Explicit Hecke eigenform product identities for Hilbert modular forms
Volume 224 / 2026
Acta Arithmetica 224 (2026), 141-157
MSC: Primary 11F41; Secondary 11F30
DOI: 10.4064/aa250926-10-4
Published online: 14 July 2026
Abstract
Let $F$ be a totally real number field, and $g,f,h$ be Hilbert modular forms over $F$ that are Hecke eigenforms satisfying $g=f\cdot h$. Under the grand Riemann hypothesis, we characterize such product identities among all real quadratic fields of narrow class number 1, 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.
