant map of Q-schemes ϕ : X P 1 which is Q unramified over the tripod P 1 \{0, 1, }. The preimage ϕ 1 (P 1 \{0, 1, }) is called an Belyi open of X. (Belyi) There exists at least one Belyi open for X. (Mochizuki) Belyi opens of X form a basis of the Zariski topology of X.

11 Hyperbolic orbicurves An orbicurve X is a smooth, geometrically connected, generically scheme-like (i.e. contains an open dense algebraic substack which is a scheme) algebraic stack of finite type of dimension 1 over a field k of char. 0. If X is a generically scheme-like algebraic stack over a field k of char. 0 that admits a finite étale cover Y with Y a hyperbolic curve over a finite extension of k, then we call X a hyperbolic orbicurve over k. (Bundgaard-Nielsen-Fox: X is a hyperbolic orbicurve if it is an orbicurve which admits a compactification X X (necessarily unique) by a proper orbicurve X over k which is scheme-like near D := X \X, and deg(ω X /k (D)) > 0.) A dominant morphism of hyperbolic orbicurves is a partial coarsification if its induced map in the associated coarse spaces is an isomorphism. A hyperbolic orbicurve is of strictly Belyi type if it is defined over an NF and is isogenous to a hyperbolic curve of genus 0. A hyperbolic orbicurve X over a field k of char. 0 is called Π-elliptically admissible for a quotient of profinite groups π 1 (X ) Π if It admits a k-core X C. C is semi-elliptic, hence admits a double-covering D δ C with D a once-punctured elliptic curve. There is a finite étale cover Y X over a fin. ext. ky /k arising from an open normal subgroup of Π so that Y C factors as Y D δ C. For every set of primes Σ, if := (π1 (X ) Π)(π 1 (X k )) is almost pro-σ then it is pro-σ. The primes dividing the degree of the covering Y CkY lie in Σ.

12 Cores of hyperbolic orbicurves, a key feature For X a normal connected algebraic stack which is generically scheme-like, denote by Loc(X ) the category with objects generically scheme-like algebraic stacks which are finite étale quotients of algebraic stacks which are finite étale over X, and with morphisms the finite étale morphisms of algebraic stacks. We say X admits a core X Z if Loc(X ) admits a terminal object Z. For k a field and X a geometrically normal, geometrically connected algebraic stack of finite type over k, we may require all the morphism appearing above be k-morphisms. Then we have the notion k-core. In the case k = C and X is a hyperbolic curve: Let X be the associated Riemann surface and X H the universal cover. Γ := Im(π top 1 (X ) Aut(H) = PSL 2 (R) 0 ). X admits a k-core iff Γ C(Γ) := C Aut(H) (Γ) has finite index. Then, there is a finite étale morphism (of stacks) X Y := H/C(Γ). This finite étale morphism gives an algebraic structure on the analytic stack Y. The resulting algebraic stack Y is a k-core of X. (Takeuchi) For the fixed type (g, r), all but finitely many hyperbolic curves over C admit k-cores. Some discussion about cores will appear in the talk of Tamás Szamuely.

13 Chains of elementary operators We shall use the following set-up frequently. Look at a slim profinite group G and a GSAFG-type extension which admits (partial) construction data with Σ. Let be a scheme-theoretic envelope. 1 Π G 1 (k, X, Σ) α : π tame 1 (X ) Π Then α determines a profinite étale covering X X and a field extension k/k.

14 An X /X -chain is a sequence of generically scheme-like stacks X = X 0 X j X n with each X j equipped with a dominant rigidifying morphism satisfying the following properties: ρ j : X X j, (a) a (uniquely determined) morphism X j Spec k j compatible with ρ j, for k j /k a finite extension over which X j is geometrically connected, (b) Each ρ j factors thru a maximal profinite étale covering X j X j so that j := ker Ä Π j := Gal( X j /X j ) G j := Gal(k j /k) ä is nontrivial slim and pro-σ, and any prime dividing the order of a finite quotient of j is invertible in k, (c) That X is a hyperbolic orbicurve over k implies that X j is a hyperbolic orbicurve over k j, (d) Each X j X j+1 is an elementary operation (compatible with ρ j, ρ j+1 ) of one of the following types:

15 (1) Type : finite étale covering ϕ : X j+1 X j. ( open immersion Π j+1 Π j.) (2) Type : finite étale quotient ϕ : X j X j+1. ( open immersion Π j Π j+1.) (3) Type (de-cuspidalization): open immersion ϕ : X j X j+1 of hyperbolic orbicurves, the complement of whose image is a single k j+1 -point of X j+1 with decomposition group in j contained in some torsion-free open normal subgroup. ( surjection Π j Π j+1.) (4) Type (de-orbification, defined only for hyperbolic orbicurves and Σ = {all primes}): a partial coarsification ϕ : X j X j+1 of hyperbolic orbicurves, which is an isomorphism over the complement of some k j+1 -point of X j+1. ( surjection Π j Π j+1.) The isomorphisms between two X /X -chains is defined in a way compatible with rigidifying morphisms and having identical types (thus any automorphism of an X /X -chain is the identity), from which we obtain a category Chain( X /X ) with objects X /X -chains and morphisms isomorphisms of such chains. A terminal morphism between two X /X -chains (X 0 X n ) (Y 0 Y m ) is a dominant k-morphism X n Y m. We then get a category Chain( X /X ) trm from these. The subcategory Chain( X /X ) iso-trm is given by terminal isomorphisms.

