Periods of Hilbert Modular forms, Kronecker series and Cohomology

TL;DR

This paper generalizes previous results on elliptic modular forms to Hilbert modular forms over totally real fields, providing a closed formula for sums involving period polynomials and Kronecker series.

Contribution

It introduces a new closed-form expression for sums of Hilbert Hecke eigenforms' period polynomials using Kronecker series, extending prior elliptic modular form results.

Findings

Derived a closed formula for Hilbert modular forms' period sums

Connected period polynomials with Kronecker series in a novel way

Extended elliptic modular form results to totally real fields

Abstract

Generalizing a result of \cite{Z1991, CPZ} about elliptic modular forms, we give a closed formula for the sum of all Hilbert Hecke eigenforms over a totally real number field with strict class number $1$, multiplied by their period polynomials, as a single product of the Kronecker series.