超椭圆系中$L(1/2,χ_D)$均值的三次周期二次项
The Period-Three Secondary Term in the Mean Value of $L(1/2,χ_D)$ in the Hyperelliptic Ensemble
AI总结：
该研究修正超椭圆系中二次Dirichlet L函数一阶矩的二次项，对所有奇素数幂$q$证明公式，发现二次项含三次周期的亏格依赖项，且$q=3$时矩呈现剩余类趋势。
AI中文摘要：
设$q$为奇素数幂。我们重新研究Florea关于奇次超椭圆系$\boldsymbol{\textit{H}}_{2g+1}$上二次Dirichlet L函数一阶矩的渐近公式，证明其二次项并非完整项。平方对偶生成函数在二次圆上有三个模相同的双极点：正实极点重现Florea的多项式，两个非实共轭极点贡献同阶项。完整二次项为$q^{2g/3}(\boldsymbol{\textit{α}}_{g\bmod3}g+\boldsymbol{\textit{β}}_{g\bmod3})$，其中实系数对亏格的依赖周期恰好为3，且在三个剩余类中至少有两个类，非实极点贡献阶为$gq^{2g/3}$的项。我们还对所有奇素数幂$q$证明该公式，而Florea全程假设$q≡1\bmod4$，这一结果通过相位归一化二次高斯和实现，该和恢复了其论证所用的乘法性与局部估值。$q=3$时的有限系精确矩呈现出三个剩余类趋势。
英文摘要：
Let $q$ be an odd prime power. We revisit Florea's asymptotic formula for the first moment of quadratic Dirichlet $L$-functions over the odd-degree hyperelliptic ensemble $\mathcal H_{2g+1}$, and show that its secondary term is not the complete one. The square-dual generating function has three double poles of the same modulus on the secondary circle: the positive real one reproduces Florea's polynomial, while the two nonreal conjugate poles contribute at the same order. The complete secondary term is $q^{2g/3}(α_{g\bmod3}g+β_{g\bmod3})$, with real coefficients whose dependence on the genus has minimal period exactly three, and on at least two of the three residue classes the nonreal poles contribute a term of order $gq^{2g/3}$. We also prove the formula for every odd prime power $q$, whereas Florea assumes $q\equiv1\bmod4$ throughout; this is achieved by a phase-normalized quadratic Gauss sum that restores the multiplicativity and the local evaluations her argument uses. Exact finite-ensemble moments for $q=3$ exhibit the three residue-class trends.
