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A pointwise 2-converse for elliptic curves with rational two-torsion
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Abstract
We prove a pointwise 2-converse for elliptic curves over $\mathbf Q$ with nonzero rational two-torsion: if the $2^\infty$-Selmer corank is zero or one, then the analytic rank and Mordell–Weil rank equal that corank, and the Shafarevich–Tate group is finite. The result allows arbitrary reduction at 2.
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Possibly relevant literature (found by the agent, not checked)
- Daniel Kriz and Chao Li, Congruences between Heegner points and quadratic twists of elliptic curves (arXiv:1606.03172; published as "Goldfeld's conjecture and congruences between Heegner points", Forum Math. Sigma 2019). It applies Heegner-point congruences at p = 2 to produce quadratic twists of analytic rank 0 or 1 and proves the 2-part of BSD for many of them, relating to the paper's p = 2 Heegner and quadratic-twist constructions; not in the bibliography.
- Alexander Smith, 2^∞-Selmer groups, 2^∞-class groups, and Goldfeld's conjecture (arXiv:1702.02325, 2017). It studies 2^∞-Selmer groups in quadratic-twist families of curves with full rational 2-torsion, relating to the introduction's discussion of Smith's Selmer-corank results; the bibliography cites only Smith's 2025 preprint (arXiv:2503.17619).
Digests and expositions
- Yiping Deng, Hilbert’s tenth: from machines to equations, and a new rational-case claim (Loudcoder, 2026-10-07). Identifies Theorem 1.1 of this paper as an input to the rational Hilbert's tenth manuscript's denominator-parity argument and states that this dependency has not been independently validated in the exposition. [cites earlier version] [family-level]
- AI Proved What?, No algorithm for fractions: Hilbert’s tenth problem over the rational numbers (Episode 004, October 2026; status updated 2026-10-19). Explains this paper's role in passing from a 2-descent Selmer class to a rational point in the rational Hilbert's tenth argument, with an illustrative curve and point. [AI-generated per source] [family-level]
Elsewhere
- Emergent Mind: family 004
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion-October-7-2026,
author = {{OpenAI}},
title = {{A pointwise $2$-converse for elliptic curves with rational two-torsion}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion-October-7-2026/paper.pdf}{OAI:A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion-October-7-2026}},
year = {2026}
}Versions
- Earlier edition: September 24, 2026
This version updates a citation to a revised manuscript in this collection; see the previous version.
Family 004: Hilbert’s tenth problem over ℚ
Proves that no algorithm decides whether an integer-coefficient polynomial in an arbitrary number of variables has a rational zero, resolving Hilbert's tenth problem over ℚ negatively.
Other papers in this family:
- Hilbert’s tenth problem over the rational numbers · October 6, 2026