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

TL;DR

This paper proves that the mass of holomorphic newforms of arbitrary level equidistributes on modular curves as the level grows, extending previous results and connecting the problem to subconvexity bounds and local hypotheses in number theory.

Contribution

It extends equidistribution results to non-squarefree levels, introduces explicit formulas for Rankin--Selberg integrals, and links mass equidistribution to subconvexity and local hypotheses.

Findings

Power savings in equidistribution rate for powerful levels

Explicit extensions of Watson's formula for non-squarefree levels

Equivalence of mass equidistribution with subconvexity and local bounds

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 qk -> infinity. 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 non-squarefree 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 the union of a global subconvexity bound and what we call a "local subconvexity bound"; a consequence of our local calculations is what we call a "local Lindelof hypothesis".