This is joint work with F. Andreatta, G. Stevens and V. Pilloni in which we construct p-adic families of overconvergent, finite slope Hilbert modular cuspforms attached to a prime integer p>2 and a totally real number field F (in which p might be ramified).
En collaboration avec B. Stroh, nous démontrons la modularité de certaines représentations p-adiques impaires de dimension 2 des groupes de Galois des corps totalement réels, à poids de Hodge-Tate nuls. Ces résultats permettent en particulier d'achever la démonstration de la conjecture d'Artin dans ce contexte.
The aim of this talk is to explain the subtle but powerful link between algebraicity results for critical/central values of automorphic L-functions and the nonvanishing of these values in families, particularly in the setting of Rankin-Selberg L-functions of GL(2) of a totally real number field. In particular, I will explain how a strategy involving p-adic L-functions can be used to reduce a natural generalization of Mazur's conjectures to the non-self dual setting to some relatively simple analytic estimates. If time permits, then I will also outline some applications (e.g. to bounding Mordell-Weil ranks) and open problems.