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On the algebraic and analytic ranks of the twin-prime elliptic curve y2=x(x−2)(x−p)

  • Ruihan Chen
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Abstract

Let p≥ 7 and suppose that p and p−2 are prime. We study Ep:y2=x(x−2)(x−p) using the classical $2$-Selmer calculation of Qiu-Zhang and the Cassels-Tate pairing. These give 2∞-Selmer corank one for p≡ 3,5pmod8 and corank zero for p≡ 7pmod8. Assuming the low-corank Birch-Swinnerton-Dyer statement announced in the October 2026 OpenAI mathematics release, we deduce equality of the analytic and algebraic ranks in these cases, finiteness of the full Tate-Shafarevich group, and the exact BSD leading-term formula. The $2$-primary Tate-Shafarevich group is trivial, and the remaining factor has odd square order. We identify precisely the obstruction left in the class p≡ 1pmod8. An appendix gives a local descent proof of the known Selmer dimensions in the coordinates used here.

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Cite

BibTeXRIS
BibTeX
@misc{chen2026algebraic,
  title = {{On the algebraic and analytic ranks of the twin-prime elliptic curve $y^2=x(x-2)(x-p)$}},
  author = {Ruihan Chen},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.09370},
  eprint = {2610.09370},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.09370},
}
RIS
TY  - UNPB
TI  - On the algebraic and analytic ranks of the twin-prime elliptic curve y^2=x(x−2)(x−p)
AU  - Ruihan Chen
PY  - 2026
DA  - 2026/10/07
PB  - arXiv
DO  - 10.48550/arxiv.2610.09370
UR  - https://arxiv.org/abs/2610.09370
AB  - Let p≥ 7 and suppose that p and p−2 are prime. We study E_p:y^2=x(x−2)(x−p) using the classical $2$-Selmer calculation of Qiu-Zhang and the Cassels-Tate pairing. These give 2^∞-Selmer corank one for p≡ 3,5pmod8 and corank zero for p≡ 7pmod8. Assuming the low-corank Birch-Swinnerton-Dyer statement announced in the October 2026 OpenAI mathematics release, we deduce equality of the analytic and algebraic ranks in these cases, finiteness of the full Tate-Shafarevich group, and the exact BSD leading-term formula. The $2$-primary Tate-Shafarevich group is trivial, and the remaining factor has odd square order. We identify precisely the obstruction left in the class p≡ 1pmod8. An appendix gives a local descent proof of the known Selmer dimensions in the coordinates used here.
ER  -