Twists and residual modular Galois representations Twists and residual modular Galois representations Samuele Anni University of Warwick Building Bridges, Bristol 10th July 2014 Twists and residual modular Galois representations Modular curves and Modular Forms 1 Modular curves and Modular Forms 2 Residual modular Galois representations 3 Local representation 4 Twist 5 Examples Twists and residual modular Galois representations Modular curves and Modular Forms Let us fix a positive integer n ∈ Z >0 . Definition The congruence subgroup Γ1 (n) of SL2 (Z ) is the subgroup given by a b Γ1 (n) = ∈ SL2 (Z ) : n | a−1 , n | c . c d The integer n is called level of the congruence subgroup. Twists and residual modular Galois representations Modular curves and Modular Forms Over the upper half plane: H = {z ∈ C| Im(z) > 0} we can define an action of Γ1 (n) via fractional transformations: Γ1 (n) × H → H (γ, z) 7→ γ(z) = az + b cz + d a b where γ = . c d Moreover, if n ≥ 4 then Γ1 (n) acts freely on H . Escher, Reducing Lizards Tessellation Twists and residual modular Galois representations Modular curves and Modular Forms Definition We define the modular curve Y1 (n)C to be the non-compact Riemann surface obtained giving on Γ1 (n)\H the complex structure induced by the quotient map. Let X1 (n)C be the compactification of Y1 (n)C . Fact: Y1 (n)C can be defined algebraically over Q (in fact over Z [1/n]). Twists and residual modular Galois representations 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 Γ1 (n). Γ1 (n) Proposition For every g ∈ GL+ 2 (Q ), the discrete groups g Γ1 (n)g −1 and Γ1 (n) are commensurable g −1 Γ1 (n)g ' Γ1 (n) H /H Y1 (n)C % Y1 (n)C g Twists and residual modular Galois representations Modular curves and Modular Forms We define operators on Y1 (n) through the correspondences given before: the Hecke operators Tp for every prime p, using 1 0 g= ∈ GL+ 2 (Q ) ; 0 p ∗ the diamond operators hdi for every d ∈ (Z /nZ ) , using a b g= ∈ Γ0 (n), where Γ0 (n) is the set of matrices in SL2 (Z ) c d which are upper triangular modulo n. Twists and residual modular Galois representations Modular curves and Modular Forms For n ≥ 5 and k positive integers, let ` be a prime not dividing n. Following Katz, we define the space of mod ` cusp forms as mod ` cusp forms S(n, k)F` = H0 (X1 (n)F` , ω ⊗k (− Cusps)). S(n, k)F` is a finite dimensional F` -vector space, equipped with Hecke operators Tn (n ≥ 1) and diamond operators hdi for every d∈(Z /nZ )∗ . Analogous definition in characteristic zero and over any ring where n is invertible. Twists and residual modular Galois representations Modular curves and Modular Forms One may think that mod ` modular forms come from reduction of characteristic zero modular forms mod `: S(n, k)Z [1/n] → S(n, k)F` . Unfortunately, this map is not surjective for k = 1. Even worse: given a character : (Z /nZ )∗ → C∗ the map S(n, k, )OK → S(n, k, )F is not always surjective even if k > 1. OK is the ring of integers of the number field where is defined, F is the residue field at ` and is the reduction of S(n, k, )OK = {f ∈ S(n, k)OK | ∀d ∈ (Z /nZ )∗ , hdi f = (d)f }. Twists and residual modular Galois representations Modular curves and Modular Forms Definition The Hecke algebra T(n, k) of S(n, k)C is the Z -subalgebra of EndC (S(Γ1 (n), k)C ) generated by Hecke operators Tp for every prime p and by diamond operators hdi for every d ∈ (Z /nZ )∗ . Fact: T(n, k) is finitely generated as Z -module. Given a character : (Z /nZ )∗ → C∗ , we associate a Hecke algebra T (n, k) to each S(n, k, )C : S(n, k, )C = {f ∈ S(n, k)C | ∀d ∈ (Z /nZ )∗ , hdi f = (d)f }. Twists and residual modular Galois representations Residual modular Galois representations 1 Modular curves and Modular Forms 2 Residual modular Galois representations 3 Local representation 