Differences between Galois representations in outer-automorphisms of - PowerPoint PPT Presentation

Differences between Galois representations in outer-automorphisms of π 1 and those in automorphisms, implied by topology of moduli spaces Makoto Matsumoto, Universit´ e de Tokyo 2010/5/12, au S´ eminaire de G´ eom´ etrie Arithm´ etique PARIS-TOKYO. email: matumoto “marque ‘at’ ”ms.u-tokyo.ac.jp This study is supported in part by JSPS Grant-In-Aid #19204002, and JSPS Core-to-Core Program No.18005. Thanks to Richard Hain for essential math- ematical ingredient. 1

• Recall the monodromy representation on π 1 of curves. • Galois monodromy often contains geometric monodromy. • Using this connection, obtain implications from topology to Galois monodromy. 2

1. Monodromy on π 1 . • K : a field ⊂ K ⊂ C . • A family of ( g, n )-curves C → B : def ⇔ B : smooth noetherian geometrically connected scheme/ K . F cpt : C cpt → B : proper smooth family of genus g curves (with geometrically connected fibers). s i : B → C cpt (1 ≤ i ≤ n ) disjoint sections, F : C → B : complement C cpt \ ∪ s i ( B ) → B . • We assume hyperbolicity 2 g − 2 + n > 0. • Π g,n : (classical) fundamental group of n -punctured genus g Riemann surface (referred to as surface group ) g,n , Π ( ℓ ) • Π ∧ g,n : its profinite, resp. pro- ℓ , completion. 3

x ¯ ↓ C ¯ b → C ↓ � ↓ ¯ b → B ¯ b, ¯ x : (geometric) base points. Gives a short exact sequence of arithmetic(=etale) π 1 : x ) → π 1 ( B, ¯ 1 → π 1 ( C ¯ x ) → π 1 ( C, ¯ b ) → 1 b , ¯ 4

The short exact sequence x ) → π 1 ( B, ¯ 1 → π 1 ( C ¯ b , ¯ x ) → π 1 ( C, ¯ b ) → 1 || GAGA Π ∧ g,n gives the pro- ℓ outer monodromy representation: π 1 ( B, ¯ 1 → π 1 ( C ¯ b , ¯ x ) → π 1 ( C, ¯ x ) → b ) → 1 ↓ Aut Π ( ℓ ) g,n 5

The short exact sequence x ) → π 1 ( B, ¯ 1 → π 1 ( C ¯ b , ¯ x ) → π 1 ( C, ¯ b ) → 1 || Π ∧ g,n gives the pro- ℓ outer monodromy representation: π 1 ( B, ¯ 1 → π 1 ( C ¯ b , ¯ x ) → π 1 ( C, ¯ x ) → b ) → 1 ↓ ↓ ↓ 1 → Inn Π ( ℓ ) Aut Π ( ℓ ) Out Π ( ℓ ) g,n → → → 1 g,n g,n 6

The short exact sequence x ) → π 1 ( B, ¯ 1 → π 1 ( C ¯ b , ¯ x ) → π 1 ( C, ¯ b ) → 1 || Π ∧ g,n gives the pro- ℓ outer monodromy representation: π 1 ( B, ¯ 1 → π 1 ( C ¯ b , ¯ x ) → π 1 ( C, ¯ x ) → b ) → 1 ↓ ↓ ρ A,C,x ↓ ρ O,C 1 → Inn Π ( ℓ ) Aut Π ( ℓ ) Out Π ( ℓ ) g,n → → → 1 g,n g,n 7

The short exact sequence x ) → π 1 ( B, ¯ 1 → π 1 ( C ¯ b , ¯ x ) → π 1 ( C, ¯ b ) → 1 || Π ∧ g,n gives the pro- ℓ outer monodromy representation: π 1 ( B, ¯ 1 → π 1 ( C ¯ b , ¯ x ) → π 1 ( C, ¯ x ) → b ) → 1 ↓ ↓ ρ A,C,x ↓ ρ O,C 1 → Inn Π ( ℓ ) Aut Π ( ℓ ) Out Π ( ℓ ) g,n → → → 1 g,n g,n (If B = Spec K , we have ρ O,C : G K = π 1 ( B ) → Out Π ( ℓ ) g,n .) 8

