We give an effective version of a result reported by Serre asserting that the images of the Galois representations attached to an abelian surface with End(A) = Z are as large as possible for almost every prime. Our algorithm depends on the truth of Serre's conjecture for two dimensional odd irreducible Galois representations. Assuming this conjecture we determine the finite set of primes with exceptional image. We also give infinite sets of primes for which we can prove (unconditionally) that the images of the corresponding Galois representations are large. We apply the results to a few examples of abelian surfaces.

We address the problem of the determination of the images of the Galois representations attached to genus 2 Siegel cusp forms of level 1 having multiplicity one. These representations are symplectic. We prove that the images are as large as possible for almost every prime, if the Siegel cusp form is not a Maass spezialform and verifies two easy to check conditions.

Abstract. In this article new cases of the inverse Galois problem are established. The main result is that for a fixed integer n, there is a positive density set of primes p such that PSL 2 (F p n ) occurs as the Galois group of some finite extension of the rational numbers. These groups are obtained as projective images of residual modular Galois representations. Moreover, families of modular forms are constructed such that the images of all their residual Galois representations are as large as a priori possible. Both results essentially use Khare's and Wintenberger's notion of good-dihedral primes. Particular care is taken in order to exclude nontrivial inner twists.

Let F be a totally real Galois number field. We prove the existence of base change relative to the extension F/Q for every holomorphic newform of weight at least 2 and odd level, under simple local assumptions on the field F .

In a previous article, we have proved a result asserting the existence of a compatible family of Galois representations containing a given crystalline irreducible odd two-dimensional representation. We apply this result to establish new cases of the Fontaine-Mazur conjecture, namely, an irreducible Barsotti-Tate λ-adic 2-dimensional Galois representation unramified at 3 and such that the traces a p of the images of Frobenii verify Q({a 2 p }) = Q always comes from an abelian variety. We also show the non-existence of irreducible Barsotti-Tate 2-dimensional Galois representations of conductor 1 and apply this to the irreducibility of Galois representations on level 1 genus 2 Siegel cusp forms.