Let p be an odd prime and g ≥ 2 an integer. We prove that a finite slope Siegel cuspidal eigenform of genus g can be p-adically deformed over the g-dimensional weight space. The proof of this theorem relies on the construction of a family of sheaves of locally analytic overconvergent modular forms.
We show that abelian surfaces (and consequently curves of genus 2) over totally real fields are potentially modular. As a consequence, we obtain the expected meromorphic continuation and functional equations of their Hasse–Weil zeta functions. We furthermore show the modularity of infinitely many abelian surfaces $A$
A
over ${\mathbf {Q}}$
Q
with $\operatorname{End}_{ {\mathbf {C}}}A={\mathbf {Z}}$
End
C
A
=
Z
. We also deduce modularity and potential modularity results for genus one curves over (not necessarily CM) quadratic extensions of totally real fields.
Dans cet article, on montre comment les idées introduites dans l'article [S4] s'appliquentà l'étude de la cohomologie cohérente des variétés de Siegel, et plus généralement des variétés de Shimura de type Hodge. Le résultat principal affirme que les classes de cohomologie cohérente supérieure sont des limites p-adique de formes modulaires cuspidales. Ceci permet dans certains cas d'associer des représentations Galoisiennesà des formes automorphes apparaissant dans la cohomologie cohérente.
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