Tilouine, J. Siegel varieties and \(p\)-adic Siegel modular forms. (English) Zbl 1136.11034 Doc. Math. Extra Vol., John H. Coates’ Sixtieth Birthday, 781-817 (2006). 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. Reviewer: Miriam Ciavarella (Torino) Cited in 9 Documents 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 Keywords:arithmetic Siegel varieties; \(q\)-expansion; bad reduction of Siegel varieties of parahoric level; overconvergent Siegel modular forms; Galois representations Citations:Zbl 0965.11019; Zbl 1076.11029; Zbl 1045.11029 PDFBibTeX XMLCite \textit{J. Tilouine}, Doc. Math. Extra Vol., 781--817 (2007; Zbl 1136.11034) Full Text: EuDML EMIS