Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Results on the Non-Vanishing of Derivatives of L-Functions of Vector-Valued Modular Forms

Subong Lim, Wissam Raji

TL;DR

The paper proves non-vanishing results for averages of derivatives of $L$-functions attached to vector-valued cusp forms of half-integer weight, by constructing a kernel function $R_{k,s,i}$ and relating its Petersson inner product with $L^*(f,s)$ to obtain non-vanishing in a prescribed critical strip. The main theorem shows that for large weight $k$ the weighted sum of derivatives does not vanish on the line $t=t_0$ in the interval $(\frac{k-1}{2},\frac{k}{2}-\epsilon)$, ensuring the existence of at least one basis element with a non-zero derivative of its $L^*$-function. The approach extends to elliptic cusp forms on $\Gamma_0(N)$, Jacobi forms via theta correspondences, and Kohnen's plus space through standard isomorphisms, yielding corresponding non-vanishing results for the relevant $L$-functions and their derivatives across these settings. This broadens non-vanishing phenomena to half-integer weight vector-valued forms and related automorphic liftings, with potential arithmetic implications for special values and derivatives of $L$-functions.

Abstract

We show a non-vanishing result for the averages of the derivatives of $L$-functions associated with the orthogonal basis of the space of vector-valued cusp forms of weight $k\in \frac12 \mathbb{Z}$ on the full group in the critical strip. We also show the existence of at least one basis element whose $L$-function does not vanish under certain conditions. As an application, we generalize our result to Kohnen's plus space and prove an analogous result for Jacobi forms.

Results on the Non-Vanishing of Derivatives of L-Functions of Vector-Valued Modular Forms

TL;DR

The paper proves non-vanishing results for averages of derivatives of -functions attached to vector-valued cusp forms of half-integer weight, by constructing a kernel function and relating its Petersson inner product with to obtain non-vanishing in a prescribed critical strip. The main theorem shows that for large weight the weighted sum of derivatives does not vanish on the line in the interval , ensuring the existence of at least one basis element with a non-zero derivative of its -function. The approach extends to elliptic cusp forms on , Jacobi forms via theta correspondences, and Kohnen's plus space through standard isomorphisms, yielding corresponding non-vanishing results for the relevant -functions and their derivatives across these settings. This broadens non-vanishing phenomena to half-integer weight vector-valued forms and related automorphic liftings, with potential arithmetic implications for special values and derivatives of -functions.

Abstract

We show a non-vanishing result for the averages of the derivatives of -functions associated with the orthogonal basis of the space of vector-valued cusp forms of weight on the full group in the critical strip. We also show the existence of at least one basis element whose -function does not vanish under certain conditions. As an application, we generalize our result to Kohnen's plus space and prove an analogous result for Jacobi forms.
Paper Structure (6 sections, 6 theorems, 56 equations)

This paper contains 6 sections, 6 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. The Kernel Function
  3. The Main Theorem
  4. The case of $\Gamma_0(N)$
  5. The case of Jacobi forms
  6. The case of Kohnen plus space

Key Result

Lemma 2.2

LR The function $R_{k,s,i}$ has the Fourier expansion where $r_{k,s,i,j}(n)$ is given by where $_1 F_1(\alpha, \beta;z)$ is Kummer's degenerate hypergeometric function.

Theorems & Definitions (10)

  • Definition 2.1
  • Lemma 2.2
  • Theorem 3.1
  • proof
  • Corollary 3.2
  • Corollary 4.1
  • Definition 5.1
  • Corollary 5.2
  • Remark 5.3
  • Corollary 6.1