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

Abstract

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 → ∞. 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 |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".

Some analytic aspects of automorphic forms on GL(2) of minimal type

“…The case of varying squarefree levels was addressed in [29], where it was shown that D f (g) → 0 as Ck → ∞ provided that C is squarefree. Finally, it was proved in [31] that D f (g) → 0 whenever Ck → ∞ (without any restriction on C). In fact, the main result of [31] gave an unconditional power savings bound D f (g) ≪ g C −δ1 0 log(Ck) −δ2 for some positive constants δ 1 , δ 2 , where as before, C 0 denotes the largest integer such that C 2 0 |C.…”

“…Finally, it was proved in [31] that D f (g) → 0 whenever Ck → ∞ (without any restriction on C). In fact, the main result of [31] gave an unconditional power savings bound D f (g) ≪ g C −δ1 0 log(Ck) −δ2 for some positive constants δ 1 , δ 2 , where as before, C 0 denotes the largest integer such that C 2 0 |C. Further extensions of this result to the case when g is not of full level were obtained in [16].…”

“…Thus, for squarefree levels, the subconvexity and QUE problems are essentially equivalent. A major point of [31] was that this equivalence is no longer true for powerful levels. For example, in the case when C is a perfect square, the results of [31] imply that |D f (g)…”

“…So in this case, the convexity bound alone is enough to imply QUE with power savings in the level aspect! More generally, as shown in [31], the QUE problem is significantly easier than the subconvexity problem in the case of newforms of powerful level (in contrast to the squarefree case, where these problems are essentially equivalent). One may ask whether the equivalence between QUE and subconvexity might be recovered for powerful levels by replacing the newform with a different choice of vector.…”

“…It is very likely that, by combining Theorem 1.2 with the arguments of [31,Sec. 3], one could establish the estimate D f (g) ≪ log(Ck) −δ for small δ > 0 and fixed g, but we do not pursue this here.…”