Abstract
Abstract 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.
Highlights
The quantum ergodicity theorem [35, 4, 44] says that on a compact Riemannian manifold with ergodic geodesic flow, almost all eigenfunctions have equidistributed mass in the large eigenvalue limit
The main results of this article may be understood as quantitative strengthenings of that result, for specific classes of eigenfunctions and observables, in the arithmetic congruence case
Well-developed techniques for analyzing averages of triple product -values and/or shifted convolution sums apply in our setting, giving nontrivial estimates in the intended direction
Summary
The quantum ergodicity theorem [35, 4, 44] says that on a compact Riemannian manifold with ergodic geodesic flow, almost all eigenfunctions have equidistributed mass in the large eigenvalue limit. When the manifold is arithmetic, additional tools become available by which one can prove quantitative strengthenings of this conclusion, to the effect that all but very few eigenfunctions (satisfying additional symmetries) have very equidistributed mass (see Section 1.2 below, or [21, 22, 17]). Well-developed techniques for analyzing averages of triple product -values and/or shifted convolution sums apply in our setting, giving nontrivial estimates in the intended direction. We instead introduce techniques involving the theta correspondence and periods of half-integral weight modular forms, which seem to give stronger results in our setting
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