Simultaneous computation of Hecke operators

TL;DR

This paper introduces a method for simultaneously computing two Hecke operators on algebraic modular forms, enabling efficient determination of the full Hecke algebra action and eigenforms for symplectic groups.

Contribution

It presents a novel approach based on Eichler's idea to compute multiple Hecke operators at once, improving computational efficiency and scope.

Findings

Successfully computes two Hecke operators simultaneously

Enables determination of the full Hecke algebra action

Computes Hecke eigenforms for compact symplectic groups

Abstract

We present a method to compute two Hecke operators acting on a space of algebraic modular forms simultaneously based on an idea of Eichler's. We show that in certain cases this method can be used to obtain the action of the full Hecke algebra with respect to a hyperspecial subgroup and use it to compute Hecke eigenforms for compact forms of symplectic groups.