On Sun's Conjectures for Truncated Jacobi-Symbol Determinants via Supersingular Elliptic Curves

Abstract

We study the truncated Jacobi-symbol determinants $$\{c,d{\}}_n=\det {\left[\left(\frac{j^2+cjk+d{k}^2}{n}\right)\right]}_{2\le j,k\le n-2}$$ 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.