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Sub-Weyl bound for $GL(2)$ via trivial delta

Roman Holowinsky, Ritabrata Munshi, Prahlad Sharma, Jakob Streipel

TL;DR

The paper proves a sub-Weyl bound for degree two $L$-functions in the $t$-aspect by refining the trivial delta method. After reducing $L(1/2+it,f)$ to short sums $S(N)=\sum_{n\sim N}\lambda(n)n^{-it}$ via the approximate functional equation, the authors separate variables with a trivial delta symbol and apply Voronoi and Poisson to obtain transformed sums. They treat the generic $N$ range near the Weyl barrier with traditional methods, then introduce a Diophantine-approximation refinement that lowers the analytic-conductor by concentrating on a single large modulus and exploiting Dirichlet approximations, followed by a second Voronoi step to extend cancellations; this yields a sub-Weyl bound $L(1/2+it,f)\ll t^{1/3-1/174+\varepsilon}$ for $SL(2,\mathbb Z)$ cusp forms, Maass forms, and Eisenstein series. The techniques open avenues for subconvexity results for GL(2) sums twisted by analytic weights and related GL(2) objects, with potential extensions to depth-aspect problems.

Abstract

For a $SL(2,\mathbb{Z})$ form $f$, we obtain the sub-Weyl bound \begin{equation*} L(1/2+it,f)\ll_{f,\varepsilon} t^{1/3-δ+\varepsilon}, \end{equation*} where $δ=1/174$, thereby crossing the Weyl barrier for the first time beyond $GL(1)$. The proof uses a refinement of the `trivial' delta method.

Sub-Weyl bound for $GL(2)$ via trivial delta

TL;DR

The paper proves a sub-Weyl bound for degree two -functions in the -aspect by refining the trivial delta method. After reducing to short sums via the approximate functional equation, the authors separate variables with a trivial delta symbol and apply Voronoi and Poisson to obtain transformed sums. They treat the generic range near the Weyl barrier with traditional methods, then introduce a Diophantine-approximation refinement that lowers the analytic-conductor by concentrating on a single large modulus and exploiting Dirichlet approximations, followed by a second Voronoi step to extend cancellations; this yields a sub-Weyl bound for cusp forms, Maass forms, and Eisenstein series. The techniques open avenues for subconvexity results for GL(2) sums twisted by analytic weights and related GL(2) objects, with potential extensions to depth-aspect problems.

Abstract

For a form , we obtain the sub-Weyl bound \begin{equation*} L(1/2+it,f)\ll_{f,\varepsilon} t^{1/3-δ+\varepsilon}, \end{equation*} where , thereby crossing the Weyl barrier for the first time beyond . The proof uses a refinement of the `trivial' delta method.

Paper Structure

This paper contains 19 sections, 16 theorems, 260 equations.

Table of Contents

  1. Introduction
  2. Proof outline
  3. Initial Setup
  4. Treating $t^{2/3-\delta}<N<t^{1-\eta}$ through "traditional" methods
  5. Treating $N\asymp t$ with a Diophantine approximation refinement
  6. Remaining obstructions
  7. Technical Obstacle: main terms in the off-diagonal
  8. Fundamental Barrier: necessary condition for Diophantine approximation
  9. Preliminaries
  10. Initial reduction via trivial delta
  11. The integral transform $\mathcal{I}_p(n,r)$
  12. Proof of Theorem \ref{['mainthm3']} and the previous "trivial delta" approach
  13. Zero frequency contribution/diagonal
  14. Off-diagonal
  15. The Weyl bound
  16. ...and 4 more sections

Key Result

Theorem 1

Let $f$ be either an $SL(2,\mathbb{Z})$ holomorphic Hecke cusp form, Hecke-Maass cusp form or the Eisenstein series $E(z,1/2)$. For $t\geq 1$,

Theorems & Definitions (34)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Remark 1
  • Remark 2
  • Remark 3
  • Definition
  • Lemma 1
  • proof
  • Lemma 2
  • ...and 24 more