具有魏尔斯特拉斯点的亏格\(g\)超椭圆曲线的局部统计与平均秩
Local statistics and average rank of genus $g$ hyperelliptic curves with a Weierstrass point
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中文总结 AI 辅助
研究具有魏尔斯特拉斯点的亏格\(g\)超椭圆曲线，通过确定其在特定素数处良好约化概率及其他约化类型概率公式，在相关猜想假设下，推导出该类曲线平均解析秩的明确上界。
中文摘要 AI 辅助
本文确定了在数域上具有魏尔斯特拉斯点的亏格\(g\)超椭圆曲线在给定剩余特征\(>2g + 1\)的素数处具有良好约化的概率。还得到了其他几种约化类型的类似概率公式，包括正环面或幂幺秩的情况。作为应用，在哈塞 - 魏尔猜想和广义黎曼假设下，推导出具有魏尔斯特拉斯点的亏格\(g\)超椭圆曲线平均解析秩的明确上界。
英文摘要
In this paper, we determine the probability that a genus $g$ hyperelliptic curve with a Weierstrass point over a number field has good reduction at a given prime of residue characteristic $>2g+1$. We also obtain analogous probability formulas for several other reduction types, including cases with positive toric or unipotent rank. As an application, assuming the Hasse--Weil conjecture and the generalized Riemann hypothesis, we derive an explicit upper bound for the average analytic rank of genus $g$ hyperelliptic curves with a Weierstrass point.
