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

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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.

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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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Authors and Affiliations

  1. Matematiska institutionen, Stockholms Universitet, 106 91 , Stockholm, Sweden

    Jonas Bergström

  2. Institutionen för matematik, KTH Royal Institute of Technology, 100 44 , Stockholm, Sweden

    Carel Faber

  3. Korteweg-de Vries Instituut, Universiteit van Amsterdam, Postbus 94248 , 1090 GE , Amsterdam, The Netherlands

    Gerard van der Geer

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  1. Jonas Bergström
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  2. Carel Faber
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  3. Gerard van der Geer
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Correspondence to Carel Faber.

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To the memory of Torsten Ekedahl

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Bergström, J., Faber, C. & van der Geer, G. Siegel modular forms of degree three and the cohomology of local systems. Sel. Math. New Ser. 20, 83–124 (2014). https://doi.org/10.1007/s00029-013-0118-6

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