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

Let $\mathbf{F} = \mathbf{Q}(\sqrt D)$ be a real quadratic field.

In this paper, we establish a large sieve inequality for the Asai lifts $ \mathrm{As} (f) $ with $f$ in a Hecke orthonormal basis of the space of Hilbert modular cusp forms of parallel weight $(k, k)$ over $ \mathbf{F} $.

As an application, for a fixed Hecke--Maass cusp form $\phi $ over $\mathbf{Q}$, we prove a non-trivial bound for the second moment of the convoluted central $L$-values $ L (1/2, \mathrm{As}(f) \times \phi) $ in the $k$-aspect.