Rankin-Cohen Brackets of Hilbert Hecke Eigenforms

TL;DR

This paper investigates the properties of Rankin-Cohen brackets of Hilbert Hecke eigenforms over totally real fields, establishing conditions under which these brackets are eigenforms and exploring related volume and dimension conjectures.

Contribution

It proves that Rankin-Cohen brackets of Hilbert Hecke eigenforms are mostly eigenforms, except in specific cases, and confirms a conjecture on the volume of Hilbert modular groups.

Findings

Rankin-Cohen brackets are eigenforms in most cases.

Confirmed Freitag's conjecture on Hilbert modular group volume.

Finiteness results on eigenform product identities under certain conjectures.

Abstract

Over any fixed totally real number field with narrow class number one, we prove that the Rankin-Cohen bracket of two Hecke eigenforms for the Hilbert modular group can only be a Hecke eigenform for dimension reasons, except for a couple of cases where the Rankin-Selberg method does not apply. We shall also prove a conjecture of Freitag on the volume of Hilbert modular groups, and assuming a conjecture of Freitag on the dimension of the cuspform space, we obtain a finiteness result on eigenform product identities.