Proefschrift

# Images of Galois representations

Promotores: S.J. Edixhoven, P.Parent

Auteur
Samuele Anni
Datum
24 oktober 2013
In this thesis we investigate $2$-dimensional, continuous, odd, residual Galois representations and their images. This manuscript consists of two parts. In the first part of this thesis we analyse a local\--global problem for elliptic curves over number fields. Let $E$ be an elliptic curve over a number field $K$, and let $\ell$ be a prime number. If $E$ admits an $\ell$-isogeny locally at a set of primes with density one then does $E$ admit an $\ell$-isogeny over $K$? The study of the Galois representation associated to the $\ell$-torsion subgroup of $E$ is the crucial ingredient used to solve the problem. We characterize completely the cases where the local\--global principle fails, obtaining an upper bound for the possible values of $\ell$ for which this can happen. In the second part of this thesis, we outline an algorithm for computing the image of a residual modular $2$-dimensional semi-simple Galois representation. This algorithm determines the image as a finite subgroup of $\GL_2(\overline{\F}_\ell)$, up to conjugation, as well as certain local properties of the representation and tabulate the result in a database. In this part of the thesis we show that, in almost all cases, in order to compute the image of such a representation it is sufficient to know the images of the Hecke operators up to the Sturm bound at the given level $n$. In addition, almost all the computations are performed in positive characteristic. In order to obtain such an algorithm, we study the local description of the representation at primes dividing the level and the characteristic: this leads to a complete description of the eigenforms in the old-space. Moreover, we investigate the conductor of the twist of a representation by characters and the coefficients of the form of minimal level and weight associated to it in order to optimize the computation of the projective image. The algorithm is designed using results of Dickson, Khare\--Wintenberger and Faber on the classification, up to conjugation, of the finite subgroups of $\PGL_2(\overline{\F}_\ell)$. We characterize each possible case giving a precise description and algorithms to deal with it. In particular, we give a new approach and a construction to deal with irreducible representations with projective image isomorphic to either the symmetric group on $4$ elements or the alternating group on $4$ or $5$ elements.