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

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.