The cubic moment of $L$-functions for specified local component families

We prove Lindel\"of-on-average upper bounds on the cubic moment of central values of $L$-functions over certain families of $\operatorname{PGL}_2/\mathbb{Q}$ automorphic representations $\pi$ given by specifying the local representation $\pi_p$ of $\pi$ at finitely many primes. Such bounds were previously known in the case that $\pi_p$ belongs to the principal series or is a ramified quadratic twist of the Steinberg representation; here we handle the supercuspidal case. Crucially, we use new Petersson/Bruggeman-Kuznetsov forumulas for supercuspidal local component families recently developed by the authors. As corollaries, we derive Weyl-strength subconvex bounds for central values of $\operatorname{PGL}_2$ $L$-functions in the square-full aspect, and in the depth aspect, or in a hybrid of these two situations. A special case of our results is the Weyl-subconvex bound for all cusp forms of level $p^2$. Previously, such a bound was only known for forms that are twists from level $p$, which cover roughly half of the level $p^2$ forms.