Siegel modular forms of degree three and the cohomology of local systems View Full Text


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Article Info

DATE

2013-03-12

AUTHORS

Jonas Bergström, Carel Faber, Gerard van der Geer

ABSTRACT

We give an explicit conjectural formula for the motivic Euler characteristic of an arbitrary symplectic local system on the moduli space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{A }_3$$\end{document} of principally polarized abelian threefolds. The main term of the formula is a conjectural motive of Siegel modular forms of a certain type; the remaining terms admit a surprisingly simple description in terms of the motivic Euler characteristics for lower genera. The conjecture is based on extensive counts of curves of genus three and abelian threefolds over finite fields. It provides a lot of new information about vector-valued Siegel modular forms of degree three, such as dimension formulas and traces of Hecke operators. We also use it to predict several lifts from genus 1 to genus 3, as well as lifts from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{G }_2$$\end{document} and new congruences of Harder type. More... »

PAGES

83-124

References to SciGraph publications

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  • 2008. Siegel Modular Forms and Their Applications in THE 1-2-3 OF MODULAR FORMS
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  • 1990. Degeneration of Abelian Varieties in NONE
  • 1971. Formes modulaires et représentations e-adiques in SÉMINAIRE BOURBAKI VOL. 1968/69 EXPOSÉS 347-363
  • 1993-12. On thel-adic cohomology of Siegel threefolds in INVENTIONES MATHEMATICAE
  • 1998-11. Motives with Galois Group of type G2>: an Exceptional Theta-Correspondence in COMPOSITIO MATHEMATICA
  • 2009. Endoscopy for GSp(4) and the Cohomology of Siegel Modular Threefolds in NONE
  • 1977. Modular forms whose fourier coefficients involve zeta-functions of quadratic fields in MODULAR FUNCTIONS OF ONE VARIABLE VI
  • 2012-09-11. Congruences for Hecke eigenvalues of Siegel modular forms in ABHANDLUNGEN AUS DEM MATHEMATISCHEN SEMINAR DER UNIVERSITÄT HAMBURG
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    http://dx.doi.org/10.1007/s00029-013-0118-6

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