Hecke operators and Hilbert modular forms

TL;DR

This paper presents a method to compute the action of Hecke operators on the cohomology related to Hilbert modular forms over real quadratic fields, enabling analysis of these forms' properties.

Contribution

It introduces a novel technique for calculating Hecke operator actions on cohomology groups associated with Hilbert modular forms over real quadratic fields.

Findings

Technique successfully computes Hecke actions on cohomology

Enables analysis of cuspidal Hilbert modular forms

Applicable to fields with class number 1

Abstract

Let F be a real quadratic field with ring of integers O and with class number 1. Let Gamma be a congruence subgroup of GL_2 (O). We describe a technique to compute the action of the Hecke operators on the cohomology H^3 (Gamma; C). For F real quadratic this cohomology group contains the cuspidal cohomology corresponding to cuspidal Hilbert modular forms of parallel weight 2. Hence this technique gives a way to compute the Hecke action on these Hilbert modular forms.