Explicit Hecke eigenform product identities for Hilbert modular forms

TL;DR

This paper characterizes when products of Hilbert modular forms are eigenforms over real quadratic fields, finding only two such identities over () and none involving Eisenstein series of different weights.

Contribution

It provides a complete classification of product identities for Hilbert modular eigenforms over real quadratic fields with narrow class number one, highlighting the special case of ().

Findings

Only two product identities occur over ().

No identities exist when both forms are Eisenstein series of different weights.

The results are specific to real quadratic fields with narrow class number one.

Abstract

Let $F$ be a totally real number field, and $g,f,h$ be Hilbert modular forms over $F$ that are Hecke eigenforms satisfying $g=f\cdot h$. We characterize such product identities among all real quadratic fields of narrow class number one, proving they occur only for $F=\mathbb Q(\sqrt{5})$, with precisely two such identities. We also shed some light on the general totally real case by showing that no such identity exists when both $f$ and $h$ are Eisenstein series of distinct weights.