Siegel modular forms of degree three and the cohomology of local systems
Siegel modular forms of degree three and the cohomology of local systems
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DOI：
10.1007/s00029-013-0118-6
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发表时间：
2011-08
期刊：
影响因子：
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通讯作者：
Jonas Bergström;C. Faber;G. Geer
中科院分区：
文献类型：
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作者：
Jonas Bergström;C. Faber;G. Geer
给出了主极化阿贝尔三叠模空间上任意辛局部系统的动力欧拉特性的一个显式推测公式。公式的主要项是某一类西格尔模形式的推测动机；其余的项承认一个令人惊讶的简单描述的动机欧拉特征为较低的属。这个猜想是基于有限域上的三属曲线和阿贝尔三折曲线的广泛计数。它提供了关于三次向量值西格尔模形式的许多新信息，如维数公式和Hecke算子的迹。我们也用它来预测从1到3的若干次提升，以及从hard类型的同余和新同余的提升。
We give an explicit conjectural formula for the motivic Euler characteristic of an arbitrary symplectic local system on the moduli spaceof 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 fromand new congruences of Harder type.