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On a Rankin-Selberg integral of three Hermitian cusp forms

Thanasis Bouganis, Rafail Psyroukis

TL;DR

The paper develops a unitary-group analogue of the Rankin–Selberg construction for a twisted spinor-type $L$-function by forming a triple product integral of a Siegel-type Hermitian Eisenstein series with two degree-2 Hermitian cusp forms and an elliptic cusp form, all of weight $k\equiv0\pmod4$. It derives a Dirichlet series $D_{F,G,h}(s)$ with analytic continuation and a functional equation, and, when $F$ lies in the Maass space, proves that $D_{F,G,h}(s)$ has an Euler product whose inert-prime factors match the twists of Gritsenko’s degree-6 $L$-function attached to $G$ by the elliptic form $h$. The inert-prime analysis yields a precise local factor $D_p^{(2)}(s)$ involving $L_p(f,k+s-2)$ and $L_p(f,k+s-2,(-4/p))$, while the split-prime case requires a GL$_4$-based factorization and yields a split-local formula in terms of Satake data and Jacobi coefficients. The work provides a pathway to algebraicity and $p$-adic interpolation phenomena for these twisted, higher-rank $L$-functions via triple products, extending Heim’s Siegel–Garrett framework to the unitary/Hermitian setting.

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, which has an analytic continuation to $\mathbb{C}$ 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.

On a Rankin-Selberg integral of three Hermitian cusp forms

TL;DR

The paper develops a unitary-group analogue of the Rankin–Selberg construction for a twisted spinor-type -function by forming a triple product integral of a Siegel-type Hermitian Eisenstein series with two degree-2 Hermitian cusp forms and an elliptic cusp form, all of weight . It derives a Dirichlet series with analytic continuation and a functional equation, and, when lies in the Maass space, proves that has an Euler product whose inert-prime factors match the twists of Gritsenko’s degree-6 -function attached to by the elliptic form . The inert-prime analysis yields a precise local factor involving and , while the split-prime case requires a GL-based factorization and yields a split-local formula in terms of Satake data and Jacobi coefficients. The work provides a pathway to algebraicity and -adic interpolation phenomena for these twisted, higher-rank -functions via triple products, extending Heim’s Siegel–Garrett framework to the unitary/Hermitian setting.

Abstract

Let . We study the Petersson inner product of a Hermitian Eisenstein series of Siegel type on the unitary group , diagonally-restricted on , against two Hermitian cuspidal eigenforms of degree and an elliptic cuspidal eigenform (seen as a Hermitian modular form of degree 1), all having weight . We obtain, through this consideration, an integral representation of a certain Dirichlet series, which has an analytic continuation to and functional equation, due to the one of the Eisenstein series. By taking to belong in the Maass space, we are able to show that the Dirichlet series possesses an Euler product. Moreover, its -factor for an inert prime can be essentially identified with the twist by of a degree six Euler factor attached to 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.
Paper Structure (15 sections, 48 theorems, 403 equations, 1 table)

This paper contains 15 sections, 48 theorems, 403 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hecke Rings and $L$-functions
  4. Integral Representation and Dirichlet Series
  5. Inert primes
  6. Hecke operators and weak rationality theorems
  7. Calculation of the Dirichlet Series
  8. Split Primes
  9. Hecke rings in $\textup{GL}_4$ and factorisation
  10. Hecke Operators and weak rationality theorems
  11. Calculation of the Dirichlet series - First Part
  12. Calculation of the Dirichlet Series - Second Part
  13. Calculation of the Dirichlet Series - Third Part
  14. Final expression for the Dirichlet series
  15. Euler Product

Key Result

Lemma 2.10

Let $\phi_{m}, \psi_{m} \in J_{k,m}^{2}$ and denote by $\tilde{\phi}_m, \tilde{\psi}_m$ the corresponding $P$-forms. Then for some (specified) constant $\beta_k$.

Theorems & Definitions (110)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • Lemma 2.10
  • ...and 100 more