On a Rankin-Selberg integral of three Hermitian cusp forms

Bouganis, Athanasios; Psyroukis, Rafail

Abstract

Let K=Q(i). We study the Petersson inner product of a Hermitian Eisenstein series of Siegel type on the unitary group U5(K), diagonally-restricted on U2(K)×U2(K)×U1(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≡0(mod4). We obtain, through this consideration, an integral representation of a certain Dirichlet series, which has an analytic continuation to C and functional equation, due to the one of the Eisenstein series. 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.

Citation

Bouganis, A., & Psyroukis, R. (in press). On a Rankin-Selberg integral of three Hermitian cusp forms. Annales mathématiques du Québec,

• Journal Article Type: Article
• Acceptance Date: Feb 28, 2025
• Deposit Date: Nov 9, 2023
• Journal: Annales mathématiques du Québec
• Print ISSN: 2195-4755
• Electronic ISSN: 2195-4763
• Publisher: Springer
• Peer Reviewed: Peer Reviewed
• Public URL: https://durham-repository.worktribe.com/output/1903066
• Publisher URL: https://link.springer.com/journal/40316
• Related Public URLs: https://arxiv.org/abs/2309.17237