Multiplier systems for Siegel modular groups

TL;DR

This paper provides an alternative proof that weights of Siegel modular forms on congruence subgroups are integral or half-integral, utilizing subgroup properties and techniques from algebraic group theory.

Contribution

It offers a new proof of Deligne's result on the weights of Siegel modular forms, using Mennicke's subgroup classification and Bass-Milnor-Serre techniques.

Findings

Weights are integral or half-integral for Siegel modular forms

Subgroups of finite index are congruence subgroups

New proof simplifies understanding of modular form weights

Abstract

Deligne proved that the weights of Siegel modular forms on any congruence subgroup of the Siegel modular group of genus g>1 must be integral or half integral. We give a different proof for this. It uses Mennicke's result that subgroups of finite index of the Siegel modular group are congruence subgroups and some techniques from a paper of Bass-Milnor-Serre.