Siegel varieties and $p$-adic Siegel modular forms

Abstract

In this paper, we present a conjecture concerning the classicality of a genus two overconvergent Siegel cusp eigenform whose associated Galois representation happens to be geometric, and more precisely, given by the Tate module of an abelian surface. This conjecture is inspired by the Fontaine-Mazur conjecture. It generalizes known results in the genus one case, due to Kisin, Buzzard-Taylor and Buzzard. The main difference in the genus two case is the complexity of the arithmetic geometry involved. This is why most of the paper consists in recalling (mostly with proofs) old and new results on the bad reduction of parahoric type Siegel varieties, with some consequences on their rigid geometry. Our conjecture would imply, in certain cases, a conjecture posed by H. Yoshida in 1980 on the modularity of abelian surfaces defined over the rationals.