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 $T\le k\le 2T$ as $T$ tends to infinity. This bound recovers the Weyl-strength subconvex bound $L(1\mathrm{/}2,f){\ll }_{\epsilon }{k}^{1\mathrm{/}3+\epsilon }$ and shows that for any $\delta >0$, the sub-Weyl subconvex bound $L(1\mathrm{/}2,f)\ll k^{1\mathrm{/}3-\delta }$ holds for all but $O_{\epsilon }(T^{12\delta +\epsilon })$ 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 $L(1\mathrm{/}2,f)$ weighted by a test function to a dual fourth moment weighted by a different test function.