On a Rankin–Selberg integral of three Hermitian cusp forms

Abstract

Let K=Q(i)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K = \mathbb {Q}(i)$$\end{document}. We study the Petersson inner product of a Hermitian Eisenstein series of Siegel type on the unitary group U5(K)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_{5}(K)$$\end{document}, diagonally-restricted on U2(K)×U2(K)×U1(K)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_2(K)\times U_2(K)\times U_1(K)$$\end{document}, 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)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k \equiv 0 \pmod 4$$\end{document}. 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.