4 Twist 5 Examples Twists and residual modular Galois representations Residual modular Galois representations Theorem (Deligne, Shimura) Let n and k be positive integers. Let F be a finite field of characteristic `, with ` not dividing n, and f : T(n, k) F a surjective morphism of rings. Then there is a continuous semi-simple representation: ρf : Gal(Q /Q ) → GL2 (F), unramified outside n`, such that for all p not dividing n` we have: Trace(ρf (Frobp )) = f (Tp ) and det(ρf (Frobp )) = f (hpi)p k−1 in F. Such a ρf is unique up to isomorphism. Computing ρf 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). Twists and residual modular Galois representations Residual modular Galois representations Question Can we compute the image of a residual modular Galois representation without computing the representation? Theorem (A.) There is a polynomial time algorithm which takes as input: n and k be positive integers; ` be a prime number not dividing n, such that 2 ≤ k ≤ ` + 1; a character : (Z /nZ )∗ → C∗ ; a morphism of ring f : T (n, k) → F` , and in particular the images of all diamond operators and of the Tp operators up to a bound B, and gives as output the image of the associated Galois representation ρf , up to conjugacy as subgroup of GL2 (F` ) without computing ρf . Twists and residual modular Galois representations Residual modular Galois representations In almost all cases, the bound B is the Sturm Bound for Γ0 (n) and weight k: Y 1 k k ·n· 1+ · n log log n 12 p 12 p|n prime In the cases when this bound is not enough, then the Sturm Bound for Γ0 (nq 2 ) and weight k, where q is the smallest odd prime not dividing n, is the required bound. Twists and residual modular Galois representations Residual modular Galois representations Some of the problems studied: ρf can arise from lower level or weight, i.e. there exists g ∈ S(m, j)F` with m ≤ n or j ≤ k such that ρg ∼ = ρf ρf can arise as twist of a representation of lower conductor, i.e. there exist g ∈ S(m, j)F` with m ≤ n or j ≤ k and a Dirichlet character χ such that ρg ⊗ χ ∼ = ρf Twists and residual modular Galois representations Residual modular Galois representations One of the principal ingredients: Theorem (Khare, Wintenberger, Dieulefait, Kisin), Serre’s Conjecture Let ` be a prime number and let ρ : Gal(Q /Q ) → GL2 (F` ) be an odd, absolutely irreducible, continuous representation. Then ρ is modular of level N(ρ), weight k(ρ) and character (ρ). N(ρ) (the level) is the Artin conductor away from `. k(ρ) (the weight) is given by a recipe in terms of ρ|I` . ∗ (ρ) : (Z /N(ρ)Z )∗ → F` is given by: det ◦ρ = (ρ)χk(ρ)−1 . Twists and residual modular Galois representations Residual modular Galois representations Setting (∗) n and k be positive integers; ` be a prime number not dividing n, such that 2 ≤ k ≤ ` + 1; : (Z /nZ )∗ → C∗ be a character; f : T (n, k) → F` be a morphism of rings; ρf : GQ → GL2 (F` ) be the unique, up to isomorphism, continuous semi-simple representation attached to f ; ∗ ∗ : (Z /nZ ) → F` be the character defined by (a) = f (hai) for all a ∈ (Z /nZ )∗ . Let p be a prime dividing n`. Let us denote by Gp = Gal(Q p /Q p ) ⊂ GQ the decomposition subgroup at p; Ip the inertia subgroup, It the tame inertia subgroup; Gi,p , with i ∈ Z >0 , the higher ramification subgroups (Ip = G0,p ). Notation: given a residual representation ρ, we will denote as Np (ρ) the valuation at p of the Artin conductor of ρ. Twists and residual modular Galois representations Local representation 1 Modular curves and Modular Forms 2 Residual modular Galois representations 3 Local representation Local representation at ` Local representation at primes dividing the level 4 Twist 5 Examples Twists and residual modular Galois representations Local representation Local representation at ` Local representation at ` Theorem (Deligne) Assume setting (∗). Suppose that f (T` ) 6= 0. Then ρf |G` is reducible, and up to conjugation in GL2 (F` ), we have k−1 ∗ χ` λ((`)/f (T` )) ρf |G` ∼ = 0 λ(f (T` )) where λ(a) is the unramified character of G` taking Frob` ∈ G` /I` to a, ∗ for any a ∈ F` . Twists and residual modular Galois representations Local representation Local representation at ` Theorem (Fontaine) Assume setting (∗). Suppose that f (T` ) = 0. Then ρf |G` is irreducible, and up to conjugation in GL2 (F` ), we have 0k−1 ϕ 0 ρf |I` ∼ = 0 ϕk−1 ∗ where ϕ0 , ϕ : It → F` are the two fundamental characters of level 2. Twists and residual modular Galois representations Local representation Local representation at primes dividing the level Local representation at primes dividing the level Theorem (Gross-Vignéras, Serre: Conjecture 3.2.6? ) Let ρ : GQ → GL(V ) be a continuous, odd, irreducible representation of the absolute Galois group over Q to a 2-dimensional F` -vector space V . Let n = N(ρ) and k = k(ρ), let f ∈ S(n, k)F` be an eigenform such that ρf ∼ = ρ. Let p be a prime divisor of `n. (1) If f (Tp ) 6= 0, then there exists a stable line D ⊂ V for the action of Gp , the decomposition subgroup at p, such that the inertia group at p acts trivially on V /D. Moreover, f (Tp ) is equal to the eigenvalue of Frobp which acts on V /D. (2) If f (Tp ) = 0, then there exists no stable line D ⊂ V as in (1). Twists and residual modular Galois representations Twist 1 Modular curves and Modular Forms 2 Residual modular Galois representations 3 Local representation 4 Twist Local representation and conductor Twisting by Dirichlet characters 5 Examples Twists and residual modular Galois representations Twist Local representation and conductor Proposition (A.) Assume setting (∗) and that ρf is irreducible and it does not arise from lower level. Let p be a prime dividing n such that f (Tp ) 6= 0. Then ρf |Gp is decomposable if and only if ρf |Ip is decomposable. This proposition is proved using representation theory. Twists and residual modular Galois representations Twist Local representation and conductor Proposition (A.) Assume setting (∗) and that ρf is irreducible and it does not arise from lower level. Let p be a prime dividing n, such that f (Tp ) 6= 0. Then: (a) ρf |Ip is decomposable if and only if Np (ρf ) = Np (); (b) ρf |Ip is indecomposable if and only if Np (ρf ) = 1 + Np (). Twists and residual modular Galois representations Twist Local representation and conductor Sketch of the proof The valuation of N(ρf ) at p is given by: Np (ρf ) = X i≥0 1 dim(V /V Gi,p ) = dim(V /V Ip ) + b(V ), [G0,p : Gi,p ] where V is the two-dimensional F` -vector space underlying the representation, V Gi,p is its subspace of invariants under Gi,p , and b(V ) is the wild part of the conductor. Since f (Tp ) 6= 0, the representation restricted to the decomposition group at p is reducible. Hence, after conjugation, k−1 1 |Ip ∗ ∼ |Ip ∗ χ ∗ ρf |Gp ∼ , ρf |Ip ∼ = 1 ` = = 0 1 0 1 0 2 where 1 and 2 are characters of Gp with 2 unramified, χ` is the mod ` cyclotomic character and ∗ belongs to F` . Twists and residual modular Galois representations Twist Local representation and conductor Remark If ρf |Ip is indecomposable then the image of inertia at p is of order divisible by ` and so the image cannot be exceptional, hence it is big. Twists and residual modular Galois representations Twist Twisting by Dirichlet characters Let n be a positive integer. Any Dirichlet character of conductor n can be decomposed into local characters, one for each prime divisor of n. With no loss of generality, we reduce ourselves to study twists of modular Galois representations with Dirichlet characters with prime power conductor. Twists and residual modular Galois representations Twist Twisting by Dirichlet characters