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

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.

MOMENTS AND HYBRID SUBCONVEXITY FOR SYMMETRIC-SQUARE L-FUNCTIONS

“…It is plausible that the methods used to prove Theorem 1.3 should also generalize to show formula (1.8) for Δ ≥ T 1/5+ , which would improve on [16], but this requires a rigorous proof. For quantum variance in the level aspect, see [28].…”

“…(Lam’s work actually involves symmetric-square L -functions attached to holomorphic Hecke eigenforms, but his method should apply equally well to Hecke–Maass forms.) Other works involving moments of symmetric square L -functions include [3, 18, 16, 19, 2, 1, 28].…”

“…In turn, by Watson’s formula [33], a sharp estimate for expression (1.6) boils down to establishing It is plausible that the methods used to prove Theorem 1.3 should also generalize to show formula (1.8) for , which would improve on [16], but this requires a rigorous proof. For quantum variance in the level aspect, see [28].…”

Theta functions, fourth moments of eigenforms and the sup-norm problem II

“…Alternatively, Blomer et al [BHMM22] have demonstrated that one may use Voronoï summation for Rankin–Selberg convolutions in place of a theta kernel. Prior to the application to fourth moments, theta kernels have played similar roles in the study of quantum variance [Nel16, Nel17, Nel19, Nel20], numerical computations [Nel15] and in the proof of Waldspurger’s formula [Wal85]. In each of these earlier works, theta kernels apparently served as a substitute for parabolic Fourier expansions, giving a tool for establishing analogues on compact quotients (where such expansions are not available) of results known already for noncompact quotients.…”