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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Bounds for Rankin–Selberg integrals and quantum unique ergodicity for powerful levels
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by Paul D. Nelson, Ameya Pitale and Abhishek Saha PDF
J. Amer. Math. Soc. 27 (2014), 147-191 Request permission


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”.

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Additional Information
  • Paul D. Nelson
  • Affiliation: École Polytechnique Fédérale de Lausanne, Mathgeom-TAN station 8, CH-1015 Lausanne, Switzerland
  • Email:,
  • Ameya Pitale
  • Affiliation: Department of Mathematics, University of Oklahoma, Norman, Oklahoma 73019
  • MR Author ID: 778555
  • Email:
  • Abhishek Saha
  • Affiliation: Department of Mathematics, University of Bristol, Bristol BS81TW, United Kingdom
  • Email:
  • 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
  • © Copyright 2013 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: J. Amer. Math. Soc. 27 (2014), 147-191
  • MSC (2010): Primary 11F11; Secondary 11F70, 22E50, 58J51
  • DOI:
  • MathSciNet review: 3110797