AI 中文总结
研究实二次域\(\mathbf{F}\)上全水平且平行权\((k,k)\)的希尔伯特尖形式，当\(k \to \infty\)时，给出中心浅井\(L\)值\(L(1/2, \mathrm{As}(f))\)二阶矩的渐近公式，改进了平均林德勒夫界。
AI 中文摘要
设\(\mathbf{F}\)为实二次域。令\(f\)遍历\(\mathbf{F}\)上全水平且平行权\((k,k)\)的希尔伯特尖形式的赫克正交基。当\(k \to \infty\)时，我们证明了中心浅井\(L\)值\(L(1/2, \mathrm{As}(f))\)二阶矩的渐近公式：\(\sum_{f } \omega_f L(1/2,\mathrm{As}(f))^2 = P_3 ( \log {k } ) k^2 + O_{\mathbf{F},\varepsilon} (k^{3/2 + \varepsilon} )\)，其中\(\omega_f\)是调和权，\(P_3 (X)\)是次数为\(3\)的显式多项式。这改进了罗文志证明的平均林德勒夫界\(O_{\mathbf{F},\varepsilon} (k^{2 + \varepsilon} )\)。
英文摘要
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 } \, ω_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 $ω_f$ 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.
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