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Selecta Mathematica

, Volume 20, Issue 1, pp 83–124 | Cite as

Siegel modular forms of degree three and the cohomology of local systems

  • Jonas Bergström
  • Carel Faber
  • Gerard van der Geer
Article

Abstract

We give an explicit conjectural formula for the motivic Euler characteristic of an arbitrary symplectic local system on the moduli space \(\mathcal{A }_3\) 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 \(\mathrm{G }_2\) and new congruences of Harder type.

Keywords

Siegel modular forms Moduli of abelian varieties Symplectic local systems Euler characteristic Lefschetz trace formula Hecke operators  Moduli of curves 

Mathematics Subject Classification (2010)

11F46 11G18 14G35 14J15 14K10 14H10 

Notes

Acknowledgments

The authors thank Pierre Deligne, Neil Dummigan, Benedict Gross, Günter Harder, Anton Mellit, and Don Zagier for their contributions, and the Max Planck Institute for Mathematics in Bonn for hospitality and excellent working conditions. We are very grateful to Maarten Hoeve for assistance with the computer programing and we thank Hidenori Katsurada for his remarks about congruences. Finally, we thank the referee. The second author was supported by the Göran Gustafsson Foundation for Research in Natural Sciences and Medicine (KVA) and grant 622-2003-1123 from the Swedish Research Council. The third author’s visit to KTH in October 2010 was supported by the Göran Gustafsson Foundation for Research in Natural Sciences and Medicine (UU/KTH).

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Copyright information

© Springer Basel 2013

Authors and Affiliations

  • Jonas Bergström
    • 1
  • Carel Faber
    • 2
  • Gerard van der Geer
    • 3
  1. 1.Matematiska institutionen, Stockholms UniversitetStockholmSweden
  2. 2.Institutionen för matematik, KTH Royal Institute of TechnologyStockholmSweden
  3. 3.Korteweg-de Vries Instituut, Universiteit van AmsterdamAmsterdamThe Netherlands

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