Eigenform Product Identities For Hilbert Modular Forms

TL;DR

This paper proves that the product of two Hecke eigenforms in Hilbert modular forms is rarely an eigenform, with only finitely many exceptions across real quadratic fields, specifically identifying two such identities for Q(√5).

Contribution

It establishes finiteness results for eigenform product identities in Hilbert modular forms over real quadratic fields, including explicit identification for Q(√5).

Findings

Product of two Hecke eigenforms is not an eigenform in most cases.

Only finitely many exceptions exist across all real quadratic fields.

Exactly two identities are found for Q(√5).

Abstract

We prove that amongst all real quadratic fields and all spaces of Hilbert modular forms of full level and of weight $2$ or greater, the product of two Hecke eigenforms is not a Hecke eigenform except for finitely many real quadratic fields and finitely many weights. We show that for $\mathbb Q(\sqrt 5)$ there are exactly two such identities.