16 Given a X /X -chain X = X 0 X j X n one obtains a Π-chain : Π = Π 0 Π j Π n. Write Π for the inverse system of open subgroups of Π. In general we can define a Π-chain of slim profinite groups equipped with open rigidifying homomorphisms satisfying: ρ j : Π Π j (a) a (uniquely determined) surjection Π j G j (with G j G an open subgroup) compatible with ρ j and the natural composite Π Π G, (b) Each j := ker (Π j G j ) is nontrivial slim, and any prime dividing the order of a finite quotient of j is invertible in k, (c) If X is a hyperbolic orbicurve over k then each j is pro-σ. A cuspidal decomposition group in j is defined to be a commensurator in j of a nontrivial image (via ρ j ) of the inverse image in Π of the decomposition group in of a cusp of X. (d) Each Π j Π j+1 is an elementary operation (compatible with ρ j, ρ j+1 ) of one of the following types:

17 (1) Type : open immersion Π j+1 Π j. (2) Type : open immersion Π j Π j+1. (3) Type (defined only if X is a hyperbolic orbicurve): surjection ϕ : Π j Π j+1 s.t. ker(ϕ) is top. normally generated by a cuspidal decomposition group in j which is contained in some torsion-free open normal subgroup of j. (4) Type (defined only if X is a hyperbolic orbicurve and Σ = {all primes}): surjection ϕ : Π j Π j+1 whose kernel is top. normally generated by a finite closed subgroup of j. Finally, we have the analogous categories Chain( Π/Π), Chain( Π/Π) trm (with a terminal homomorphism of two Π-chains defined as an open outer homomorphism between the last groups in the chains) and Chain( Π/Π) iso-trm. Fixing certain types, we define DLoc( X /X ) = Chain( X /X ) trm {, }, DLoc( Π/Π) = Chain( Π/Π) trm {, }, EtLoc( X /X ) = Chain( X /X ) Iso trm {, }, EtLoc( Π/Π) = Chain( Π/Π) iso trm {, }.

18 Decomposition groups of hyperbolic orbicurves Recall [Absolute anabelian cuspidalizations of proper hyperbolic curves, Lemma 2.1] and [A combinatorial version of the Grothendieck conjecture, 1.2]. Given: Σ, a nonempty set of primes numbers., a pro-σ group of GFG-type admitting base-prime (i.e. every prime dividing the order of a finite quotient group of is invertible in k) partial construction data (k, X, Σ) with X a hyperbolic orbicurve and k = k. x A x B, two closed points or cusps of X, with decomposition groups D A and D B.

19 Have D A and D B are pro-cyclic groups with D A D B = {1}. If x A is a closed point with D A nontrivial, then D A is a finite and normally terminal subgroup of. If x A is a cusp, then D A is a torsion-free, commensurably terminal and infinite subgroup of. The nontrivial decomposition groups of closed points of X are characterized as the maximal finite nontrivial closed subgroups of. X is a hyperbolic curve iff is torsion-free. Suppose the quotient ψ A : A by the closed normal subgroup top. generated by D A in is nontrivial and slim. In the case that x A is a closed point, assume further Σ = {all primes}; in the case that x A is a cusp, assume D A is contained in a torsion-free open normal subgroup of. Then Then A is a profinite group of GFG-type admitting base-prime partial construction data (k, X A, Σ) with X A a hyperbolic orbicurve equipped with a dominant k-morphism ϕ A : X X A, which is uniquely determined (up to a unique isomorphism) by the property of inducing ψ A (up to composition with inner automorphisms). If xa is a closed point then ϕ A is a partial coarsification which is an isomorphism either over X A or over X A \{point determined by x A }. If x A is a cusp then ϕ A is an open immersion with Im(ϕ A ) = X A \{point determined by x A }. D B {1} ψ A (D B ) {1}.

20 Group-theoretic characterization of cuspidal decomposition groups of hyperbolic orbicurves Suppose further that there is a prime l Σ such that the cyclotomic character χ cyc G : G Z l has open image. For M a fin. dim. Q l -v.s. carrying a continuous G-action, look at any filtration M n M 0 = M of Q l [G]-modules with M j /M j+1 either having finite G-action or having no nontrivial subquotients, and define the quasi-trivial rank of M to be τ(m) = dim M j /M j+1. Write for χ : G Z l a character d χ (M) = τ(m(χ 1 )) τ(hom Ql (M, Q l )). Results from [A combinatorial version of the Grothendieck conjecture] (summarized in [TAAG-I, 4.5]): The hyperbolic orbicurve X is non-proper iff every torsion-free pro-σ open subgroup of is free pro-σ. Suppose X is non-proper. Let H be a torsion-free pro-σ characteristic open subgroup with H (l) the maximal pro-l quotient. Then the decomposition groups of cusps H (l) are the maximal closed subgroups I H (l) isomorphic to Z l s.t. for every characteristic open subgroup J H (l), Ä d χ cyc (Jab Q G l ) + 1 = [IJ : J] d χ cyc (IJ) ab ä Q G l + 1. For X, H as above, the following correspondence is functorial in H hence gives a characterization of decomposition groups of cusps in Π. Cusps of the covering of X k corresponding to H Conj. classes of maximal closed subgroups I H (l) as above

