Action of Hecke operators on Siegel theta series II

TL;DR

This paper investigates how Hecke operators act on Siegel theta series, demonstrating that the average theta series becomes an eigenform and calculating its eigenvalues, enhancing understanding of their algebraic structure.

Contribution

It extends previous work by explicitly analyzing the action of Hecke operators on Siegel theta series and computing eigenvalues for the average theta series.

Findings

Average theta series is an eigenform under Hecke operators

Eigenvalues of the average theta series are explicitly computed

Hecke operators generate the local Hecke algebra at p

Abstract

Given a Siegel theta series and a prime p not dividing the level of the theta series, we apply to the theta series the n+1 Hecke operators that generate the local Hecke algebra at p. We show that the average theta series is an eigenform and we compute the eigenvalues.