Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Zeros of Rankin-Selberg $L$-functions in families

Peter Humphries, Jesse Thorner

Abstract

Let $\mathfrak{F}_n$ be the set of all cuspidal automorphic representations $π$ of $\mathrm{GL}_n$ with unitary central character over a number field $F$. We prove the first unconditional zero density estimate for the set $\mathcal{S}=\{L(s,π\timesπ')\colonπ\in\mathfrak{F}_n\}$ of Rankin-Selberg $L$-functions, where $π'\in\mathfrak{F}_{n'}$ is fixed. We use this density estimate to establish (i) a hybrid-aspect subconvexity bound at $s=\frac{1}{2}$ for almost all $L(s,π\timesπ')\in \mathcal{S}$, (ii) a strong on-average form of effective multiplicity one for almost all $π\in\mathfrak{F}_n$, and (iii) a positive level of distribution for $L(s,π\times\tildeπ)$, in the sense of Bombieri-Vinogradov, for each $π\in\mathfrak{F}_n$.

Zeros of Rankin-Selberg $L$-functions in families

Abstract

Let be the set of all cuspidal automorphic representations of with unitary central character over a number field . We prove the first unconditional zero density estimate for the set of Rankin-Selberg -functions, where is fixed. We use this density estimate to establish (i) a hybrid-aspect subconvexity bound at for almost all , (ii) a strong on-average form of effective multiplicity one for almost all , and (iii) a positive level of distribution for , in the sense of Bombieri-Vinogradov, for each .

Paper Structure

This paper contains 18 sections, 23 theorems, 160 equations.

Table of Contents

  1. Introduction and statement of the main result
  2. Applications
  3. Bounds for Rankin--Selberg L-functions
  4. Effective multiplicity one
  5. Automorphic level of distribution
  6. Properties of L-functions
  7. Standard L-functions
  8. Rankin--Selberg L-functions
  9. Rankin--Selberg combinatorics
  10. A new mean value estimate
  11. Proof of Theorem \ref{['thm:ZDE']}
  12. Bounds for Rankin--Selberg $L$-functions
  13. Effective multiplicity one on average
  14. Automorphic level of distribution
  15. Preliminaries
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $n,n'\geq 1$ and $\varepsilon>0$. Let $\mathcal{S}\subseteq\mathfrak{F}_n$ and $\mathcal{S}(Q)=\{\pi\in\mathcal{S}\colon C(\pi)\leq Q\}$. If $0\leq\sigma\leq 1$, $\pi'\in\mathfrak{F}_{n'}$, and $Q,T\geq 1$, then

Theorems & Definitions (46)

  • Theorem 1.1
  • Theorem 2.1
  • Remark 2.2
  • Theorem 2.3
  • Remark 2.4
  • Theorem 2.5
  • Remark 2.6
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 36 more