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

Jonas Bergström ^[1] ; Carel Faber ^[2] ; Gerard van der Geer ^[3]
1. [1] Stockholm University
  
  Stockholm University
  
  Suecia
2. [2] Royal Institute of Technology
  
  Royal Institute of Technology
  
  Suecia
3. [3] University of Amsterdam
  
  University of Amsterdam
  
  Países Bajos
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Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 20, Nº. 1, 2014, págs. 83-124
Idioma: inglés
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Resumen
- We give an explicit conjectural formula for the motivic Euler characteristic of an arbitrary symplectic local system on the moduli space A3A3 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 G2G2 and new congruences of Harder type.