The Twelfth Moment of Hecke $L$-Functions in the Weight Aspect

We prove an upper bound for the twelfth moment of Hecke $L$-functions associated to holomorphic Hecke cusp forms of weight $k$ in a dyadic interval $T \leq k \leq 2T$ as $T$ tends to infinity. This bound recovers the Weyl-strength subconvex bound $L(1/2,f) \ll_{\varepsilon} k^{1/3 + \varepsilon}$ and shows that for any $\delta > 0$, the sub-Weyl subconvex bound $L(1/2,f) \ll k^{1/3 - \delta}$ holds for all but $O_{\varepsilon}(T^{12\delta + \varepsilon})$ Hecke cusp forms $f$ of weight at most $T$. Our result parallels a related result of Jutila for the twelfth moment of Hecke $L$-functions associated to Hecke-Maass cusp forms. The proof uses in a crucial way a spectral reciprocity formula of Kuznetsov that relates the fourth moment of $L(1/2,f)$ weighted by a test function to a dual fourth moment weighted by a different test function.