Explicit computation of Galois representations occurring in families of curves

We extend our method to compute division polynomials of Jacobians of curves over Q to curves over Q(t), in view of computing mod ell Galois representations occurring in the \'etale cohomology of surfaces over Q. Although the division polynomials which we obtain are unfortunately too complicated to achieve this last goal, we still obtain explicit families of Galois representations over P^1_Q, and we study their degeneration at places of bad reduction of the corresponding curve.Comment: Comments welcom

Certification de représentations galoisiennes modulaires

International audienceWe show how the output of the algorithm to compute modular Galois representations described in our previous article can be certified. We have used this process to compute certified tables of such Galois representations obtained thanks to an improved version of this algorithm, including representations modulo primes up to 31 and representations attached to a newform with non-rational (but of course algebraic) coefficients, which had never been done before. These computations take place in the Jacobian of modular curves of genus up to 26. The resulting data are available on the author's webpage

Calcul de représentations galoisiennes modulaires

It was conjectured in the late 60's by J.-P. Serre and proved in the early 70's by P.Deligne that to each newform f = q +Σn ⩾2 anqn 2 Sk(N; "), k ⩾2, and each primel of the number field Kf = Q(an; n ⩾ 2), is attached an l-adic Galois representationPf;l : Gal(Q=Q) ! GL2(ZKf;l ), which is unrami fied outside ℓN and such the characteristicpolynomial of the Frobenius element at p ∤ ℓN is X2 apX +"(p)pk1. Reducing modulo land semi-simplifying, one gets a mod l Galois representation Pf;l : Gal(Q=Q) ! GL2(Fl),which is unrami filed outside ℓN and such that the characteristic polynomial of the Frobeniuselement at p ℓN is X2 apX +"(p)pk1 mod l. In particular, its trace is ap mod l, whichgives a quick way to compute ap mod l for huge p.The goal of this thesis is to study and implement an algorithm based on this idea(originally due to J.-M. Couveignes and B. Edixhoven) which computes the coefficients apmodulo l by computing the mod l Galois representation first, relying on the fact that ifk < ℓ, this representation shows up in the ℓ-torsion of the jacobian of the modular curveX1(ℓN).Thanks to several improvements, such as the use of K. Khuri-Makdisi's methods tocompute in the modular Jacobian J1(ℓN) or the construction of an arithmetically well-behaved function alph 2 Q(J1(ℓN)), this algorithm performs very well, as illustrated bytables of coefficients. This thesis ends by the presentation of a method to formally provethat the output of the algorithm is correct.J.-P. Serre a conjecturé à la fin des années 60 et P. Deligne a prouvé au début des années 70 que pour toute newform f = q + ∑ n⩾2 a n q n 2 S k (N; "), k ⩾ 2, et tout premier l du corps de nombres Kf = Q(a n ; n ⩾ 2), il existe une représentation galoisienne l-adique pf;l : Gal(Q=Q) ! GL2 (ZKf;l) qui est non-ramifiée en dehors de ℓN et telle que le polynôme caractéristique du Frobenius en p ∤ ℓN est X2 a pX + "(p)p k 1 .Après réduction modulo l et semi-simplification, on obtient une représentation galoisienne pf;l : Gal(Q=Q) ! GL2 (Fl) modulo l, non-ramifiée en dehors de ℓN et telle que lepolynôme caractéristique du Frobenius en p ∤ ℓN est X 2 a pX + "(p)p k 1mod l, d'où un moyen de calcul rapide de ap mod l pour p gigantesque.L'objet de cette thèse est l'étude et l'implémentation d'un algorithme reposant sur cette idée (initialement due à J.-M. Couveignes and B. Edixhoven), qui calcule les coefficients ap modulo l en calculant d'abord cette représentation modulo l, en s'appuyant sur le fait que pour k < ℓ, cette représentation est réalisée dans la ℓ-torsion de la jacobienne de la courbe modulaire X1 (ℓN ).Grâce à plusieurs améliorations, telles que l'utilisation des méthodes de K. KhuriMakdisi pour calculer dans la jacobienne modulaire J1(ℓN ) ou la construction d'une fonction a 2 Q (J1(ℓN )) au bon comportement arithmétique, cet algorithme est très efficace, ainsi qu'illustré par des tables de coefficients. Cette thèse se conclut par la présentation d'une méthode permettant de prouver formellement que les résultats de ces calculs sont corrects

Rigorous computation of the endomorphism ring of a Jacobian

We describe several improvements to algorithms for the rigorous computation of the endomorphism ring of the Jacobian of a curve defined over a number field

A method to prove that a modular Galois representation has large image

Let $\rho$ be a mod $\ell$ Galois representation attached to a newform $f$. Explicit methods are sometimes able to determine the image of $\rho$, or even the number field cut out by $\rho$, provided that $\ell$ and the level $N$ of $f$ are small enough; however these methods are not amenable to the case where $\ell$ or $N$ are large. The purpose of this short note is to establish a sufficient condition for the image of $\rho$ to be large and which remains easy to test for moderately large $\ell$ and $N$