Geodesic
Siegel varieties and $p$-adic Siegel modular forms
Documenta mathematica, John H. Coates' Sixtieth Birthday (2006), pp. 781-817.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

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.
Classification : 11F33, 11F46, 11F80, 11G18
Keywords: arithmetic Siegel varieties, $q$-expansion, bad reduction of Siegel varieties of parahoric level, overconvergent Siegel modular forms, Galois representations 
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     author = {Tilouine, J.},
     title = {Siegel varieties and $p$-adic {Siegel} modular forms},
     journal = {Documenta mathematica},
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Tilouine, J. Siegel varieties and $p$-adic Siegel modular forms. Documenta mathematica, John H. Coates' Sixtieth Birthday (2006), pp. 781-817. http://geodesic.mathdoc.fr/item/DOCMA_2006__S5__a1/