Question What is the conductor of the twist? Shimura gave an upper bound: lcm(cond(χ)2 , n) where n is the level of the form and χ is the character used for twisting. Twists and residual modular Galois representations Twist Twisting by Dirichlet characters Proposition (A.) Assume setting (∗). Let p be a prime not dividing n`. Let ∗ χ : (Z /p i Z )∗ → F` , for i > 0, be a non-trivial character. Then Np (ρf ⊗ χ) = 2Np (χ). Twists and residual modular Galois representations Twist Twisting by Dirichlet characters Proposition (A.) Assume setting (∗) and that ρf is irreducible and it does not arise from lower level. Let p be a prime dividing n and suppose that f (Tp ) 6= 0. ∗ Let χ : (Z /p i Z )∗ → F` , for i > 0, be a non-trivial character. Then Np (ρf ⊗ χ) = Np (χ) + Np (χ). Twists and residual modular Galois representations Twist Twisting by Dirichlet characters It is also possible to know what is the system of eigenvalues associated to the twist: Proposition (A.) Assume setting (∗). Suppose that ρf is irreducible and that N(ρf ) = n. Let p be a prime dividing n and suppose that f (Tp ) 6= 0. Let χ from ∗ (Z /p i Z )∗ to F` , with i > 0, be a non-trivial character. Then (a) if ρf |Ip is decomposable then the representation ρf ⊗ χ restricted to Gp , the decomposition group at p, admits a stable line with unramified quotient if and only if Np (ρf ⊗ χ) = Np (ρf ); (b) if ρf |Ip is indecomposable then the representation ρf ⊗ χ restricted to Gp does not admit any stable line with unramified quotient. Twists and residual modular Galois representations Twist Twisting by Dirichlet characters Proposition (A.) Assume setting (∗). Suppose that ρf is irreducible and that N(ρf ) = n. Let p be a prime dividing n and suppose that f (Tp ) = 0. Then: (a) if ρf |Gp is reducible then there exists a mod ` modular form g of weight k and level at most np and a non-trivial character ∗ χ : (Z /p i Z )∗ → F` with i > 0 such that g (Tp ) 6= 0 and ∼ ρg = ρf ⊗ χ; (b) if ρf |Gp is irreducible then for any non-trivial character ∗ χ : (Z /p i Z )∗ → F` with i > 0 the representation ρf ⊗ χ restricted to Gp does not admit any stable line with unramified quotient. Twists and residual modular Galois representations Examples 1 Modular curves and Modular Forms 2 Residual modular Galois representations 3 Local representation 4 Twist 5 Examples Example 1 Example 2 Example 3 Twists and residual modular Galois representations Examples Example 1 Example 1 Let n = 135 = 33 5. Let be the Dirichlet character modulo 135 of 9 . conductor 5 mapping 56 → 1, 82 → ζ36 two Galois orbits: the two Hecke eigenvalue fields are: S(135, 3, )new C Q (x 16 + 217x 12 + 9264x 8 + 59497x 4 + 28561) and Q (x 16 + 286x 12 + 16269x 8 + 85684x 4 + 62500). In the first one, applying the reduction map for all prime ideals above 7, we obtain the following eigenvalue systems defined over F72 ∼ = F7 [x]/[x 2 + 6x + 4] ∼ = F7 [α]: Twists and residual modular Galois representations Examples Example 1 p f1 (Tp ) f2 (Tp ) f3 (Tp ) 2 3 5 7 11 13 17 19 23 α 0 5α + 4 0 α+3 6α 6α 6α + 4 3α + 4 6α 0 2α + 3 0 6α + 4 6α α 6α + 4 4α + 3 α+6 0 5α + 5 0 α+3 α+6 6α + 1 α+3 3α f1 ,f2 and f3 are mod 7 modular forms of level 135 and weight 3. It is easy to verify that the corresponding representations are of minimal level and weight. Twists and residual modular Galois representations Examples Example 1 Let us focus on f1 . Since 5 divides the level and f1 (T5 ) 6= 0: 2 1 χ7 ∗ ∼ −1 χ2 ∼ ρf1 |G5 = = 2 f1 7 0 2 0 0 2 The character of f1 has conductor 5 hence the representation is reducible and decomposable, so ∗ = 0. Moreover 2 (5) = f1 (T5 ). If we twist by −1 f1 then we have: ∼ (ρf1 ⊗ −1 f1 )|G5 = 2 −1 2 χ7 0 0 2 −1 f1 hence we know the eigenvalue at 5. The computation on the conductor −1 2 gives that the conductor is 135 and (−1 2 χ7 )(5) = 42 (5) = 5α + 5. Twists and residual modular Galois representations Examples Example 1 p f1 (Tp ) f2 (Tp ) f3 (Tp ) ρf1 ⊗ −1 f1 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 α 0 5α + 4 0 α+3 6α 6α 6α + 4 3α + 4 2 4 α+6 5α + 1 α 2α 6α 0 2α + 3 0 6α + 4 6α α 6α + 4 4α + 3 5 4 α+6 2α + 6 α 5α α+6 0 5α + 5 0 α+3 α+6 6α + 1 α+3 3α 5 4 6α 5α + 1 6α + 1 2α + 5 α+6 0 5α + 5 0 α+3 α+6 6α + 1 α+3 3α 5 4 6α 5α + 1 6α + 1 2α + 5 Twists and residual modular Galois representations Examples Example 1 The level is also divisible by 3. But f1 (T3 ) = 0. The local representation at 3 is irreducible: in order to prove this we have to check all lover levels and possible twist. In this case it is easy since the newforms space is empty in most cases. The argument is similar for all levels, so we will just show what happens for level 15. p f1 (Tp ) g1 (Tp ) g2 (Tp ) g3 (Tp ) g4 (Tp ) 2 3 5 7 11 13 α 0 5α + 4 0 α+3 6α 6α + 6 6α 3α + 5 4α + 5 6α + 1 4α + 5 α+5 α+6 4α + 1 3α + 2 α 3α + 2 4α + 1 α 2α + 1 2 α 5 3α + 5 6α + 1 5α + 3 2 6α + 1 5 Twists and residual modular Galois representations Examples Example 1 It is possible to show that the image of the Galois representation up to conjugation is ρf1 (GQ ) ∼ =< α > SL2 (F7 ) ⊂ GL2 (F72 ) (f (Tp ))2 (f (p)p 2 )−1 7→ 6, 0, 6, 4, 5, 4, 5, 3, 1, 1, 6, 4, 3 Twists and residual modular Galois representations Examples Example 2 Example 2 Let us consider S(40, 2, τ )new where τ is the quadratic character C conductor 40 mapping 31 → 1,21 → −1, 17 → −1. As before, by reduction get mod 7 eigenvalue systems, for example fi for i = 1, 2, 3, 4: p f1 (Tp ) f2 (Tp ) f3 (Tp ) f4 (Tp ) 2 3 5 7 11 13 17 19 23 6 3 2 6 4 0 2 3 1 1 4 1 1 4 0 5 3 6 5 3 6 1 3 0 5 4 6 2 4 5 6 3 0 2 4 1 Twists and residual modular Galois representations Examples Example 2 Let χ be the character over F7 of conductor 8 such that χf1 has conductor 4. Since f1 (T2 ) 6= 0, then we have that: N2 (ρf ⊗ χ) = N2 (χf1 ) + N2 (χ) = 2 + 3 = 5 6= 6. Hence, let us check the twist and the eigenvalue systems at level 160: N2 (ρf ⊗ χ) = 25 5 = 160: p g1 (Tp ) g2 (Tp ) g3 (Tp ) g4 (Tp ) ρf1 ⊗ χ ρf2 ⊗ χ ρf3 ⊗ χ ρf4 ⊗ χ 2 3 5 7 11 13 17 19 23 0 4 6 6 4 0 5 3 1 0 3 5 1 4 0 2 3 6 0 4 2 1 3 0 2 4 6 0 3 1 6 3 0 5 4 1 0 3 5 1 4 0 2 3 6 0 4 6 6 4 0 5 3 1 0 3 1 6 3 0 5 4 1 0 4 2 1 3 0 2 4 6 Twists and residual modular Galois representations Examples Example 2 All the systems given by gi are reduction of the same form. Also in this case it is possible to prove that the image of the Galois representation is big: in this case, it is isomorphic to GL2 (F7 ), up to conjugation. Twists and residual modular Galois representations Examples Example 3 Example 3 S(7, 3)new , S(49, 3)new C C The Hecke eigenvalue fields in newformspace at level 49 are given by: Q (x 2 − 3x + 9) Q (x 4 − 4x 3 + 22x 2 − 44x + 167) Q (x 8 + 4x 7 − 6x 6 − 96x 5 + 225x 4 + 1336x 3 − 514x 2 − 5948x + 7399) Number Field with defining polynomial x 54 + 5x 53 + 43x 52 + 169x 51 + · · · + 248413945171829320x 16 + · · · + 348605788594202619x 8 + · · · + 10909749546081 Number Field with defining polynomial x 96 + 13x 95 + 69x 94 + 174x 93 + 16x 92 − 1672x 91 + . . . while in level 7 . . . the Hecke eigenvalue field is Q . Twists and residual modular Galois representations Examples Example 3 Let F52 ∼ = F5 [β]. = F5 [x]/[x 2 + 2x + 4] ∼ Let χ be Dirichlet character over F52 modulo 49 of conductor 7 mapping 3 → 3β. p f (Tp ) g (Tp ) ρf ⊗ χ 2 3 5 7 11 13 17 19 23 29 2 0 0 3 4 0 0 0 3 1 β 0 0 0 3β+1 0 0 0 4β 1 β 0 0 0 3β+1 0 0 0 4β 1 In this case we have that Pρf ∼ = A4 and ρf (GQ ) ∼ = F∗5 π −1 (A4 ), where A4 is the alternating group of 4 elements. Twists and residual modular Galois representations Examples Example 3 Twists and residual modular Galois representations Samuele Anni University of Warwick Building Bridges, Bristol 10th July 2014 Thanks!