Residual modular Galois representations and their images Samuele - PowerPoint PPT Presentation

Residual modular Galois representations and their images Residual modular Galois representations and their images Samuele Anni University of Warwick University of Warwick, Number Theory Seminar 2nd December 2013

Residual modular Galois representations and their images Modular curves and Modular Forms 1 Modular curves and Modular Forms 2 Residual modular Galois representations 3 Image 4 Algorithm 5 The old-space 6 Local representation 7 Twist 8 Projective image S 4 : a construction

Residual modular Galois representations and their images Modular curves and Modular Forms Let us fix a positive integer n ∈ Z > 0 . Definition The congruence subgroup Γ 1 ( n ) of SL 2 ( Z ) is the subgroup given by �� � � a b Γ 1 ( n ) = ∈ SL 2 ( Z ) : n | a − 1 , n | c . c d The integer n is called level of the congruence subgroup.

Residual modular Galois representations and their images Modular curves and Modular Forms Over the upper half plane: H = { z ∈ C | Im( z ) > 0 } we can define an action of Γ 1 ( n ) via fractional transformations : Γ 1 ( n ) × H → H γ ( z ) = az + b ( γ, z ) �→ cz + d � � a b where γ = . c d Moreover, if n ≥ 4 then Γ 1 ( n ) acts freely Escher, Reducing Lizards Tessellation on H .

Residual modular Galois representations and their images Modular curves and Modular Forms Definition We define the modular curve Y 1 ( n ) C to be the non-compact Riemann surface obtained giving on Γ 1 ( n ) \ H the complex structure induced by the quotient map. Let X 1 ( n ) C be the compactification of Y 1 ( n ) C . Fact: Y 1 ( n ) C can be defined algebraically over Q (in fact over Z [1 / n ]).

�� � �� � � � � Residual modular Galois representations and their images Modular curves and Modular Forms The group GL + 2 ( Q ) acts on H via fractional transformation, and its action has a particular behaviour with respect to Γ 1 ( n ). Γ 1 ( n ) Γ 1 ( n ) Proposition g g − 1 Γ 1 ( n ) g H H For every g ∈ GL + 2 ( Q ) , the discrete groups g Γ 1 ( n ) g − 1 and Γ 1 ( n ) are commensurable Y 1 ( n ) C Y 1 ( n ) C

Residual modular Galois representations and their images Modular curves and Modular Forms We define operators on Y 1 ( n ) through the correspondences given before: the Hecke operators T p for every prime p , using � 1 � 0 ∈ GL + g = 2 ( Q ) ; 0 p the diamond operators � d � for every d ∈ ( Z / n Z ) ∗ , using � � a b g = ∈ Γ 0 ( n ), where Γ 0 ( n ) is the set of matrices in SL 2 ( Z ) c d which are upper triangular modulo n .

Residual modular Galois representations and their images Modular curves and Modular Forms For n ≥ 5 and k positive integers, let ℓ be a prime not dividing n . Following Katz, we define the space of mod ℓ cusp forms as mod ℓ cusp forms S ( n , k ) F ℓ = H 0 ( X 1 ( n ) F ℓ , ω ⊗ k ( − Cusps)) . S ( n , k ) F ℓ is a finite dimensional F ℓ -vector space, equipped with Hecke operators T n ( n ≥ 1) and diamond operators � d � for every d ∈ ( Z / n Z ) ∗ . Analogous definition in characteristic zero and over any ring where n is invertible.

Residual modular Galois representations and their images Modular curves and Modular Forms One may think that mod ℓ modular forms come from reduction of characteristic zero modular forms mod ℓ : S ( n , k ) Z [1 / n ] → S ( n , k ) F ℓ . Unfortunately, this map is not surjective for k = 1. Even worse: given a character ǫ : ( Z / n Z ) ∗ → C ∗ the map S ( n , k , ǫ ) O K → S ( n , k , ǫ ) F is not always surjective even if k > 1, where O K is the ring of integers of the number field where ǫ is defined, F ℓ ⊆ F and S ( n , k , ǫ ) O K = { f ∈ S ( n , k ) O K | ∀ d ∈ ( Z / n Z ) ∗ , � d � f = ǫ ( d ) f } .

Residual modular Galois representations and their images Modular curves and Modular Forms Definition The Hecke algebra T ( n , k ) of S ( n , k ) C is the Z -subalgebra of End C ( S (Γ 1 ( n ) , k ) C ) generated by Hecke operators T p for every prime p and by diamond operators � d � for every d ∈ ( Z / n Z ) ∗ . Fact: T ( n , k ) is finitely generated as Z -module. Given a character ǫ : ( Z / n Z ) ∗ → C ∗ , we associate a Hecke algebra T ǫ ( n , k ) to each S ( n , k , ǫ ) C : S ( n , k , ǫ ) C = { f ∈ S ( n , k ) C | ∀ d ∈ ( Z / n Z ) ∗ , � d � f = ǫ ( d ) f } .