21 Grothendieck conjecture-type hypotheses Let i = 1, 2. Look at a slim profinite group G i and a GSAFG-type extension 1 i Π i G i 1 which admits partial construction data (k i, X i, Σ i ), with a scheme-theoretic envelope. α i : π tame 1 (X i ) Π i Let D be a set of tripes of isomorphism classes of algebric stacks, isomorphism classes of fields, and non-empty set of some primes. Assume ([X i ], [k i ], Σ i ) D, so do those triples ([X j ], [k j ], Σ) arising from all the X i /X i -chains. (A D with this last property is called chain-full.) Assume k 1 k2, which induces (via α i ) an outer isomorphism G 1 G2. We say rel-isom-gc for D holds if for any such pair above, we have the following bijection induced by α i : Isom k1,k 2 (X 1, X 2 ) Isom out G 1,G 2 (Π 1, Π 2 ). Similarly, we say rel-hom-gc for D holds if Hom dom k 1,k 2 (X 1, X 2 ) Hom out-open G 1,G 2 (Π 1, Π 2 ). [The local pro-p anabelian geometry of curves]: Both rel-isom-gc and rel-hom-gc hold for hyperbolic orbicurves X i over sub-p-adic fields k i with Σ i {p}.

22 Semi-absoluteness of chains of elementary operations Assume rel-isom-gc for D holds. Assume (1) if one of the X i is hyperbolic orbicurve then both are, (2) if one of the X i is a non-proper hyperbolic orbicurve then l Σ 1 Σ 2 s.t. χ cyc G i : G i Z l has open image. An isomorphism of profinite groups ϕ : Π 1 Π2, which induces isomorphisms 1 2 and G 1 G2, (i) Natural (functorial w.r.t. type-chains) equivalences: Chain( X i /X i ) Chain(Π i ), Chain( X i /X i ) iso-trm Chain(Πi ) iso-trm, EtLoc( X i /X i ) EtLoc(Π i ). (ii) Natural (functorial w.r.t. type-chains and ϕ) equivalences: Chain(Π 1 ) Chain(Π 2 ), Chain((Π 1 )) iso-trm Chain(Π2 ) iso-trm, EtLoc((Π 1 )) EtLoc(Π 2 ). (iii) Assume further that rel-hom-gc for D holds, and assume X i s are hyperbolic orbicurves. Then (functorially w.r.t. type-chains) Chain( X i /X i ) trm Chain(Πi ) trm, DLoc( X i /X i ) DLoc(Π i ). (iv) Keep the assumptions in (iii). Then (functorially w.r.t. type-chains and ϕ) Chain(Π 1 ) trm Chain(Π2 ) trm, DLoc(Π 1 ) DLoc(Π 2 ).

23 Characterizations of some profinite groups Let 1 Π G 1 be an extension of profinite groups of AFG-type or of GSAFG-type with (partial) construction data (k, X, Σ) s.t. k is MLF or NF. (Recall any profinite group of GFG-type is tfg.) In the case that the base field k is NF, is characterized as the maximal closed normal subgroup of Π which is tfg. (G F has the property that any of its tfg closed normal subgroup is trivial.) Let k be an MLF over Q p and (assume for simplicity) X a hyperbolic orbicurve over k. Then is characterized as the intersection of all open subgroups H Π s.t. dim Qp H 1 (H, Q p ) dim Qp H 1 (H, Q p ) = [Π : H][k : Q p ] for p a prime different from p.

24 Tempered chains of elementary operations, end of [TAAG-I] Let i = 1, 2. Let k i be an MLF of residue char. p i, and X i a hyperbolic orbicurve over k i. Look at an extension of top. groups 1 i Π i G i 1 which is isomorphic to the natural extension with a scheme-theoretic envelope 1 π tp 1 ((X i) ki ) π tp 1 (X i) G ki 1 α i : π tp 1 (X i) Π i. Suppose we have an isomorphism of top. groups ϕ : Π 1 Π2. Then (i) Natural (functorial w.r.t. type-chains) equivalences: Chain( X i /X i ) Chain(Π i ), Chain( X i /X i ) iso-trm Chain(Πi ) iso-trm, Chain( X i /X i ) trm Chain(Πi ) trm, EtLoc( X i /X i ) EtLoc(Π i ), DLoc( X i /X i ) DLoc(Π i ). (ii) The two residue characteristics p 1 = p 2. The isomorphism ϕ : Π 1 Π 1 induces isomorphisms 1 2 and G 1 G2. Have natural (functorial w.r.t. type-chains and ϕ) equivalences: Chain(Π 1 ) Chain(Π 2 ), Chain(Π 1 ) iso-trm Chain(Π2 ) iso-trm, Chain(Π 1 ) trm Chain(Π2 ) trm, DLoc(Π 1 ) DLoc(Π 2 ), EtLoc(Π 1 ) EtLoc(Π 2 ).

