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

We establish the first moment bound $$\sum_{\varphi} L(\varphi \otimes \varphi \otimes \Psi, \tfrac{1}{2}) \ll_{\varepsilon} p^{5/4+\varepsilon}$$ for triple product $L$-functions, where $\Psi$ is a fixed Hecke-Maass form on $\operatorname{SL}_2(\mathbb{Z})$ 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 $\operatorname{SL}_2(\mathbb{Z}) \backslash \mathbb{H}$. Our main result turns out to be closely related to estimates such as $$\sum_{|n| < p} L(\Psi \otimes \chi_{n p},\tfrac{1}{2}) \ll p,$$ where the sum is over $n$ 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.