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We prove Serre's conjecture for the case of Galois representations of Serre's weight 2 and level 1. We do this by combining the potential modularity results of Taylor and lowering the level for Hilbert modular forms with a Galois descent argument, properties of universal deformation rings, and the non-existence of p-adic Barsotti-Tate conductor 1 Galois representations that we proved in a previous paper.

In this article we give a proof of Serre's conjecture for the case of odd level and arbitrary weight. Our proof will not depend on any generalization of Kisin's modularity lifting results to characteristic 2(moreover, we will not consider at all characteristic 2 representations in any step of our proof). The key tool in the proof is Kisin's recent modularity lifting result, which is combined with the methods and results of previous articles on Serre's conjecture by Khare, Wintenberger, and the author, and modularity results of Schoof for semistable abelian varieties of small conductor. Assuming GRH, infinitely many cases of even level will also be proved.

In their paper [9], P. Paillier and J. Villar make a conjectur e about the malleability of an RSA modulus. In this paper we present an ex plicit algo- rithm refuting the conjecture. Concretely we can factorize an RSA modulus n using very little information on the factorization of a conc rete n ′ coprime to n . However, we believe the conjecture might be true, when impo sing some extra conditions on the auxiliary n ′ allowed to be used. In particular, the paper shows how subtle the notion of malleability is

Using results of Ribet on the images of Galois representations attached to modular forms, we realize linear groups as Galois groups over Q. In particular, combining with results of Brumer we prove that there exist infinitely many exponents r for which PXL_2(p^r) are Galois groups over Q for infinitely many primes p, where PXL means PSL or PGL. (Note: this paper dates from 1998, it is the author's first research work as a PhD student, before the thesis).

In this article we study the behavior of inertia groups for modular Galois mod ℓn representations and in some cases we give a generalization of Ribet’s lowering the level result.

We prove existence of conjugate Galois representations, and we use it to derive a simple method of weight reduction. As a consequence, an alternative proof of the level 1 case of Serre's conjecture follows., very minor changes

We prove the following uniformity principle: if one of the Galois representations in the family attached to a genus two Siegel cusp form of weight $k>3$, "semistable" and with multiplicity one, is reducible (for an odd prime $p$),then all the representations in the family are reducible. This, combined with Serre's conjecture (which is now a theorem) gives a proof of the Endoscopy Conjecture., revised version, to appear in Bol. Soc. Mat. Mexicana