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 ${\mathrm{\Gamma }}_0(4)$ that is an eigenfunction of all Hecke operators $T_n$, so that $T_{n}f={\mathrm{\Lambda }}_f(n)n^{\frac{\kappa -1}{2}}f$. Let $\mathrm{\parallel }f\mathrm{\parallel }$ 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 $S_{\kappa }({\mathrm{\Gamma }}_0(4))$. This corresponds to studying the following sum: $$\sum _{f\in H_{\kappa }(4)}\frac{{\mathrm{\Lambda }}_f(n)\mathrm{\mid }L(1\mathrm{/}2,f){\mathrm{\mid }}^2}{\mathrm{\parallel }f{\mathrm{\parallel }}^2}\mathrm{.}$$ 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 $$L(1\mathrm{/}2,f){\ll }_{\epsilon }({\kappa }^2{)}^{\frac{1}{4}-\frac{1}{40}+\epsilon }\mathrm{.}$$ 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.