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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.
We study the distribution of 2-Selmer ranks in the family of quadratic twists of an elliptic curve 
$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}E$ over an arbitrary number field 
$K$. Under the assumption that 
${\rm Gal}(K(E[2])/K) \ {\cong }\ S_3$, we show that the density (counted in a nonstandard way) of twists with Selmer rank 
$r$ exists for all positive integers 
$r$, and is given via an equilibrium distribution, depending only on a single parameter (the ‘disparity’), of a certain Markov process that is itself independent of 
$E$ and 
$K$. More generally, our results also apply to 
$p$-Selmer ranks of twists of two-dimensional self-dual 
${\bf F}_p$-representations of the absolute Galois group of 
$K$ by characters of order 
$p$.
