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Siegel varieties and \(p\)-adic Siegel modular forms. (English) Zbl 1136.11034

In this paper the author presents an interesting conjecture concerning the classicality and the level of a genus 2 overconvergent Siegel cusp eigenform whose associated local Galois representation is isomorphic to the contragredient of the representation on the \(p\)-adic Tate module of an abelian surface defined over the rationals. Several results in the analogous situation for genus 1 [see K. Buzzard and R. Taylor, “Companion forms and weight one forms”, Ann. Math. (2) 149, No. 3, 905–919 (1999; Zbl 0965.11019); K. Buzzard, “Analytic continuation of overconvergent eigenforms”, J. Am. Math. Soc. 16, No. 1, 29–55 (2003; Zbl 1076.11029); M. Kisin, “Overconvergent modular forms and the Fontaine-Mazur conjecture”, Invent. Math. 153, No. 2, 373–454 (2003; Zbl 1045.11029)] lead the author to make this conjecture.
Moreover in this paper the author gathers useful geometric facts about Siegel threefolds with parahoric level \(p\), which are necessary for the study of the analytic continuation of such overconvergent cusp eigenforms to the whole compactified Siegel threefold. Thanks to the rigid GAGA principle, this would imply the claimed classicality. The conjecture would imply, in certain cases, a conjecture of Yoshida (1980) on the modularity of abelian surfaces defined over the rationals.

MSC:

11F33 Congruences for modular and \(p\)-adic modular forms
11F46 Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms
11F80 Galois representations
11G18 Arithmetic aspects of modular and Shimura varieties
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