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A note on the twisted degree $6$ $L$-function for Hermitian cusp forms of degree $2$

Rafail Psyroukis

TL;DR

This paper addresses the analytic properties of the twisted degree six $L$-function attached to a Hermitian cusp form $F$ of degree two over $\mathbb{Q}(i)$. By combining Gritsenko's Hermitian framework with the twisted Dirichlet-series methods of Das–Jha and exploiting Fourier–Jacobi expansions, it constructs a completed twisted $L$-function $Z_{F}^{*}(s,\chi)$ and derives its Euler product structure. The main accomplishment is establishing meromorphic continuation to $\mathbb{C}$ and a functional equation $Z_{F}^{*}(2k-3-s,\chi)=Z_{F}^{*}(s,\chi)$, with specific possible simple poles when $\chi$ is principal, by relating twisted Dirichlet series to $Z_{F}^{*}$ through an inner-product identity involving a Jacobi cusp form in the Maass space. This advances the understanding of twisted $L$-functions for Hermitian modular forms and provides a bridge between parallel Hermitian and Siegel frameworks, enabling potential arithmetic applications.

Abstract

Let $F$ be a cuspidal Hermitian eigenform of degree two over $\mathbb{Q}(i)$, with first Fourier-Jacobi coefficient not identically zero. Building on a paper by Das and Jha, we prove the meromorphic continuation to $\mathbb{C}$ and the functional equation of a degree six $L$-function attached to $F$ by Gritsenko, twisted by a Dirichlet character.

A note on the twisted degree $6$ $L$-function for Hermitian cusp forms of degree $2$

TL;DR

This paper addresses the analytic properties of the twisted degree six -function attached to a Hermitian cusp form of degree two over . By combining Gritsenko's Hermitian framework with the twisted Dirichlet-series methods of Das–Jha and exploiting Fourier–Jacobi expansions, it constructs a completed twisted -function and derives its Euler product structure. The main accomplishment is establishing meromorphic continuation to and a functional equation , with specific possible simple poles when is principal, by relating twisted Dirichlet series to through an inner-product identity involving a Jacobi cusp form in the Maass space. This advances the understanding of twisted -functions for Hermitian modular forms and provides a bridge between parallel Hermitian and Siegel frameworks, enabling potential arithmetic applications.

Abstract

Let be a cuspidal Hermitian eigenform of degree two over , with first Fourier-Jacobi coefficient not identically zero. Building on a paper by Das and Jha, we prove the meromorphic continuation to and the functional equation of a degree six -function attached to by Gritsenko, twisted by a Dirichlet character.

Paper Structure

This paper contains 4 sections, 4 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hecke Rings and the degree 6 $L$-function
  4. Main Theorem

Key Result

Lemma 3.1

Let $(\Gamma_0, S_0)$ and $(\Gamma, S)$ be two Hecke pairs. We assume that Then, given an arbitrary element $X \in H(\Gamma, S)$, according to the second condition, we can write it as with $g_i \in S_0$. Then, if we set then $\epsilon$ does not depend on the selection of the elements $g_i \in S_0$ and is an embedding (as a ring homomorphism) of the Hecke algebra $H(\Gamma, S)$ to $H(\Gamma_0, S

Theorems & Definitions (18)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Lemma 3.1
  • proof
  • Definition 3.1
  • ...and 8 more