25 Elliptic and Belyi cuspidalizations; the goal of [TAAG-II] If k is MLF or NF, then the extension of profinite groups we use below can be replaced by the single profinite group Π X.

26 Scheme-theoretic Belyi cuspidalizations Let k be a field of char. 0 and k /k a finite Galois extension. Assume Gal(k/k) is slim. Let P be a tripod over k, Y an arbitrary hyperbolic curve over k, U Y a nonempty open subscheme. Suppose U and Y are defined over a number field so that, by [Noncritical Belyi maps, 2004] there is some nonempty open subscheme W U, and a finite étale Belyi map β : W P. WMA the cusps of W are defined over k. Now we can apply the construction of chains of elementary operations to 1 = π 1 (P k ) Π = π 1 (P) G = Gal(k/k ) 1 to get a chain of type,,, : P W W n W n 1 W 1 = U U m U m 1 U 1 = Y. Keep the notation above. Let X be a hyperbolic orbicurve of strictly Belyi type over k. Suppose we are given finite étale coverings Y X, Y Q and an open immersion Q P. (Thus Q is a hyperbolic curve of genus 0 over k.) For simplicity, assume Y X is Galois, U Y descends to an open subscheme U X X, and the cusps of Q are defined over k.

27 In a word, we have the following (non-commutative!) diagram: W fét Belyi U fét Y Q P X fét We then get a chain of type,,,,,,,,, : X Y Q Q l Q 1 = P W W n W 1 = U U m U 1 = Y X.

28 Profinite Belyi cuspidalization Recall that as in [TAAG-I] D denote a set of tripes of isomorphism classes of algebraic stacks, isomorphism classes of fields, and non-empty set of some primes (with the property that if ([X ], [k], Σ) D, so do those triples ([X j ], [k j ], Σ) arising from the X /X -chains.) Look at a slim profinite group G and a GSAFG-type extension 1 Π G 1 which admits partial construction data (k, X, Σ) consisting of a filed k of char. 0, a hyperbolic orbicurve of strictly Belyi type X, and Σ = {all primes}. The scheme-theoretic envelope α : π 1 (X ) Π gives rise to the profinite étale covering X X and the field extension k/k. Assume that (1) α is an isomorphism. (2) There is a prime l Σ such that the cyclotomic character χ cyc G : G Z l has open image. (3) The rel-isom-gc for D holds. Recall we have constructed in [TAAG-I] group-theoretically from the data above the categories Chain(Π), Chain(Π) iso-trm, EtLoc(Π).

29 Given a nonempty open subscheme U X X defined over a number field, we can construct the natural surjection (i.e. the cuspidalization of Π) Π UX = π 1 (U X ) π 1 (X ) α Π as follows: (a) For some open normal subgroup Π Y Π corresponding to a finite cover Y X of hyperbolic orbicurves, there is some (not necessarily unique) Π-chain, which admits an entirely group-theoretic description, of type,,,,,,,,, which admits a terminal isomorphism to the trivial Π-chain, s.t. the surjection Π U := Π Y Π Π UX Π Y can be recovered from the second portion of the Π-chain consisting of s. (b) One may recover the surjection Π UX Π from Π U Π Y by forming out w.r.t. the unique lifting (rel. to Π U Π Y ) to a group of outer automorphisms of Π U of the outer action of the finite group Π/Π Y on Π Y : 1 Π U Π U out Π/Π Y Π/Π Y 1 = 1 Π U Aut(Π U ) Out(Π U ) 1 (c) The decomposition groups of the closed points in X \U X are obtained as the images of the cuspidal decomposition groups of Π UX (recall [TAAG-I, 4.5]) under Π UX Π.

30 The group-theoretic construction above has the following immediate Grothendieck Conjecture-style consequence. Given two sets of data 1 i Π i G i 1 with partial construction (k i, X i, Σ i ) D (together with the assumptions) as above, we look at an isomorphism of profinite groups ϕ : Π 1 Π2 with the property We have ϕ( 1 ) = 2. For any nonempty open subscheme U X1 X 1 defined over an NF, there is a nonempty open subscheme U X2 X 2 defined over an NF, and an isomorphism of profinite groups ϕ U : Π UX1 ΠUX2 which is compatible with ϕ, via the natural surjections Π UXi Π i. The isomorphism ϕ is unique up to compositions with an inner automorphism coming from an element in ker(π UXi Π i ).

