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

2025/12/23 by Steven Creech, Creech, Steven, Henry Twiss +5
Mathematics · #Advanced Algebra and Geometry #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2512.20483

openalex publication_date 2025/12/23 · openalex created_date 2025/12/25 · openalex updated_date 2026/07/28

Abstract

We let f be a half-integral weight modular form of weight κ>4 on Γ₀(4) that is an eigenfunction of all Hecke operators T_n, so that T_nf = Λ_f(n)n^(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(Γ₀(4)). This corresponds to studying the following sum: ∑_{f∈ Hκ(4)}(Λ_f(n)\vert L(1/2,f)\vert²)/(‖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)≪_ε (κ²)^{(1)/(4)-(1)/(40)+ε}. 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.

Citations

• A note on a classical relative trace formula
• Relative Trace Formula and Uniform non-vanishing of Central L-values of Hilbert Modular Forms
• Moments of L-functions and Liouville-Green method
• Epstein zeta-functions, subconvexity, and the purity conjecture
• Non-vanishing of L-functions associated to cusp forms of half-integral weight
• Hecke operators in half-integral weight
• Integral moments of L-functions
• Integral moments of L-functions
• Determination of modular forms by twists of critical L-values
• On Calculations of Zeros of Various L-functions
• Remarks on the generalized Lindelöf Hypothesis

Related

• The twisted second moment of modular half integral weight L--functions
• Signs of Fourier coefficients of half-integral weight modular forms
• On Hecke L-functions attached to half-integral weight modular forms
• On Modular Forms of Half Integral Weight
• The first moment of central values of symmetric square L-functions in the weight aspect