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 Lindelöf bound $ O_{\mathbf{F},\varepsilon} (k^{2 + \varepsilon} ) $ proved by Wenzhi Luo.