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arXiv · 2607.13490

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

Changlin Li·Zhi Qi
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Abstract

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.

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Changlin Li, Zhi Qi. 2026-07-15. On the Second Moment of $L (1/2, \mathrm{As} (f))$. https://arxiv.org/abs/2607.13490

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