有限域上高亏格曲线的点计数乘积
Products of point counts of higher genus curves over finite fields
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中文总结 AI 辅助
本文针对有限域上亏格至少为2的光滑射影曲线，提出类似椭圆曲线Birch-Swinnerton-Dyer猜想的渐近式，结合Kurokawa的相关猜想给出分析基础，并提供了数值证据。
中文摘要 AI 辅助
设$E/\mathbb Q$为一条椭圆曲线，对每个素数$p$，令$N_p$表示$E$模$p$的点数。Birch和Swinnerton-Dyer猜想的原始版本断言，当$x \to \infty$时，$\prod \limits _{p \leq x} \frac{N_p}{p} \sim C (\log x) ^{\text{rank}(E(\mathbb Q))}$。本文中，我们针对亏格至少为2的光滑射影曲线，构建了一个类似的猜想渐近式，该渐近式的贡献不仅来自其雅可比簇的秩，还来自曲线的Sato–Tate群。构建我们猜想时的关键分析依据是Kurokawa（2012年）关于临界线上整$L$-函数欧拉乘积收敛性的猜想。我们还针对不同情形为该猜想提供了一些数值证据。
英文摘要
Let $E/\mathbb Q$ be an elliptic curve and for each prime $p$, let $N_p$ denote the number of points of $E$ modulo $p$. The original version of the conjecture of Birch and Swinnerton-Dyer asserts that $\prod \limits _{p \leq x} \frac{N_p}{p} \sim C (\log x) ^{\text{rank}(E(\mathbb Q))}$ as $x \to \infty$. In this paper, we formulate a similar conjectural asymptotic for smooth projective curves of genus at least 2, in which the contributions to the conjectured asymptotic come not only from the rank of the Jacobian but also from the Sato--Tate group of the curve. The key analytic input in formulating our conjecture is a conjecture due to Kurokawa (2012) on the convergence of Euler products of entire $L$-functions on the critical line. We also provide some numerical evidence for our conjecture in various cases.
