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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.
