简单二维亏格雅可比簇上的有理挠点
Rational torsion on simple genus two Jacobians
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中文总结 AI 辅助
该研究针对有理数域上的二维亏格曲线，构造出阶数96等的新有理挠点子群，还记录了该类雅可比簇的所有已知有理挠点子群。
中文摘要 AI 辅助
我们展示了定义在有理数域$\boldsymbol{\text{Q}}$上的二维亏格曲线的几何简单雅可比簇中有理挠点的新子群。阶数为96、不变量为[2,2,2,12]的最大子群，由满足$a^2 + b^2 + c^2 = u^2 + v^2$且$a^4 + b^4 + c^4 = u^4 + v^4$的正整数$a,b,c,u,v$构成的曲线$y^2 = x(x-a^2)(x-b^2)(x-c^2)(x-u^2)(x-v^2)$实现。我们还得到了[2,2,20]、[2,2,4,4]、[2,2,2,8]、[2,4,8]和[6,6]等子群的实现。最后，我们记录了据我们所知，有理数域上二维亏格雅可比簇中出现的所有已知子群，包括几何简单情形和一般情形。
英文摘要
We exhibit new subgroups of rational torsion points in geometrically simple Jacobians of genus-two curves over $\mathbb Q$. The largest group, which has order 96 and invariants [2,2,2,12], is realized by curves of the form $y^2 = x(x-a^2)(x-b^2)(x-c^2)(x-u^2)(x-v^2)$ where $a,b,c,u,v$ are positive integers that satisfy $a^2 + b^2 + c^2 = u^2 + v^2$ and $a^4 + b^4 + c^4 = u^4 + v^4$. We also find realizations of the groups [2,2,20], [2,2,4,4], [2,2,2,8], [2,4,8], and [6,6]. Finally, we record, to the best of our knowledge, all known subgroups that arise in genus-two Jacobians over $\mathbb Q$, in the geometrically simple case and in general.
