#### DOCUMENTA MATHEMATICA,
Vol. Extra Volume: John H. Coates' Sixtieth Birthday (2006), 781-817

** J. Tilouine **
Siegel Varieties and p-Adic Siegel Modular Forms

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.

2000 Mathematics Subject Classification: 11F33, 11F46, 11F80, 11G18

Keywords and Phrases: Arithmetic Siegel varieties, $q$-expansion, Bad reduction of Siegel varieties
of parahoric level, Overconvergent Siegel modular forms, Galois representations

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