Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Dimension formulas for spaces of vector-valued Siegel modular forms of degree two and level two

Jonas Bergström, Fabien Cléry

TL;DR

The paper determines the $\mathfrak{S}_6$-isotypical decomposition of vector-valued Siegel modular forms of degree two and level two by combining cohomology of local systems on $\mathcal{A}_2[2]$ with Euler-characteristic calculations. It provides explicit dimension formulas and generating series for both scalar- and vector-valued cases, decomposing spaces into Eisenstein, Saito-Kurokawa, Yoshida, and general-type (G) packets, with detailed isotypical content under $\mathfrak{S}_6$. The main approach hinges on stratifying $\mathcal{A}_2[2]$ by automorphism groups of genus-two curves and pairs of elliptic curves, computing eigenvalues on $H^1$ and actions on Weierstrass points, and then translating these into $\mathfrak{S}_6$-representations via the Arthur packet framework. The results include proven decompositions (via Rösner) and practical, computable formulas for multiplicities, accompanied by Sage code and tables for implementation and further study in automorphic-representation contexts.

Abstract

Using a description of the cohomology of local systems on the moduli space of abelian surfaces with a full level two structure, together with a computation of Euler characteristics we find the isotypical decomposition, under the symmetric group on 6 letters, of spaces of vector-valued Siegel modular forms of degree two and level two.

Dimension formulas for spaces of vector-valued Siegel modular forms of degree two and level two

TL;DR

The paper determines the -isotypical decomposition of vector-valued Siegel modular forms of degree two and level two by combining cohomology of local systems on with Euler-characteristic calculations. It provides explicit dimension formulas and generating series for both scalar- and vector-valued cases, decomposing spaces into Eisenstein, Saito-Kurokawa, Yoshida, and general-type (G) packets, with detailed isotypical content under . The main approach hinges on stratifying by automorphism groups of genus-two curves and pairs of elliptic curves, computing eigenvalues on and actions on Weierstrass points, and then translating these into -representations via the Arthur packet framework. The results include proven decompositions (via Rösner) and practical, computable formulas for multiplicities, accompanied by Sage code and tables for implementation and further study in automorphic-representation contexts.

Abstract

Using a description of the cohomology of local systems on the moduli space of abelian surfaces with a full level two structure, together with a computation of Euler characteristics we find the isotypical decomposition, under the symmetric group on 6 letters, of spaces of vector-valued Siegel modular forms of degree two and level two.
Paper Structure (14 sections, 6 theorems, 75 equations, 1 table)

This paper contains 14 sections, 6 theorems, 75 equations, 1 table.

Table of Contents

  1. Introduction
  2. Siegel modular forms
  3. Isotypical decompositions in the scalar-valued case
  4. Isotypical decomposition of M 2k+1(Gamma[2])
  5. Isotypical decomposition of M 2k(Gamma[2])
  6. Isotypical decomposition of $E_{2k}(\Gamma[2])$
  7. Isotypical decomposition of $S_{2k}(\Gamma[2])$
  8. Euler characteristics of local systems
  9. Formulas for the Euler characteristics
  10. Smooth curves of genus two
  11. Pairs of elliptic curves
  12. Isotypical decomposition in the vector-valued case
  13. Isotypical decomposition of M k,j(Gamma[2])
  14. An isotypical dimension formula for S k,j(Gamma[2])

Key Result

Theorem 3.1

We have

Theorems & Definitions (17)

  • Remark 2.1
  • Remark 2.2
  • Theorem 3.1: Igusa
  • Theorem 3.2: Igusa
  • proof
  • Remark 3.3
  • Corollary 3.4
  • proof
  • Corollary 3.5
  • proof
  • ...and 7 more