31 Scheme-theoretic elliptic cuspidalizations Let k be a field of char. 0 and k /k a finite Galois extension. Assume Gal(k/k) is slim. N Z 0 ; D = E\{o E } a once-punctured elliptic curve over k s.t. the points E[N] is defined over k ; D C a semi-elliptic k -core. Then we have U := E\E[N] fét cover [N] D Now we can apply the construction of chains of elementary operations to 1 = π 1 (D k ) Π = π 1 (D) G = Gal(k/k ) 1 to get a chain of type,,, : D D U U n=n 2 1 U n 1 U 1 = D. Suppose we have an elliptically admissible hyperbolic orbicurve X /k and fét covering V X, V D with V hyperbolic curve over k. Suppose (for simplicity) V X is Galois and preserves the open subscheme U V := V D U V (it thus descends to an open subscheme U X X ).

32 Consider the following diagram U fét cover D V fét cover X V fét cover [N] D fét quotient fét cover X We get a chain of type,,,,,,, : X V D U U n U n 1 U 1 = D V X.

33 Pro-Σ elliptic cuspidalizations For an elliptically admissible hyperbolic orbicurve (hence of strictly Belyi type if it is defined over a number field) X, we have the finer pro-σ cuspidalization, not just profinite cuspidalization. Recall that D denote a set of tripes of isomorphism classes of algebraic stacks, isomorphism classes of fields, and non-empty set of some primes (with the property that if ([X ], [k], Σ) D, so do those triples ([X j ], [k j ], Σ) arising from the X /X -chains.) Look at a slim profinite group G and a GSAFG-type extension 1 Π G 1 which admits partial construction data (k, X, Σ) consisting of a filed k of char. 0, a Π-elliptically admissible hyperbolic orbicurve X. The scheme-theoretic envelope α : π 1 (X ) Π gives rise to the profinite étale covering X X and the field extension k/k. Assume that (1) There is a prime l Σ such that the cyclotomic character χ cyc G : G Z l has open image. (2) The rel-isom-gc for D holds. Recall we have constructed in [AAG, I] group-theoretically from the data above the categories Chain(Π), Chain(Π) iso-trm, EtLoc(Π).

34 (i) Let G G an open normal subgroup corresponding to a finite extension k /k. Write Π = Π G G ; C a k -core of X k. Then the fét cover X k C determines a chain in Chain(Π ) which is characterized (up to isomorphisms in Chain(Π )) as the unique chain of length 1 in Chain(Π ) with type s.t. the resulting object of EtLoc(Π ) forms a terminal object of EtLoc(Π ). (ii) The open subgroups Π D Π C arising from fét covers D C that exhibit C as semi-elliptic is characterized as the open subgroups J Π C of index 2 such that J C is torsion-free. (iii) Let N be a product of primes in Σ, and let U, V X, V D, U V, U X be as in the scheme-theoretic setting, with open normal subgroup Π V Π X corresponding to V X and V D arising from an open immersion Π V Π D. We then have the extensions of GSAFG-type 1 U Π U G 1, 1 UV Π UV G 1, 1 UX Π UX G 1. For any G G sufficiently small (rel. to N), we can construct the natural surjection (i.e. the cuspidalization of Π) Π UX Π as follows:

35 (a) There is some (not necessarily unique) Π-chain, which admits an entirely group-theoretic description, of type,,,,,,, admitting a terminal isomorphism to the trivial Π-chain, whose final three groups consist of Π D Π V Π, s.t. the surjection Π U Π D can be recovered from the portion of the Π-chain consisting of s, and the surjection Π UV Π V can be recovered by furthering forming fiber product with Π V Π D. (b) One may recover the surjection Π UX Π from Π UV Π V by forming out w.r.t. the unique lifting (rel. to Π UV Π V ) to a group of outer automorphisms of Π UV of the outer action of the finite group Π/Π V on Π V. (c) The decomposition groups of the closed points in X \U X are obtained as the images of the cuspidal decomposition groups of Π UX (recall [TAAG-I, 4.5]) under Π UX Π.

36 The group-theoretic construction above has the following immediate Grothendieck Conjecture-style consequence. Given two sets of data 1 i Π i G i 1 with partial construction (k i, X i, Σ i ) D (together with the assumptions) as above, we look at an isomorphism of profinite groups ϕ : Π 1 Π2 with the property We have ϕ( 1 ) = 2. For any nonempty open subscheme U X1 X 1 defined over an NF, there is a nonempty open subscheme U X2 X 2 defined over an NF, and an isomorphism of profinite groups ϕ U : Π UX1 ΠUX2 which is compatible with ϕ, via the natural surjections Π UXi Π i. The isomorphism ϕ is unique up to compositions with an inner automorphism coming from an element in ker(π UXi Π i ).

