Introduction to Combinatorial Anabelian Geometry III

Transcription

1 Introduction to Combinatorial Anabelian Geometry III Introduction Semi-graphs of Anabelioids 3 Application of CombGC 4 Proof of CombGC

2 K: annf or anmlf K: an alg closure of K G K = Gal(K/K) (g, r): a pair of nonnegative integers s.t. g +r>0 C: a hyperbolic curve of type (g, r) /K (g: thegenus, r: the number of cusps) π (( )): the étale fundamental gp of ( ) Recall: The homotopy ext seq π (C K K) π (C) G K induces an outer representation ρ : G K Out(π (C K K)) Belyĭ, Voevodskiĭ, Matsumoto If r>0, then ρ is injective. Today, Thm (Hoshi-Mochizuki) For any (g, r), ρ is injective. Method: Combinatorial Anabelian Geometry

3 Notations: k: an alg closed field of char 0 X: a hyperbolic curve of type (g, r) /k X n = {(x,,x n ) Π n = π (X n ) In particular, the projections n {}}{ X k k X x i x j if i j } X n X n X X (x,,x n ) (x,,x n ) (x,x ) x induce a standard sequence of [outer] surjections Π n Π n Π Π Write K m =Ker(Π n Π m ), Π 0 = {}. {} = K n K n K K 0 = Π n α Aut(Π n )isf-admissible For any fiber subgroup J Π n, it holds that α(j) =J. Recall: J Π n is a fiber subgroup J =Ker(Π n Π n ) whereπ n Π n is induced by a proj X n X n.

4 α Aut(Π n )isc-admissible (i) α(k m )=K m (0 m n) (ii) α : K m /K m+ Km /K m+ induces a bijection between the set of cusp l inertia subgps K m /K m+ α Aut(Π n )isfc-admissible α is F-admissible and C-admissible Aut FC (Π n ) = { FC-admissible automorphisms of Π n } Out FC (Π n ) =Aut FC (Π n )/Inn(Π n ) Observe: X n+ X n forgetting the last factor induces φ n :Out FC (Π n+ ) Out FC (Π n ) Thm φ n is injective for n. Remark: (i) φ n is bijective for n 4. (ii) Various X n+ X n forgetting a factor induce the same Out FC (Π n+ ) Out FC (Π n ). 3

5 Thm Thm Let X = C K K, k = K ( π (C K K) = π (X) = Π ) Note: Theouterrep n ρ : G K Out(Π ) factors as G K Out FC (Π ) Out(Π ). Thus, to show that ρ is injective, it suffices to show that G K Out FC (Π ) is injective. This follows from the commutativity of the diagram G K Out FC (Π ) φ G K Out FC (Π ) φ G K Out FC (Π 3 ) Out(π (P K \{0,, })) injective by Belyĭ 4

6 Today, for simplicity, we consider the proof of the injectivity of φ :Out FC (Π ) Out FC (Π ). It suffices to verify: Prop 3 Let Aut IFC (Π ) = α Aut FC (Π ) α Π p p Π Π α =id α =id, Ξ = Ker(p ) Ker(p ) [ Π ]. Then the injection Ξ conj. Aut IFC (Π ) [cf. the slimness of Π ] is bijective. Indeed, let σ Aut FC (Π ) s.t. p Π σ = Inn(g ) σ Π p Π σ 5

7 Observe: Let D Π be a decomp. gp assoc. to the diagonal X k X. Then it holds that σ(d) = π D π (π Π ). Thus, since D Π (p,p ) {(a, a)} Π Π we conclude that σ = Inn(g ) (g Π ). Let g Π s.t. p (g) =g and p (g) =g. Inn(g) σ Aut IFC (Π ) Ξ σ Inn(Π ) In the following, we consider the proof of Prop 3. 6

8 e e A pointed stable curve / k v v e A semi-graph of anabelioids of PSC-type B(π (v )) B(π (v )) B(π (e )) B(π (e)) B(π (e )) where irreducible component vertex node closed edge cusp open edge B(G) : the connected anabelioid assoc. to G 7

9 G: a semi-graph of anabelioids of PSC-type A natural notion of finite étale coverings of G One can ine the [profinite] fundamental gp Π G Example: (A finite étale covering of degree 4) 8

10 Notation: G: a semi-graph of anabelioids of PSC-type G: the underlying semi-graph of G V(G): the set of vertices of G N (G): the set of nodes of G C(G): the set of cusps of G E(G) = N (G) C(G) In particular, z V(G) (resp. N (G); C(G); E(G)) determines a(n) verticial (resp. nodal; cuspidal; edge-like) subgroup Π z Π G [up to Π G -conjugacy]. G: a profinite group H: a closed subgroup Z G (H) = {g G g h = h g for h H } N G (H) = {g G g H g = H } g H g C G (H) = g G g H g H open open H 9

