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Bounds for twisted symmetric square L-functions via half-integral weight periods

Part of: Partial differential equations on manifolds; differential operators Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  09 November 2020

Paul D. Nelson
Paul D. Nelson*
Affiliation:
ETH Zürich, Department of Mathematics, Rämistrasse 101, CH-8092, Zürich, Switzerland; E-mail: paul.nelson@math.ethz.ch

Abstract

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We establish the first moment bound

$$\begin{align*}\sum_{\varphi} L(\varphi \otimes \varphi \otimes \Psi, \tfrac{1}{2}) \ll_\varepsilon p^{5/4+\varepsilon} \end{align*}$$
for triple product L-functions, where $\Psi $ is a fixed Hecke–Maass form on $\operatorname {\mathrm {SL}}_2(\mathbb {Z})$ and $\varphi $ runs over the Hecke–Maass newforms on $\Gamma _0(p)$ of bounded eigenvalue. The proof is via the theta correspondence and analysis of periods of half-integral weight modular forms. This estimate is not expected to be optimal, but the exponent $5/4$ is the strongest obtained to date for a moment problem of this shape. We show that the expected upper bound follows if one assumes the Ramanujan conjecture in both the integral and half-integral weight cases.

Under the triple product formula, our result may be understood as a strong level aspect form of quantum ergodicity: for a large prime p, all but very few Hecke–Maass newforms on $\Gamma _0(p) \backslash \mathbb {H}$ of bounded eigenvalue have very uniformly distributed mass after pushforward to $\operatorname {\mathrm {SL}}_2(\mathbb {Z}) \backslash \mathbb {H}$ .

Our main result turns out to be closely related to estimates such as

$$\begin{align*}\sum_{|n| < p} L(\Psi \otimes \chi_{n p},\tfrac{1}{2}) \ll p, \end{align*}$$
where the sum is over those n for which $n p$ is a fundamental discriminant and $\chi _{n p}$ denotes the corresponding quadratic character. Such estimates improve upon bounds of Duke–Iwaniec.

Keywords

L-functionsfor each term replaceand give space andmodular formshalf-integral weightquadratic twists

MSC classification

Primary: 11F27: Theta series; Weil representation; theta correspondences
Secondary: 11F37: Forms of half-integer weight; nonholomorphic modular forms 58J51: Relations between spectral theory and ergodic theory, e.g. quantum unique ergodicity
Type
Number Theory
Information
Forum of Mathematics, Sigma , Volume 8 , 2020 , e44
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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