Residual modular Galois representations and their images Residual modular Galois representations 1 Modular curves and Modular Forms 2 Residual modular Galois representations 3 Image 4 Algorithm 5 The old-space 6 Local representation 7 Twist 8 Projective image S 4 : a construction

Residual modular Galois representations and their images Residual modular Galois representations Theorem (Deligne, Shimura) Let n and k be positive integers. Let F be a finite field of characteristic ℓ , with ℓ not dividing n , and f : T ( n , k ) ։ F a surjective morphism of rings. Then there is a continuous semi-simple representation: ρ f : Gal( Q / Q ) → GL 2 ( F ) , unramified outside n ℓ , such that for all p not dividing n ℓ we have: Trace( ρ f (Frob p )) = f ( T p ) and det( ρ f (Frob p )) = f ( � p � ) p k − 1 in F . Such a ρ f is unique up to isomorphism. Computing ρ f is “difficult”, but theoretically it can be done in polynomial time in n , k , # F : Edixhoven, Couveignes, de Jong, Merkl, Bruin, Bosman (# F ≤ 32); Mascot, Zeng, Tian (# F ≤ 41).

Residual modular Galois representations and their images Residual modular Galois representations Question Can we compute the image of a residual modular Galois representation without computing the representation?

Residual modular Galois representations and their images Image 1 Modular curves and Modular Forms 2 Residual modular Galois representations 3 Image 4 Algorithm 5 The old-space 6 Local representation 7 Twist 8 Projective image S 4 : a construction

Residual modular Galois representations and their images Image Main ingredients: Theorem (Dickson) Let ℓ be an odd prime and H a finite subgroup of PGL 2 ( F ℓ ). Then a conjugate of H is one of the following groups: a finite subgroup of the upper triangular matrices; SL 2 ( F ℓ r ) / {± 1 } or PGL 2 ( F ℓ r ) for r ∈ Z > 0 ; a dihedral group D 2 n with n ∈ Z > 1 , ( ℓ, n ) = 1; or it is isomorphic to A 4 , S 4 or A 5 . Definition If G := ρ f (Gal( Q / Q )) has order prime to ℓ we call the image exceptional .

� � Residual modular Galois representations and their images Image The field of definition of the representation is the smallest field F ⊂ F ℓ over which ρ f is equivalent to all its conjugate. The image of the representation ρ f is then a subgroup of GL 2 ( F ). Let P ρ f : Gal( Q / Q ) → PGL 2 ( F ) be the projective representation associated to the representation ρ f : ρ f Gal( Q / Q ) GL 2 ( F ) π �� P ρ f PGL 2 ( F ) . The representation P ρ f can be defined on a different field than the field of definition of the representation. This field is called the Dickson’s field for the representation.

Residual modular Galois representations and their images Image Theorem (Khare, Wintenberger, Dieulefait, Kisin), Serre’s Conjecture Let ℓ be a prime number and let ρ : Gal( Q / Q ) → GL 2 ( F ℓ ) be an odd, absolutely irreducible, continuous representation. Then ρ is modular of level N ( ρ ), weight k ( ρ ) and character ǫ ( ρ ). N ( ρ ) (the level) is the Artin conductor away from ℓ . k ( ρ ) (the weight) is given by a recipe in terms of ρ | I ℓ . ∗ ǫ ( ρ ): ( Z / N ( ρ ) Z ) ∗ → F ℓ is given by: det ◦ ρ = ǫ ( ρ ) χ k ( ρ ) − 1 .

Residual modular Galois representations and their images Algorithm 1 Modular curves and Modular Forms 2 Residual modular Galois representations 3 Image 4 Algorithm 5 The old-space 6 Local representation 7 Twist 8 Projective image S 4 : a construction

Residual modular Galois representations and their images Algorithm Algorithm Input: n positive integer; ℓ prime such that ( n , ℓ ) = 1; k positive integer such that 2 ≤ k ≤ ℓ + 1; a character ǫ : ( Z / n Z ) ∗ → C ∗ ; a morphism of ring f : T ǫ ( n , k ) → F ℓ ; Output: Image of the associated Galois representation ρ f , up to conjugacy as subgroup of GL 2 ( F ℓ ).

Residual modular Galois representations and their images Algorithm Algorithm Problems Step 1 Iteration “down to ρ f can arise from lower level or weight, top”, i.e. considering all i.e. there exists g ∈ S ( m , j ) F ℓ with divisors of n : creation of a m ≤ n or j ≤ k such that ρ g ∼ = ρ f database ρ f can arise as twist of a Step 2 Determine minimality representation of lower conductor, i.e. with respect to level and with there exist g ∈ S ( m , j ) F ℓ with m ≤ n respect to weight. or j ≤ k and a Dirichlet character χ Step 4 Determine minimality such that ρ g ⊗ χ ∼ = ρ f up to twisting.