11 Prop 4 (i) If z V(G) (resp. E(G)), then Π z is slim (resp. = Ẑ). (ii) Let {z,z } be a subset of either V(G) ore(g). open If Π z Π z Π z,thenz = z. (iii) If z V(G) E(G), then C ΠG (Π z )=Π z. (iv) Let z E(G). Then Π z Π G is cuspidal (resp. nodal) Π z is contained in precisely one (resp. two) verticial subgroup(s). Definition G, H: semi-graphs of anabelioids of PSC-type α :Π G ΠH : an isomorphism of profinite groups (i) α is graphic α arises from an isom G H (ii) α is group-theoretically verticial (resp. nodal; cuspidal; edge-like) α maps each verticial (resp. nodal; cusp l; edge-like) subgp Π G onto a verticial (resp. nodal; cusp l; edge-like) subgp Π H, and, every verticial (resp. nodal; cusp l; edge-like) subgp Π H arises in this fashion. 0

12 Prop 5 α :Π G ΠH : an isomorphism of profinite groups Then α is graphic α is group-theoretically verticial and group-theoretically edge-like. Moreover, in this case, α arises a unique isom G H. (Proof) Prop 5 follows from Prop 4, (ii), (iii). Observe: Π G is topologically finitely generated A profinite topology on Out(Π G ) Since the natural hom. Aut(G) Out(Π G )isan injection with closed image [cf. Prop 5], A profinite topology on Aut(G) Definition I: a profinite group We refer to a continuous hom. I Aut(G) [ Out(Π G )] as an outer representation of PSC-type.

13 Thm 6 (A comb. ver. of the Grothendieck Conj.) G, H: semi-graphs of anabelioids of PSC-type ρ I : I Aut(G), ρ J : J Aut(H): outer representations of PSC-type α :Π G ΠH : an isomorphism of profinite groups which fits into a commutative diagram ρ I I Aut(G) Out(Π G ) Out(α) ρ J J Aut(H) Out(Π H ) wherei J is an isomorphism. Suppose that (i) ρ I, ρ J are of NN-type [cf. 4]. (ii) C(G) and α is group-theoretically cusp l. Then α is graphic [cf. Prop 5].

14 3 (Proof of Prop 3) Remark: At last year s seminar, we have already discussed the affine case of Prop 3! For simplicity, let X = To verify Prop 3, we may replace X Spec(k) bya stable log curve v v Z log = e over a log point S log. [cf. the ormation theory of stable log curves; the specialization isomorphism Π n = π (X n ) Ker(π log (Zlog n ) π log (Slog ))] 3

15 Z log a semi-graph of anabelioids of PSC-type G Observe: Thefiber (Z log ) e of pr : Z log Z log at e is v v0 v e 0 e e a semi-graph of anabelioids of PSC-type G /e In particular, we have Π / Π Π p Π G/e Π G whereπ / =Ker(Π Π ); the horizontal seq. is exact. an outer rep n ρ :Π Out(Π / ) 4

16 Y Z: the irreducible component corr. to v Y log :thesmooth log curve over S log determined by the hyperbolic curve Y \{e} Π sub n Π sub / = Ker(π (Y log n ) π (S log )) = Ker(Π sub Π sub ) Note: The natural closed imm. Y Z induces a commutative diagram Y Z pr pr Y Z This diagram induces a commutative diagram Π sub / Π sub Π sub p Π / Π Π where the horizontal seq. is exact; the vertical arrows are injective. an outer rep n ρ sub :Π sub Out(Π sub / ) 5

17 Π sub / Π / v v0 v e 0 e e (Y log ) e (Z log ) e Π sub Π v v e Y log Z log 6

18 Let α Aut IFC (Π ). Π e Π : an edge-like subgp assoc. to e We have a comm. diag. Π e Π Out(Π / ) ρ Out(α Π/ ) Π e Π Out(Π / ) ρ Note: Since the composite Π e Π Out(Π / ) factors as the composite of Π e Aut(G /e ) with Aut(G /e ) Out(Π / ) where the resulting outer rep n is of NN-type it follows from Thm 6 that α Π/ is graphic. Moreover, since α p Π Π α =id α Π/ induces identity automorphism on G /e [cf. Prop 4, (i), (ii)]. 7

