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

We let $f$ be a half-integral weight modular form of weight $\kappa>4$ on $\Gamma_0(4)$ that is an eigenfunction of all Hecke operators $T_n$, so that $T_nf = \Lambda_f(n)n^{\frac{\kappa-1}{2}}f$. 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 $H_\kappa(4)$ of $S_{\kappa}(\Gamma_0(4))$. This corresponds to studying the following sum: $$\sum_{f\in H_\kappa(4)}\frac{\Lambda_f(n)\vert L(1/2,f)\vert^2}{\|f\|^2}.$$ 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 $$L(1/2,f)\ll_{\varepsilon} (\kappa^2)^{\frac{1}{4}-\frac{1}{40}+\varepsilon}.$$ 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.