37 Point-theoreticity implies geometricity, the main result in [TAAG-II] The preservation of decomposition groups of closed points is an input for the following result ( point-theoreticity implies geometricity ) [TAAG-II, 2.9]: For i = 1, 2, let k i be an MLF with residue char. p i. Σ i a set of primes of cardinality 2, containing p i. X i a hyperbolic curve over k i with X i the proper smooth geometrically connected curve over k i determined by the function field of X i. Ξ i X i (k i ) a Galois-dense subset. i.e. for any finite extension k /k, Ξ X i (k i ) is dense in X i (k i ). Xi the maximal pro-σ i quotient of π 1 ((X i ) ki ), and π 1 (X i ) Π Xi the quotient by its kernel. Let ϕ : Π X1 ΠX2 be an isomorphism of profinite groups s.t. a closed subgroup of Π X1 is a decomposition group of a point in Ξ 1 iff it corresponds (via ϕ) to a decomposition group of a point in Ξ 2. Then ϕ is geometric, i.e. comes from a unique isomorphism of schemes X 1 X2. The Section Conjecture over MLFs implies the absolute isomorphism version of the Grothendieck Conjecture over MLFs. We explain the strategy of the proof below:

38 Geometric uniformly toral neighborhoods of a local Galois group Recall the version of Grothendieck Conjecture for p-adic local fields: Let k 1, k 2 be MLFs. Then we have a natural bijection Isom Qp (k 1, k 2 ) Isom outer Fil (G k1, G k2 ) where Fil on G ki is given by the higher ramification groups in the upper numbering. Look at an open subgroup H G, with corresponding MLF k H. Identify via LCFT O k H Tor(H) := Im(Ik H H ab ). The p-adic logarithm O k H k H induces an isomorphism of groups λ H : Tor(H) Q p kh. A uniformly toral neighborhood of G k is a collection of subgroups N H Tor(H) Q p with {H G k } a collection of open subgroups which form a basis of topology of G k, such that there is a, b Z 0 (uniform for H) so that p a O kh λ H (N H ) p b O kh.

39 Criteria for geometricity of an isomorphisms of local Galois groups An open homomorphism ϕ : G k1 G k2 is called uniformly toral if G k1 admits a UTN which image under ϕ is a UTN of G k2. An open homomorphism ϕ : G k1 G k2 is called of Hodge-Tate type if, via ϕ, k 2 k 1 as topological G k1 -modules. An open homomorphism ϕ : G k1 G k2 is called geometric if it arises from a field isomorphism k 2 k1 which maps k 2 into k 1. An open homomorphism ϕ : G k1 G k2 is called RF-preserving if it is compatible with ramification filtrations on both sides. The four conditions are equivalent if ϕ is an isomorphism. Explanation of the equivalence: For each open subgroup H G ki picking an appropriate higher ramification group (related to the ramification index of k H ) and multiplying it by a suitable p-power one gets a UTN of G ki. Then ϕ is uniformly toral if it is assumed to be RF-preserving. Assume that ϕ is uniformly toral. Set N i to be the subgroup of I i = lim H Tor(H) Q p generated by the N H for G ki. Then the topology on I i given by p c N i (c Z 0 ) coincides with the p-adic topology on k i. Thus Î i k i, which implies that ϕ is of HT-type. One can then show that if ϕ is of HT-type then it is geometric, by studying Hodge-Tate characters.

40 Graph-theoreticity for hyperbolic curves Let i = 1, 2 and 0 i Π i G i 0 an extension of AFG-type or of GSAFG-type, with partial construction data (k i, X i, Σ i ). Assume k i is MLF with residue characteristic p i, X i is a hyperbolic curve with stable reduction X i over O ki, and Σ i contains a prime different from p i. Suppose we have an isomorphism of profinite groups ϕ : Π 1 Π2. Then ϕ induces an isomorphism of the two extensions. Now we only need to show that ϕ G : G 1 G2 is uniformly toral! (Recall the relative anabelilan conjecture.) Σ 1 = Σ 2. ( Σ i is the minimal set Σ of primes s.t. i is almost pro-σ.) The existence of a prime different from p in Σ allows us to use the stable reduction criterion. WMA X i admits a log stable model X i over Spec O log k i.

41 Let Γ i denote the dual semi-graph with compact structure of the geometric special fiber of X i, i.e. the dual graph together with additional open edges associated to the cusps. Then ϕ induces an isomorphism of semi-graphs ϕ Γ : Γ 1 Γ2 which is functorial in ϕ. This tells us that ϕ preserves cuspidal decomposition groups. WMA X i are proper. WMA the dual graphs of the special fibers X i are loop-ample and not k i -smooth. (A connected graph is loop-ample if it remains connected after removal of an edge.)

42 Prime-power cyclic coverings and log-modifications Let X be a stable curve over O k with k MLF. Denote its generic fiber as X. Assume X is loop-ample. Then a stable curve Y over O k with generic fiber Y and a cyclic finite étale cover Y X of positive p-power degree, s.t. at least one of the following two conditions is satisfied: (a) (Loopification) Y X is loopifying (i.e. the loop-rank of Y is strictly bigger than that of X ) and wildly ramified at some stable irreducible component C Y which is potentially base-semi-stable: there exists a log-modification W log V log of Y log X log s.t. C is the image of a stable irreducible component of W which is base-semi-stable, i.e. mapping finitely to a stable irreducible component of V. Y X is wildly ramified at C Y iff Gal(Y /X ) stabilizes and induces the identity on C iff Gal(Y /X ) stabilizes (the conjugacy class of) and induces the identity (outer automorphism) on C. (b) (Component crushing) some stable irreducible component C Y which is not potentially base-semi-stable. Group-theoretic characterization: Let l p be a prime and Y log := ker(π 1 (Y log ) G k log) (l) = ker(π 1 (Y ) G k ) (l). Then (b) is equivalent to the condition that the image of the decomposition group C Y log in X log is trivial. we may choose Y i, C i for X i (assuming Y i is split by replacing k i by a finite extension) as above and assume that they satisfy (a) (resp. (b)) simultaneously and that C 1 is compatible with C 2 via ϕ.