2. Universal monodromy. Grothendieck, Takayuki Oda, . . . • M g,n : the moduli stack of ( g, n )-curves over Q . • C g,n → M g,n be the universal family of ( g, n )-curves. Applying the previous construction, we have: π 1 ( M g,n , ¯ 1 → π 1 ( C ¯ b , ¯ x ) → π 1 ( C g,n , ¯ x ) → b ) → 1 ↓ ↓ ρ A,univ, ¯ ↓ ρ O,univ x 1 → Inn Π ( ℓ ) Aut Π ( ℓ ) Out Π ( ℓ ) g,n → → → 1 g,n g,n This representation is universal, since any ( g, n )-family C → B has classifying morphism, C → C g,n ↓ � ↓ B → M g,n , 9

2. Universal monodromy. Grothendieck, Takayuki Oda, . . . • M g,n : the moduli stack of ( g, n )-curves over Q . • C g,n → M g,n be the universal family of ( g, n )-curves. Applying the previous construction, we have: π 1 ( M g,n , ¯ 1 → π 1 ( C ¯ b , ¯ x ) → π 1 ( C g,n , ¯ x ) → b ) → 1 ↓ ↓ ρ A,univ, ¯ ↓ ρ O,univ x 1 → Inn Π ( ℓ ) Aut Π ( ℓ ) Out Π ( ℓ ) g,n → → → 1 g,n g,n This representation is universal, since any ( g, n )-family C → B has classifying morphism, choose ¯ b , b → C → C g,n C ¯ ↓ � ↓ � ↓ ¯ b → B → M g,n , then universality as follows. 10

π 1 ( B, ¯ 1 → π 1 ( C ¯ b , ¯ x ) → π 1 ( C, ¯ x ) → b ) → 1 || ↓ ↓ π 1 ( M g,n , ¯ 1 → π 1 ( C ¯ b , ¯ x ) → π 1 ( C g,n , ¯ x ) → b ) → 1 ↓ ↓ ρ A,univ, ¯ ↓ ρ O,univ x 1 → Inn Π ( ℓ ) Aut Π ( ℓ ) Out Π ( ℓ ) g,n → → → 1 g,n g,n where the vertical composition is ρ A,C,x (middle), ρ O,C (right). In particular, if C → B = b = Spec K , we have ρ O,univ Out Π ( ℓ ) ρ O,C : G K = π 1 ( b, ¯ b ) → π 1 ( M g,n /K, ¯ b ) → g,n and hence ρ O,C ( G K ) ⊂ ρ O,univ ( π 1 ( M g,n /K )) ⊂ Out Π ( ℓ ) g,n . Definition If the equality holds for the left inclusion, the curve C → b is called monodromically full . 11

Theorem (Tamagawa-M, 2000) The set of closed points in M g,n corresponding to monodromically full curves is infinite, and dense in M g,n ( C ) with respect to the complex topology. Remark As usual, the π 1 of M g,n is an extension 1 → π 1 ( M g,n ⊗ Q ) → π 1 ( M g,n ) → G Q → 1 . The left hand side is isomorphic to the profinite completion of the mapping class group Γ g,n . (Topologists studied a lot.) Monodromically full ⇔ Galois image contains Γ g,n . Scketch of Proof of Theorem goes back to Serre, Terasoma, . . . Hilbert’s irreducibility + almost pro- ℓ ness. 12

Proposition If P is a finitely generated pro- ℓ group, then take H := [ P, P ] P ℓ ⊳ P . Then P/H is a finite group (flattini quotient). If a morphism of profinite groups Γ → P is surjective modulo H , namely Γ → P → P/H is surjective, then Γ → P is surjective. Definition A profinite group G is almost pro- ℓ if it has a pro- ℓ open subgroup P . Corollary Suppose in addition G is finitely generated. Put H := [ P, P ] P ℓ . Then [ G : H ] < ∞ . If Γ → G → G/H is surjective, so is Γ → G . 13

