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


Authors: Paul D. Nelson, Ameya Pitale and Abhishek Saha
Journal: J. Amer. Math. Soc. 27 (2014), 147-191
MSC (2010): Primary 11F11; Secondary 11F70, 22E50, 58J51
Published electronically: August 6, 2013
MathSciNet review: 3110797
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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 \vert f\vert^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''.


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Additional Information

Paul D. Nelson
Affiliation: École Polytechnique Fédérale de Lausanne, Mathgeom-TAN station 8, CH-1015 Lausanne, Switzerland
Email: paul.nelson@epfl.ch, nelson.paul.david@gmail.com

Ameya Pitale
Affiliation: Department of Mathematics, University of Oklahoma, Norman, Oklahoma 73019
Email: apitale@math.ou.edu

Abhishek Saha
Affiliation: Department of Mathematics, University of Bristol, Bristol BS81TW, United Kingdom
Email: abhishek.saha@gmail.com

DOI: https://doi.org/10.1090/S0894-0347-2013-00779-1
Received by editor(s): May 30, 2012
Received by editor(s) in revised form: January 8, 2013, April 4, 2013, and June 9, 2013
Published electronically: August 6, 2013
Additional Notes: The first author was supported by NSF grant OISE-1064866 and partially supported by grant SNF-137488
The second author was supported by NSF grant DMS 1100541
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.