同余数曲线的表示缺陷与Cassels配对
Representation Defects and Cassels Pairings for Congruent Number Curves: Rédei Symbols and Governing Fields
- Nanjing University（南京大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究针对同余数曲线，建立了Qin归一化表示缺陷奇偶性与Cassels配对Pfaffian的关系，推导了特定条件下的同余数相关同余式并推广了相关结论。
AI中文摘要：
Qin通过三元二次型表示数的差值表示了同余数曲线$E_n:y^2=x^3-n^2x$的中心值，Zhang则给出了纯2-Selmer群上Cassels配对的显式公式。我们证明，Qin的归一化表示缺陷的奇偶性是Cassels配对的Pfaffian。对于满足$s_2=4$的$n=pq$，我们计算了$4\times4$的Cassels矩阵，得到不变量$\Pi(p,q)$，使得$r_a(pq)\equiv h(-pq)+16\Pi(p,q)\pmod{32}$。我们还证明了一个偶类比结果和高维推广。
英文摘要：
Using BSD results for CM elliptic curves, we relate Qin's quadratic form representation defects to the Cassels pairing on the pure $2$-Selmer group of a congruent number elliptic curve. When the pure $2$-Selmer dimension is even, we prove that the normalized representation defect modulo $2$ is the Pfaffian of the Cassels pairing matrix; the dimension of its radical yields sharper $2$-adic divisibility and information on the $2$-primary Shafarevich--Tate group. For products of primes congruent to $1$ modulo $8$ that are pairwise quadratic residues, we give explicit Cassels pairing matrices for both $E_n$ and $E_{2n}$ and express their entries in terms of quartic and Rédei symbols. For each fixed prime $p$, we construct a governing field of degree $256$ and determine the exact joint distribution of the two Pfaffians by Chebotarev's theorem. In particular, there is a set of primes $q$ of natural density $5/128$ for which both $E_{17q}$ and $E_{34q}$ have rank zero and $2$-primary Shafarevich--Tate group isomorphic to $(\Z/2\Z)^4$.
