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The subconvexity bound for standard L-function in level aspect

Yueke Hu, Paul Nelson

TL;DR

This work establishes a new subconvexity bound for the standard L-function L(π, 1/2) of a unitary cuspidal automorphic representation π on GL_n over Q in the level aspect, allowing a varying finite ramification set S under a uniform archimedean growth condition. The authors develop an S-adic framework combining localized Whittaker analysis, p-adic test vectors, and Archimedean Eisenstein-growth control, introducing an auxiliary parameter R to balance disparate contributions and a denominator-control mechanism for finite places. The main result yields |L(π, 1/2)| ≤ C_1 (T^n)^{1/4 - δ} C(π^S)^{1/2} for some δ > 0 and all twists by square-conductor characters, with a corresponding corollary for L(π ⊗ χ, 1/2 + it) and conductor N^2. This advances higher-rank subconvexity in the horizontal variation (varying ramification sets) by unifying previous archimedean/subconvex techniques with new S-adic localization and Rankin–Selberg analyses, providing a framework potentially adaptable to broader level-aspect problems in automorphic L-functions.

Abstract

In this paper we prove a new subconvexity result for the standard L-function of a unitary cuspidal automorphic representation $π$ of $\text{GL}_n$, where the finite set of places $S$ with large conductors is allowed to vary, provided that the local parameters at every place in $S$ satisfy certain uniform growth condition.

The subconvexity bound for standard L-function in level aspect

TL;DR

This work establishes a new subconvexity bound for the standard L-function L(π, 1/2) of a unitary cuspidal automorphic representation π on GL_n over Q in the level aspect, allowing a varying finite ramification set S under a uniform archimedean growth condition. The authors develop an S-adic framework combining localized Whittaker analysis, p-adic test vectors, and Archimedean Eisenstein-growth control, introducing an auxiliary parameter R to balance disparate contributions and a denominator-control mechanism for finite places. The main result yields |L(π, 1/2)| ≤ C_1 (T^n)^{1/4 - δ} C(π^S)^{1/2} for some δ > 0 and all twists by square-conductor characters, with a corresponding corollary for L(π ⊗ χ, 1/2 + it) and conductor N^2. This advances higher-rank subconvexity in the horizontal variation (varying ramification sets) by unifying previous archimedean/subconvex techniques with new S-adic localization and Rankin–Selberg analyses, providing a framework potentially adaptable to broader level-aspect problems in automorphic L-functions.

Abstract

In this paper we prove a new subconvexity result for the standard L-function of a unitary cuspidal automorphic representation of , where the finite set of places with large conductors is allowed to vary, provided that the local parameters at every place in satisfy certain uniform growth condition.

Paper Structure

This paper contains 56 sections, 57 theorems, 223 equations, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Uniform growth at finite places
  4. Main results
  5. Sketch of proof
  6. Organization of the paper
  7. Acknowledgment
  8. Basic notations
  9. Global
  10. Local
  11. Localized Whittaker functions
  12. Linear-algebraic preliminaries
  13. Stability
  14. Concentration of localized Whittaker functions
  15. Multiplicity one for localized vectors in generic representations
  16. ...and 41 more sections

Key Result

Theorem 1.1

Let $n$ be a natural number, and let $c_0 > 0$. For each $\delta < \delta_n^\sharp$, there exists $C_1 \geq 0$ with the following property. Let $\pi$ be a unitary cuspidal automorphic representation of $\mathop{\mathrm{GL}}\nolimits_n$ over $\mathbb{Q}$ such that each archimedean $L$-function parame Then

Theorems & Definitions (127)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Remark 1.4
  • Definition 3.1
  • Example 3.2
  • Definition 3.3
  • Proposition 3.4
  • Definition 3.5
  • Lemma 3.6
  • ...and 117 more