On the Second Moment of $L (1/2, \mathrm{As} (f))$

Let $\mathbf{F}$ be a real quadratic field. Let $f$ traverse a Hecke orthonormal basis of Hilbert cusp forms over $\mathbf{F}$ of full level and parallel weight $(k,k)$. As $k \rightarrow \infty$, we prove an asymptotic formula for the second moment of central Asai $L$-values $L (1/2, \mathrm{As} (f))$: \begin{equation*} {\sum}_{f } \, \omega_f L(1/2,\mathrm{As}(f))^2 = P_3 ( \log {k } ) k^2 + O_{\mathbf{F},\varepsilon} (k^{3/2 + \varepsilon} ), \end{equation*} where $\omega_f$ are the harmonic weights and $P_3 (X)$ is an explicit polynomial of degree $3$. This refines the mean Lindel\"of bound $O_{\mathbf{F},\varepsilon} (k^{2 + \varepsilon} )$ proved by Wenzhi Luo.