Claim C → B be a family of ( g, n )-curves over a smooth variety B over a NF K . Then the image of π 1 ( B ) → Out Π ( ℓ ) g,n is a finitely generated almost pro- ℓ group. • Out(fin.gen.pro- ℓ ) is almost pro- ℓ . • a closed subgroup of almost pro- ℓ group is again so. • finitely generatedness: π 1 ( B ⊗ ¯ K ) is finitely generated. G K not. But take L ⊃ K so that C ( L ) � = ∅ and G L → Out Π ( ℓ ) has pro- ℓ image. Only finite number of places of O L ramifies, and class field theory tells that Im( G L ) has finite flattini quotient. Corollary ∃ H < Im( π 1 ( B )) such that Γ ։ Im( π 1 ( B )) /H implies Γ ։ Im( π 1 ( B )) . 14

Corollary Take a subgroupthe above H for the image of π 1 ( B ). H ′ the inverse image in π 1 ( B ). Let B ′ → B be the etale cover corresponding to H ′ . If b ∈ B has a connected fiber (i.e. one point) in B ′ , Then the composition G k ( b ) → Im( π 1 ( B )) → Im( π 1 ( B )) /H is surjective, hence the left arrow is surjective. Last Claim Existence of many such b follows from Hilbertian property: Take a quasi finite dominating ratl. map B → P dim B . K Apply Hilbertian property to B ′ → B → P dim B . K 15

3. Aut and Out. Again consider C → b = Spec K . Take a closed point x in C , and ¯ x a geometric point. This yields G k ( x ) ↓ x ∗ 1 → π 1 ( C ¯ b , ¯ x ) → π 1 ( C, ¯ x ) → G K → 1 ↓ ↓ ↓ 1 → Inn Π ( ℓ ) g,n → Aut Π ( ℓ ) g,n → Out Π ( ℓ ) g,n → 1 . Vertical composition gives ρ A,x : G k ( x ) → Aut Π ( ℓ ) g,n ∩ ↓ → Out Π ( ℓ ) ρ O : G K g,n Question: Is the map AO ( C, x ) : ρ A,x ( G k ( x ) ) → ρ O ( G K ) injective? (Do we lose information in Aut → Out?) 16

Definition I ( C, x ) := the statement “ AO ( C, x ) is injective.” Remark If C = P 1 −{ 0 , 1 , ∞} / Q and x is a canonical tangen- tial base point, then AO ( C, x ) is an isom (hence I ( C, x ) holds: Belyi, Ihara, Deligne, 80’s). Main Theorem (M, 2009) Suppose g ≥ 3 and ℓ divides 2 g − 2. Let C → Spec K be a monodromically full ( g, 0)-curve ([ K : Q ] < ∞ ). Then, for every closed point x in C such that ℓ � | [ k ( x ) : K ], I ( C, x ) does not hold. In this case, the kernel of AO ( C, x ) is infinite. 17

A topological result. Proof reduces to a topological result. Γ g,n := π orb ( M an g,n ) . 1 Γ g := Γ g, 0 , Π g, 0 = Π g . Topological version of universal family yields 1 → Π g → Γ g, 1 → Γ g → 1 g =: Π g / Π ′ and by putting H := Π ab g 1 → H → Γ g, 1 / Π ′ g → Γ g → 1 Theorem (S. Morita 98, Hain-Reed 00). Let g ≥ 3. The cohomology class of the above extension [ e ] ∈ H 2 (Γ g , H ) has the order 2 g − 2. 18

Proof of Main Theorem. Suppose ℓ | (2 g − 2), x ∈ C with ℓ � | [ k ( x ) : K ]. Suppose I ( C, x ), namely the image of G k ( x ) in the middle G k ( x ) → Aut Π ( ℓ ) → Out Π ( ℓ ) g g is same with the image in the third. Let S be this image. This gives a restricted section from S to the middle group: 1 → Inn Π ( ℓ ) Aut Π ( ℓ ) Out Π ( ℓ ) → → → 1 g g g || ∪ ∪ 1 → Inn Π ( ℓ ) → Im ρ A,univ,x → Im ρ O,univ → 1 g ∨ S 19