3/5 同余数问题中的高互反律、Cassels 配对与 Selmer 塔

中文摘要 AI 辅助

我们研究附属于 $3/5$ 同余数问题的椭圆曲线 $A_m:y^2=x(x-m)(x+4m)$ 的算术性质。三元表示数之差控制着相关的中心 $L$-值。当 $p\equiv11\pmod{40}$ 时，通常的 Cassels 配对退化；我们构造了一个显式的 $4$-覆盖，并证明下一个 Cassels--Tate 配对由归一化表示缺陷所控制，等价地由阶乘特征、Pell 符号和类数同余所控制。对于复合参数，我们计算了四个扭曲的通常 Cassels 矩阵，并且在两个素数情形下，计算了一个次数为 $1024$ 的控制其联合分布的域。相同的高次下降可推广到更大的根式：第二个有理推出确定了下一个配对的完整 $\Lambda'$ 行。我们还确定了两个具有四维通常根式的显式 Selmer 塔，以及当通常根式为一维时所有有限的 $2$-幂 Selmer 群。

英文摘要

We study the arithmetic of the elliptic curves \[ A_m:y^2=x(x-m)(x+4m) \] attached to the $3/5$ congruent number problem. A difference of ternary representation numbers controls the relevant central $L$-values. For $p\equiv11\pmod{40}$ the ordinary Cassels pairing degenerates; we construct an explicit $4$-cover and show that the next Cassels--Tate pairing is governed by the normalized representation defect, equivalently by a factorial character, a Pell symbol, and a class number congruence. For composite parameters we compute the ordinary Cassels matrices of four twists and, in the two prime case, a degree $1024$ governing field for their joint distribution. The same higher descent extends to larger radicals: a second rational pushout determines the full $Λ'$ row of the next pairing. We also determine two explicit Selmer towers with four dimensional ordinary radical, and all finite $2$-power Selmer groups when the ordinary radical is one dimensional.

发表机构

• Nanjing University（南京大学）

机构由 AI 辅助整理，请以论文原文为准。