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The Weyl bound for Rankin-Selberg $L$-functions with Joint Ramification

Yunjian Peng

TL;DR

The paper proves a Weyl-type subconvex bound for the Rankin-Selberg L-function in a joint ramification setting by combining a refined Petersson trace formula with a bespoke Voronoi summation formula and a $p$-adic stationary phase analysis of local Whittaker integrals. Central technical contributions include establishing epsilon-factor invariance within a small family and obtaining square-root cancellation for a crucial local integral, enabling sharp control of the off-diagonal terms in the first moment. The approach yields explicit first-moment bounds in the level-aspect and, under suitable nonnegativity assumptions, translates these into the desired subconvex bounds in terms of the analytic conductor $N$, achieving a Weyl bound in the regime $1/3\le\delta\le 2/3$. This work advances subconvexity in joint ramification regimes and provides a framework potentially applicable to other joint-ramification L-functions and local-integral analyses.

Abstract

In this paper, we establish the Weyl bound for the Rankin-Selberg $L$-function in a certain joint ramification setting. To achieve this result, we employ the refined Petersson trace formula and develop a special Voronoï summation formula. Additionally, we obtain the sharp bound for the integral of products of Whittaker functions via the $p$-adic stationary phase method.

The Weyl bound for Rankin-Selberg $L$-functions with Joint Ramification

TL;DR

The paper proves a Weyl-type subconvex bound for the Rankin-Selberg L-function in a joint ramification setting by combining a refined Petersson trace formula with a bespoke Voronoi summation formula and a -adic stationary phase analysis of local Whittaker integrals. Central technical contributions include establishing epsilon-factor invariance within a small family and obtaining square-root cancellation for a crucial local integral, enabling sharp control of the off-diagonal terms in the first moment. The approach yields explicit first-moment bounds in the level-aspect and, under suitable nonnegativity assumptions, translates these into the desired subconvex bounds in terms of the analytic conductor , achieving a Weyl bound in the regime . This work advances subconvexity in joint ramification regimes and provides a framework potentially applicable to other joint-ramification L-functions and local-integral analyses.

Abstract

In this paper, we establish the Weyl bound for the Rankin-Selberg -function in a certain joint ramification setting. To achieve this result, we employ the refined Petersson trace formula and develop a special Voronoï summation formula. Additionally, we obtain the sharp bound for the integral of products of Whittaker functions via the -adic stationary phase method.

Paper Structure

This paper contains 15 sections, 42 theorems, 173 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation and prerequisites
  4. Rankin-Selberg L-function
  5. Compact induction and local Langlands correspondence
  6. Small families and the refined trace formula
  7. Local epsilon-factors
  8. Bounds for local integrals
  9. Basics of Whittaker functions
  10. The p-adic stationary phase method
  11. Main local result
  12. Voronoi summation formula
  13. Setting up the left side
  14. Computation the right hand side
  15. Complete the proof

Key Result

Theorem 1

Let $M=p^{\mathfrak c_1}$ with $\mathfrak c_1\ge 3$ and Let $g$ be a holomorphic cuspidal newform with level $N= p^{\mathfrak c_2}$, fixed weight $\iota \ge 4$. Then Furthermore suppose $L(f\times g,1/2) \ge 0$ for all $f\in \mathcal{F}_\theta [l]$. Suppose $\mathfrak c_1 = \delta \mathfrak c_2$ for $0<\delta<1$. By picking $l= l_0 \in \{ 0, 1\}$, we get the subconvex bound By picking $l$ to be

Theorems & Definitions (79)

  • Theorem 1
  • Remark 2
  • Lemma 3
  • proof
  • Definition 4
  • Definition 5
  • Lemma 6
  • Definition 7
  • Proposition 8
  • Definition 9
  • ...and 69 more