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

  • Guo Li
  • Xiaojun Yan
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Abstract

We study the truncated Jacobi-symbol determinants {c,d}n=det [((j2+cjk+dk2)/n)]2≤ j,k≤ 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 𝔽p× 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.

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Cite

BibTeXRIS
BibTeX
@misc{li2026suns,
  title = {{On Sun's Conjectures for Truncated Jacobi-Symbol Determinants via Supersingular Elliptic Curves}},
  author = {Guo Li and Xiaojun Yan},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2608.08509},
  eprint = {2608.08509},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.08509},
}
RIS
TY  - UNPB
TI  - On Sun's Conjectures for Truncated Jacobi-Symbol Determinants via Supersingular Elliptic Curves
AU  - Guo Li
AU  - Xiaojun Yan
PY  - 2026
DA  - 2026/10/23
PB  - arXiv
DO  - 10.48550/arxiv.2608.08509
UR  - https://arxiv.org/abs/2608.08509
AB  - We study the truncated Jacobi-symbol determinants {c,d}_n=det [((j^2+cjk+dk^2)/n)]_2≤ j,k≤ 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 𝔽_p^× 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.
ER  -