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Second Moment of Central Values of Half-Integral Weight Modular Forms and Subconvexity

Steven Creech, Henry Twiss, Zhining Wei, Peter Zenz

TL;DR

This work proves a subconvexity bound in the weight aspect for L-functions attached to half-integral weight modular forms on $\Gamma_0(4)$ by establishing a half-integral Kuznetsov-type second-moment formula via a relative trace formula. The authors decompose the geometric side into singular and regular orbital integrals, compute main terms and bounds for each, and apply amplification using Shimura correspondence to derive a subconvexity bound $L(1/2,f)\ll_{\varepsilon}(\kappa^{2})^{1/4-1/40+\varepsilon}$ for $f$ in the Kohnen plus space (hence for all $f$). They also obtain a quantitative simultaneous non-vanishing result for central L-values, showing many half-integral weight forms yield $L(1/2,f)L(1/2,F\times\chi_D)\neq0$. The methods extend the relative trace formula approach to half-integral weight settings without Euler products and illustrate an amplification strategy via Shimura correspondences, contributing a first subconvexity result in this weight regime with explicit error terms.

Abstract

We let $f$ be a half-integral weight modular form of weight $κ>4$ on $Γ_0(4)$ that is an eigenfunction of all Hecke operators $T_n$, so that $T_nf = Λ_f(n)n^{\frac{κ-1}{2}}f$. Let $\|f\|$ denote the Petersson norm of $f$. We study a weighted second moment of the central value of the $L$-function associated to $f$ over an orthogonal basis $H_κ(4)$ of $S_κ(Γ_0(4))$. This corresponds to studying the following sum: $$\sum_{f\in H_κ(4)}\frac{Λ_f(n)\vert L(1/2,f)\vert^2}{\|f\|^2}.$$ Using the relative trace formula, we obtain an asymptotic formula for the second moment. We then use the method of amplification to get the subconvexity bound $$L(1/2,f)\ll_{\varepsilon} (κ^2)^{\frac{1}{4}-\frac{1}{40}+\varepsilon}.$$ This is the first subconvexity result for half-integral weight modular forms in the weight aspect. We also apply our second moment result to get a quantitative simultaneous non-vanishing result for central values of $L$-functions.

Second Moment of Central Values of Half-Integral Weight Modular Forms and Subconvexity

TL;DR

This work proves a subconvexity bound in the weight aspect for L-functions attached to half-integral weight modular forms on by establishing a half-integral Kuznetsov-type second-moment formula via a relative trace formula. The authors decompose the geometric side into singular and regular orbital integrals, compute main terms and bounds for each, and apply amplification using Shimura correspondence to derive a subconvexity bound for in the Kohnen plus space (hence for all ). They also obtain a quantitative simultaneous non-vanishing result for central L-values, showing many half-integral weight forms yield . The methods extend the relative trace formula approach to half-integral weight settings without Euler products and illustrate an amplification strategy via Shimura correspondences, contributing a first subconvexity result in this weight regime with explicit error terms.

Abstract

We let be a half-integral weight modular form of weight on that is an eigenfunction of all Hecke operators , so that . Let denote the Petersson norm of . We study a weighted second moment of the central value of the -function associated to over an orthogonal basis of . This corresponds to studying the following sum: Using the relative trace formula, we obtain an asymptotic formula for the second moment. We then use the method of amplification to get the subconvexity bound This is the first subconvexity result for half-integral weight modular forms in the weight aspect. We also apply our second moment result to get a quantitative simultaneous non-vanishing result for central values of -functions.
Paper Structure (28 sections, 30 theorems, 240 equations)

This paper contains 28 sections, 30 theorems, 240 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Strategy of the Proof
  4. Notation Guide
  5. The half-integral weight modular forms
  6. The $L$-functions
  7. The transformations of matrices
  8. Hypergeometric functions and the beta function
  9. The upper and lower triangular matrices
  10. Preliminaries
  11. The Complex Logarithm and Complex Power Functions
  12. The Hecke Operators
  13. The Kernel Function
  14. The Spectral Side
  15. The Geometric Side
  16. ...and 13 more sections

Key Result

Theorem A

Let $\kappa>4$ be a half-integer, and $n$ an odd square integer. Then where and where $g_*$ is the squarefree part of $g,$$g_0=\frac{\mathop{\mathrm{rad}}\nolimits(g)}{g_*}$ are divisors of $g$, and $C_{\kappa}$ is defined in eq. kappa constant. $J_{\operatorname{Reg}}(\textbf{0},n)$ is given in Proposition prop, the error term in RTF and satisfies

Theorems & Definitions (66)

  • Theorem A
  • Remark 1.2
  • Remark 1.3
  • Theorem B
  • Theorem C
  • Lemma 2.1
  • Proposition 2.2
  • Remark 2.3
  • Lemma 2.4
  • proof
  • ...and 56 more