19 Fix an edge-like subgroup Π e Π / assoc. to e. Claim: γ Ξ s.t. α(π e ) = γ Π e γ. (Proof of Claim) By the above Note, there exists γ Π / s.t. α(π e ) = γ Π e γ. Thus, we have p (Π e ) = p (γ ) p (Π e ) p (γ ). By Prop4, (iii), we conclude that p (γ ) p (Π e ). In particular, by multiplying γ by a suitable Π e, we obtain an element γ Ξ, as desired. In light of Claim, to verify that α Ξ, we may assume that () α(π e ) = Π e. 8

20 Π v,π v 0 Π / :theunique verticial subgps assoc. to v, v0 that contain Π e [cf. Prop 4, (iv)] Π sub / Π /: theunique Π / -conj. of the image of Π sub / Π / that contains and is topologically generated by Π v,π v 0 By () and the graphicity of α Π/, we conclude: () α(π v )=(Π v ); (3) α(π v 0 )=(Π v 0 ); (4) α(π sub / )=(Πsub / ). Observe: Since C Π/ (Π sub / )=Πsub / the diagram [cf. Prop 4, (iii)], Π sub Out(Π sub Π sub ρ sub ρ sub / ) Out(Π sub Out(α Π sub ) / / ) [cf. (4)] commutes. Thus, α Π sub arises from an / α sub Aut(Π sub ) [cf. the slimness of Π sub / ]. 9

21 Moreover, it follows from the construction that α sub Aut IFC (Π sub ). On the other hand, by the affine case of Prop 3 [cf. Remark at the beginning of the proof], we have Aut IFC (Π sub ) Ξ sub = Ξ Π sub. α Π sub / is a Ξ-inner automorphism! Thus,toverifythatα Ξ, we may assume that (5) α Π sub / = id. (6) α Πv 0 = id [cf. (3)] Π e Π / : an edge-like subgp assoc. to e which is contained in Π v 0 Π v Π / :theunique verticial subgp assoc. to v that contains Π e [cf. Prop 4, (iv)] (7) α(π e ) = Π e By (7) and the graphicity of α Π/, we conclude: (8) α(π v ) = Π v. 0

22 By (8), we obtain a comm. diag. Π v Π v α Πv p (Π v ) p (Π v ) Hence, we have (9) α Πv = id. Since Π / is topologically generated by Π sub / and Π v, it follows from (5), (9) that (0) α Π/ = id. Finally, it follows from (0) and the assumption α p Π Π α =id that α =id[cf. theslimness of Π / ].

23 4 Fix a universal covering G G. V( N ( C( E( G) = lim V(G ) G) = lim N (G ) G) = lim C(G ) G) = N ( G) C( G) where the proj. limits are over all conn. fin. étale subcoverings G G of G G Let {V, E}. If z ( G), then we write z(g ) for the image of z via the natural map ( G) (G ). Π z Π G :aunique -like subgroup assoc. to z(g) s.t. for every conn. fin. étale subcovering G G of G G, the subgroup Π z Π G Π G is a -like subgroup assoc. to z(g )

24 ρ I : I Aut(G): an outer rep n of PSC-type out Π I =Π G I: the profinite group obtained by pulling back the exact sequence conj. Π G Aut(Π G ) Out(Π G ) by the composite I ρ I Aut(G) Out(Π G ). we have a comm. diag. Π G Π I I Π G Aut(Π G ) Out(Π G ) where the horizontal seq. are exact. ρ I Definition (i) If z V( G) or N ( G), then we shall write I z = Z ΠI (Π z ) D z = N ΠI (Π z ). (ii) If z C( G), then we shall write I z =Π z D z = N ΠI (Π z ). 3

25 Lem 7 (i) Let ṽ V( G). Then the composite Iṽ Π I I is injective. (ii) Let ẽ N( G) that abuts to ṽ V( G). Then it holds that Iṽ Iẽ. (Proof) (i) follows from Prop 4, (i), (iii). (ii) is easy. Definition ρ I : I Aut(G): an outer rep n of PSC-type ρ I is of NN-type (resp. SNN-type) () I = Ẑ. () For every ṽ V( G), the image of the composite Iṽ Π I I is open (resp. I) [cf. Lem 7, (i)]. (3) For every ẽ N( G), the natural inclusions Iṽ,Iṽ Iẽ whereẽ abuts to ṽ,ṽ V( G) [cf. Lem 7, (ii)] induces an open injection Iṽ Iṽ Iẽ. 4

