Cohomology and L-values

TL;DR

This paper extends Shimura's cohomological approach to compute critical L-values from elliptic modular forms to Hilbert modular forms over real quadratic fields, providing explicit numerical examples.

Contribution

It introduces a method using second cohomology groups to calculate L-values for Hilbert modular forms, expanding on Shimura's original approach.

Findings

Explicit numerical examples demonstrating the method

Extension of cohomological calculation to Hilbert modular forms

Validation of the approach with concrete data

Abstract

In a paper published in 1959, Shimura presented an elegant calculation of the critical values of L-functions attached to elliptic modular forms using the first cohomology group. We will show that a similar calculation is possible for Hilbert modular forms over real quadratic fields using the second cohomology group. We present explicit numerical examples calculated by this method.