AI 中文总结
该研究针对孙智伟提出的截断雅可比符号行列式，通过统一方法结合超奇异椭圆曲线，证明了孙的多个相关猜想及强化形式，揭示了这类恒等式的通用机制。
AI 中文摘要
我们研究由孙智伟在文献[ZWSun]中提出的截断雅可比符号行列式$$\{c,d\}_n=\det\\!\n\left[\left(\frac{j^2+cjk+dk^2}{n}\right)\right]_{2\le j,k\le n-2}$$，并证明了[ZWSun]中的猜想5.1(i)、5.2、5.3、5.4、5.5、5.6(i)、5.7、5.8、5.6(ii)的一个特例，[Sun2019]中的猜想4.8(i)以及若干强化形式。所有结果均来自一种统一方法：对素数$p$，我们对角化由$\mathbb{F}_p^\times$索引的非剩余矩阵，将行列式的消失性归结为某些复乘椭圆曲线的超奇异约化。该框架可自然推广到更多参数族，暗示这类恒等式背后存在一种通用机制。
英文摘要
We study the truncated Jacobi-symbol determinants $$\{c,d\}_n=\det\!\left[\left(\frac{j^2+cjk+dk^2}{n}\right)\right]_{2\le j,k\le n-2}$$ proposed by Zhi-Wei Sun in \cite{ZWSun} and prove Conjectures 5.1(i), 5.2, 5.3, 5.4, 5.5, 5.6(i), 5.7, 5.8, a case of 5.6(ii) of \cite{ZWSun}, Conjecture 4.8(i) of \cite{Sun2019} and some strengthened forms. All results follow from a single unified approach: for a prime $p$, we diagonalize the nonzero-residue matrix indexed by $\mathbb{F}_p^\times$ and reduce the vanishing of determinants to the supersingular reduction of certain CM elliptic curves. The same framework extends naturally to further families of parameters, suggesting a general mechanism behind identities of this type.