43 Uniformly toral neighborhood via cyclic coverings By the previous slide, we have the following commutative diagram: W log V log split base-field-isomorphic log modification Y log X log log modification (A base-field-isomorphic log modification is a log étale proper birational morphism of log schemes. It is split if Gal(k/k) acts trivially on the dual graph of the special fiber of the domain of the log modification.) s.t. In case (a): the unique irreducible component C W W mapping finitely to C Y maps finitely to V. In case (b): the unique irreducible component C W maps to a closed point of Some notation: U [1] V := {x X log (M X /O X )gp x is of rank 1}. U [1] W (O k) y x U [1] V (O k) so that y C W. U y := W (W (C W U [1] W )), U x =... J Y the Jacobian of Y and J Y the unique semi-abelian scheme over S = Spec O k extending J Y. Have J X, J X. ι Y : Y J Y the morphism determined by y. In case (a): J := (1 σ)(j Y ) the image abelian scheme (σ a generator of Gal(Y /X )), and J S the unique semi-abelian scheme extending J. Have the dominant morphism induced by (1 σ): κ : J Y J. In case (b): have κ : J Y J := J X induced by Y X.

44 Suppose further X is not smooth over k. (i) s.e.s. of formal schemes over S 0 J J Ĵ 0 (completion at 0) with J G m. Reason: In case (a), the torsion portion of J is of rank loop-rk(y) loop-rk(x ). In case (b), it is of rank loop-rk(x ). (ii) ι Y extends to ι Y : U y J Y, and β k : Y Y g Y copies of ι Y J Y κ J extends to β : U y U y J. (iii) For any finite extension k /k, define Å ã I k := Im U y (O k ) β J (O k ). Then M Z >0 (independent of k ) s.t. M I k J (O k ). The subgroup of J (O k ) it determines is denoted by Îk. Then N k = Image of (Îk J (O k )) in J (O k ) Q p form a uniformly toral neighborhood of G k as k varies.

45 Constructing uniformly toral neighborhoods To finish the proof of point-theoreticity implies geometricity, we just need to construct uniformly toral neighborhoods for G i which are compatible via ϕ. Look at the decomposition group (l) ( C i (= π 1 (C i U [1] ) Y i ) k i ) in (l) Y i. It is slim hence the action of G i on it factors thru G ki. We get an extension of prifinite groups 0 (l) C i We have similarly another extension of G log ki them: Π (l) C i G ki 0. by (l) Y i, and a morphism between 1 (l) C i Π (l) C i Gki G ki log G ki log 1 1 (l) Y i id Π (l) Y i G log ki 1 A point in Y i (k i ) is uniquely determined by the conjugacy class of its decomposition group in Π Yi ; By our assumption, ϕ determines a bijection between Y 1 (k 1, Ξ 1 ) and Y 2 (k 2, Ξ 2 ), the subsets of Y i (k i ) lying over Ξ i. Pick y i Y i (k i, Ξ i ) compatible ( with C i (i.e. compatible with ϕ), which is a group-theoretic condition: Π Yi Π (l) ) Y i (D yi [Π Yi ]) determines a subgroup of Π (l) ( C i Gki G log ki which contains ker Π (l) C i Gki G log ki Π (l) C i ).

46 Set T Y i = ( (p) Y i ) ab, T X i = ( (p) X i ) ab. In case (a): Choose generator σ Gal(Y i /X i ) (compatible with ϕ). Have In case (b): Have T i := Ä (1 σ i )(T Y i Q p ) ä T Y i, κ Ti : T Y i T i. κ Ti : T Y i T i := T X i. A result of Tate: G i -submodule T i T i isomorphic to Z p (1). For another pair of points y i Y i (k i ) (compatible with ϕ), consider the difference of the two Galois sections given by y i, y i in H 1 (G i, T Y i ). Denote by δ yi,y i its image under The image of κ Ti : H 1 (G i, T Y i ) H 1 (G i, T i )( J i (k i ) Z p ). Ä ä closure of gyi elements of the form M i δ yi,y i H 1 (G i, T i )( O k i Z p ) (intersection taken inside H 1 (G i, T i )) in H 1 (G i, T i Q p ) O k i Q p, for varying k i, form UTN of G i, compatible with ϕ!

47 Absolute Grothendieck conjecture for hyperbolic curves of Belyi type Want to show: Let X (resp. Y ) be a hyperbolic curve over MLF k X (resp. k Y ) with étale fundamental group Π X (resp. Π Y ). If either (hence both) of them is of (strictly) Belyi type, then any isomorphism of profinite groups α : Π X ΠY arises from a uniquely determined commutative diagram of schemes X Y Π X X Y Π Y Reduce to showing that α is point-theoretic. That is, it preserves decomposition groups of closed points. The point-theoreticity is the main result of [The absolute anabelian geometry of hyperbolic curves].