26 Lem 8 ρ I : I Aut(G): of SNN-type (i) Let ṽ V( G). Then Dṽ =Πṽ Iṽ. (ii) Let ṽ V( G); ẽ E( G) an element that abuts to ṽ. ThenDẽ =Πẽ Iṽ. (iii) Πṽ = Z ΠI (Iṽ) Π G = N ΠI (Iṽ) Π G. Notation: (i) Let v, w V(G). We shall write δ(v, w) n if the following conditions are satisfied: If n =0,thenv = w. If n, then there exist {e,...,e n } N(G); {v 0,...,v n } V(G) wherev 0 = v; v n = w; e i abuts to v i, v i. We shall write δ(v, w) =n if δ(v, w) n and δ(v, w) n. (ii) Let ṽ, w V( G). We shall write δ(ṽ, w) = sup G {δ(ṽ(g ), w(g ))}. 5

27 Lem 9 ρ I : I Aut(G): an outer rep n of PSC-type Let ṽ,ṽ V( G). Consider the following 8 conditions: () δ(ṽ, ṽ )=0. () δ(ṽ, ṽ )=. (3) δ(ṽ, ṽ )=. (4) δ(ṽ, ṽ ) 3. ( ) Dṽ = Dṽ. ( ) Dṽ Dṽ ; Dṽ Dṽ Π G {}. (3 ) Dṽ Dṽ {}; Dṽ Dṽ Π G = {}. (4 ) Dṽ Dṽ = {}. Then we have equivalences () ( ); () ( ); (3) (3 ); (4) (4 ). Moreover, suppose that ρ I is of SNN-type. Then if (3 ) is satisfied, then there exists a unique ṽ 3 V( G) s.t. δ(ṽ, ṽ 3 )=δ(ṽ, ṽ 3 )= and Dṽ Dṽ = Iṽ3. 6

28 (Proof of Thm 6) To verify Thm 6, by replacing Π I by an open subgroup Π I, we may assume that G, H are sturdy, andthat ρ I, ρ J are of SNN-type [cf. Prop 4, (iii)]. sturdy Every irr. component of the pointed stable curve that gives rise to G satisfies the following: The genus of the normalization is. Thus, to verify Thm 6, it suffices to verify Claim A: G, H: thecompactifications of G, H, respectively Then the isom. α :Π G ΠH which is induced by α is graphic. [We apply Claim A to various conn. fin. ét. coverings!] = G G 7

29 Lem 0 Gp-theoretically verticial Gp-theoretically nodal. It follows from Prop 5, Lem 0 that Claim A Claim B: α :Π G ΠH is gp-theoretically verticial. (Proof of Claim B) First, let us prove that: There exists a verticial subgroup Π G α( ) is a verticial subgroup Π H. s.t. Write I Out(Π G ) (resp. J Out(Π H )) for the outer rep n of PSC-type determined by ρ I (resp. ρ J )and Π I = Π G out I, Π J = Π H out J α :Π G ΠH [and α] induces a comm. diag. ( ) Π I =Π G out I β Π H out J =Π J Π I =Π G out I β Π H out J = Π J where the vertical arrows are the surj. induced by Π G Π G, Π H Π H. 8

30 By assumption (ii), e G C(G), e H C(H) s.t. β(d eg )=D eh. Write v G V(G) (resp. v H V(H)) for the vertex to which e G (resp. e H ) abuts. By Lem 8, (ii), the diag. ( ) induces a diag. D eg β D eh β(i vg )=I vh! I vg β I vh Thus, we conclude from Lem 8, (iii), that α(π vg ) = β(n ΠI (I vg ) Π G ) = N ΠI (β(i vg )) Π H = N ΠI (I vh ) Π H = Π vh. =Π vg Therefore, to verify Claim B, it suffices to show that: Let ṽ,ṽ V( G) s.t. δ(ṽ (G), ṽ (G)). Then if α(πṽ ) is verticial, then α(πṽ ) is verticial. 9

31 If ṽ (G) =ṽ (G), then it is immediate. Suppose that ṽ (G) ṽ (G) andthat α(πṽ ) is verticial. Observe: There exist w,ũ, w V( G) s.t. (a) ṽ (G) = w (G) =ũ (G); ṽ (G) = w (G). (b) δ( w, ũ )=. (c) δ( w, w )=δ( w, ũ ) =. [cf. the next page] (a) There exist w,ũ V( H) s.t. β(d w )=D w, β(dũ )=Dũ. (b), (c), Lem 9 D w Dũ = I w D w Dũ {}, D w Dũ Π H = {} w V( H) s.t. D w Dũ = I w [cf. Lem 9] β(i w )=I w Thus, it follows from Lem 8, (iii), that α(π w )=Π w. We conclude from (a) that α(πṽ ) is verticial! 30

32 G G: a connected finite étale covering of degree = w (G ) w (G ) ũ (G ) ṽ (G) ṽ (G) 3