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

2011/08/18 by Jonas Bergström, Carel Faber, Bergström, Jonas +3
Mathematics · #11F46 #11G18 #14G35 #14H10 #14J15 #14K10 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1108.3731

openalex publication_date 2011/08/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We give an explicit conjectural formula for the motivic Euler characteristic of an arbitrary symplectic local system on the moduli space A₃ 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 G₂ and new congruences of Harder type.

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