Bounds for Rankin–Selberg integrals and quantum unique ergodicity for powerful levels

Abstract

Let $f$ be a classical holomorphic newform of level $q$ and even weight $k$. We show that the pushforward to the full level modular curve of the mass of $f$ equidistributes as $q k \rightarrow \infty$. This generalizes known results in the case that $q$ is squarefree. We obtain a power savings in the rate of equidistribution as $q$ becomes sufficiently “powerful” (far away from being squarefree) and in particular in the “depth aspect” as $q$ traverses the powers of a fixed prime. We compare the difficulty of such equidistribution problems to that of corresponding subconvexity problems by deriving explicit extensions of Watson’s formula to certain triple product integrals involving forms of nonsquarefree level. By a theorem of Ichino and a lemma of Michel–Venkatesh, this amounts to a detailed study of Rankin–Selberg integrals $\int |f|^2 E$ attached to newforms $f$ of arbitrary level and Eisenstein series $E$ of full level. We find that the local factors of such integrals participate in many amusing analogies with global $L$-functions. For instance, we observe that the mass equidistribution conjecture with a power savings in the depth aspect is equivalent to knowing either a global subconvexity bound or what we call a “local subconvexity bound”; a consequence of our local calculations is what we call a “local Lindelöf hypothesis”.