48 Preservation of decomposition groups for hyperbolic curves of strictly Belyi type Let K, L be two MLF and X K, Y L two hyperbolic curves. Have natural s.e.s. 0 X Π X := π 1 (X K ) G K 0. Let X K X K be the canonical compactification. For a closed point x X K we have decomposition group D x and the inertia I x := D x X, which is Ẑ(1) if x is a cusp (and it trivial if x is not a cusp). A closed subgroup of Π X which is isomorphic to some I x with x a cusp is called a cuspidal geometric decomposition group.

49 (i) ([The local pro-p anabelian geometry of curves, Theorem A]) The étale fundamental group functor determines a bijection between the set of dominant morphisms of schemes and the set of open outer homomorphisms ϕ : Π XK Π YL that fit into the diagram can. Π XK G K ϕ open imm. arising from embeddings of fields L K can. Π YL G L (ii) ([The local pro-p anabelian geometry of curves, Theorem C]) A closed point x is completely determined by the conjugacy classes of D x Π XK. If x is a cusp, then it is completely determined by the conjugacy classes of I x Π XK. (iii) ([The absolute anabelian geometry of hyperbolic curves, Lemma 1.3.9]) Any isomorphism Π XK ΠYL sends inertia groups of cusps to inertia groups of cusps. (This uses the fact that I x X and I x (l) terminal.) (iv) (i,ii,iii) (l) X are commensurably Every isomorphism of profinite groups ΠXK ΠYL preserves cuspidal decomposition groups and cuspidal geometric decomposition groups. No noncuspidal decomposition group of Π XK is contained in a cuspidal decomposition group of Π XK.

50 Since X is tfg, sequence of characteristic open subgroups (j Z 1 ) X [j + 1] X [j] X such that X [j] = {1}. Given any section σ : G K Π XK, have open subgroups Π XK [j,σ] := Im(σ) X [j] Π XK. (v) (Criterion for Galois sections associated to rational points) Suppose X K is defined over a number field F K, and that Im(σ) is not contained in any cuspidal decomposition group of Π XK. Then the following conditions are equivalent: (a) σ comes from a point x X K (K) (i.e. Im(σ) = D x ). (b) For every j 1, Π XK [j,σ] contains a decomposition group (rel. to Π XK ) of an (closed) NF-point of X K surjecting onto G K. (vi) If X K, Y L are of strictly Belyi type, then any isomorphism of profinite groups Π XK ΠYL preserves the decomposition groups of (closed) NF-points. (iv, v,vi) If X K, Y L are of strictly Belyi type (with K, L MLF), then any isomorphism of profinite groups Π XK ΠYL preserves the decomposition groups of the closed points.

51 Proof of (vi): First reduce to the case that X, Y are of genus 0. Then take a closed NF-point x X. We have the following commutative diagram: V U = X \{x} X β fét β fét X P = P 1 \{0, 1, } X = P 1 Look at the following two categories, which are equivalent by (i): β dominant DLoc(X ): the objects are partial compactifications (lying in the canonical compactifications) V Z with Z a hyperbolic curve over K and V a hyperbolic curve over some field that arises as a finite étale cover V X. The morphisms are the dominant maps on the partial compactifications Z. DLoc(Π X ): the objects are the quotients H J with H an open subgroup of Π X whose kernel is the closed subgroup topologically generated by a collection of some geometric cuspidal decomposition groups of H. The morphisms are open outer homomorphisms between the H s, compatible with the various natural surjections to the Galois group G K. We apply the equivalence above to the inclusion V X obtained via the Belyi map β. Then under the equivalence DLoc(X ) DLoc(Π X ) we get the quotient H V J X whose kernel is topologically generated by the I w with w X \V. Moreover, it again follows directly from (i) that an isomorphism α : Π XK ΠYL induces an equivalence DLoc(Π XK ) DLoc(Π YL ). It follows from the definitions that such an equivalence sends the decomposition groups of the above type to the decomposition groups of the same type.

52 Preservation of decomposition groups of torsion closed points for once-punctured elliptic curves Again let K, L be two MLF, and X K, Y L two once-punctured elliptic curves defined over them, respectively. Let Σ K (resp. Σ L ) be a set of primes containing the residue characteristic of K (resp. of L) and Π XK (resp. Π YL ) the maximal pro-σ K (resp. pro-σ L ) quotient of π 1 (X K ) (resp. of π 1 (Y L )). Then any isomorphism of profinite groups α : Π XK ΠYL preserves the decomposition groups of the closed points arising from the torsion points of the underlying elliptic curves. Moreover, the bijection between torsion closed points is compatible with the induced map on Tate modules: ab X ab Y. The proof is similar to that in the general Belyi case: We have the following object in DLoc(X K ): U open immersion X K finite étale cover Then the proof goes as before. X K