On a Rankin-Selberg integral of three Hermitian cusp forms

TL;DR

This paper develops an integral representation for a Dirichlet series associated with Hermitian cusp forms on a unitary group, establishing its Euler product structure and analyzing local factors at inert primes.

Contribution

It introduces a new integral representation linking Hermitian cusp forms and Dirichlet series, demonstrating the Euler product property and analyzing local factors at inert primes.

Findings

Dirichlet series has an Euler product when F is in the Maass space.

p-factor at inert primes matches a twist of a degree six Euler factor.

Partial understanding of local factors at split primes achieved.

Abstract

Let $K = \mathbb{Q}(i)$. We study the Petersson inner product of a Hermitian Eisenstein series of Siegel type on the unitary group $U_{5}(K)$, diagonally-restricted on $U_2(K)\times U_2(K)\times U_1(K)$, against two Hermitian cuspidal eigenforms $F, G$ of degree $2$ and an elliptic cuspidal eigenform $h$ (seen as a Hermitian modular form of degree 1), all having weight $k \equiv 0 \pmod 4$. We obtain, through this consideration, an integral representation of a certain Dirichlet series, together with an additional residue term. By taking $F$ to belong in the Maass space, we are able to show that the Dirichlet series possesses an Euler product. Moreover, its $p$-factor for an inert prime $p$ can be essentially identified with the twist by $h$ of a degree six Euler factor attached to $G$ by Gritsenko. The question of whether the same holds for the primes that split remains unanswered here, even though we make considerable steps in that direction too. Our paper is inspired by a work of Heim, who considered a similar question in the case of Siegel modular forms.