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Weight aspect asymptotic and simultaneous non-vanishing for Rankin-Selberg $L$-functions

Aritra Ghosh

TL;DR

The article proves a weight-aspect asymptotic for the average of two central Rankin-Selberg L-values, establishing simultaneous non-vanishing in a GL(4) context without averaging over the critical line. The approach combines the approximate functional equation with the Petersson trace formula, reduces the problem to a diagonal main term and an off-diagonal that is controlled via a GL(2) Voronoi transform and shifted-convolution bounds. Central technical feats include a precise Voronoi summation analysis, handling of Bessel kernels, and a delta/Circle-method framework to bound shifted sums, culminating in an error term of size O(K^{1/2+ε}). This yields an explicit main-term polynomial in log K and confirms non-vanishing phenomena with improved quantitative strength, relevant for GL(4) automorphic lifts.

Abstract

In this article we show simultaneous non-vanishing of two Rankin-Selberg $L$-functions by proving an asymptotic result in weight aspect. The main input of this paper is to remove the $t$-integral dependence from the result of Blomer-Harcos (see \cite{BH2}) and getting a square root exponent for the error term.

Weight aspect asymptotic and simultaneous non-vanishing for Rankin-Selberg $L$-functions

TL;DR

The article proves a weight-aspect asymptotic for the average of two central Rankin-Selberg L-values, establishing simultaneous non-vanishing in a GL(4) context without averaging over the critical line. The approach combines the approximate functional equation with the Petersson trace formula, reduces the problem to a diagonal main term and an off-diagonal that is controlled via a GL(2) Voronoi transform and shifted-convolution bounds. Central technical feats include a precise Voronoi summation analysis, handling of Bessel kernels, and a delta/Circle-method framework to bound shifted sums, culminating in an error term of size O(K^{1/2+ε}). This yields an explicit main-term polynomial in log K and confirms non-vanishing phenomena with improved quantitative strength, relevant for GL(4) automorphic lifts.

Abstract

In this article we show simultaneous non-vanishing of two Rankin-Selberg -functions by proving an asymptotic result in weight aspect. The main input of this paper is to remove the -integral dependence from the result of Blomer-Harcos (see \cite{BH2}) and getting a square root exponent for the error term.
Paper Structure (25 sections, 13 theorems, 138 equations)

This paper contains 25 sections, 13 theorems, 138 equations.

Table of Contents

  1. Introduction
  2. Sketch of the proof
  3. Preliminaries
  4. Voronoi summation formula
  5. Bounds for the Bessel functions
  6. The Ramanujan bound on average
  7. $\mathrm{GL}(2)$ holomorphic cusp forms
  8. Rankin-Selberg $L$-function
  9. Proof of theorem \ref{['main']}
  10. Diagonal terms
  11. Off-diagonal terms
  12. Estimation of $R_{1}$
  13. Estimation of $R_{2}$
  14. Simplification of $R_{2}$
  15. Further estimation
  16. ...and 10 more sections

Key Result

Theorem 1

Let $f,h$ be two distinct fixed automorphic forms for $\mathrm{SL}(2,\mathbb{Z})$. Let $u \in \mathcal{C}_{c}^{\infty}(0,\infty)$. Then for $K$ large, we have where $P(\log x)$ is a polynomial in $\log x$, depending on $f,h$ only. $P(\log x)$ is a degree $2$ polynomial if $h\neq \overline{f}$, atleast one of $f,h$ is a cusp form; $P(\log x)$ is a degree $3$ polynomial if $h= \overline{f}$, $f$ is

Theorems & Definitions (26)

  • Theorem 1
  • Corollary 1
  • Remark 1
  • Remark 2
  • Theorem 2
  • Remark 3
  • Remark 4
  • Proposition 3.1.1
  • Proposition 3.2.1
  • Lemma 3.3.1
  • ...and 16 more