Low-corank Selmer converse for elliptic curves over Q
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For an elliptic curve A/Q and any prime p with full p-power Selmer corank r∈{0,1}, must analytic rank and Mordell–Weil rank both equal r and the whole Tate–Shafarevich group be finite? The target imposes no reduction, CM, residual-representation, torsion or isogeny restrictions.
Why it matters and what counts as progress
Why it matters: It asks whether low Selmer corank determines analytic rank and arithmetic finiteness without restrictions on the prime or elliptic curve.
What counts as progress: Deduce analytic and Mordell–Weil ranks and finiteness of the full Tate–Shafarevich group from low Selmer corank without extra curve or prime restrictions.
How the problem developed
- 1986 · Partial result [1]
Benedict Gross and Don Zagier Established the Heegner-point height formula that yields a non-torsion point in the analytic-rank-one setting.
- 1989 · Partial result [2]
Victor Kolyvagin Used Euler systems to derive Mordell–Weil rank and Tate–Shafarevich finiteness in the rank-at-most-one setting.
- 7 Oct 2026 · Theorem claim [3]
OpenAI Claimed that for every elliptic curve over Q and every prime q, q-power Selmer corank zero or one forces analytic rank and Mordell–Weil rank to equal that corank, with finite Tate–Shafarevich group.
AI contribution
The Selmer converse for elliptic curves at every prime
For every elliptic curve A/Q, every prime p and full p-power Selmer corank r in {0,1}, the manuscript claims analytic rank and Mordell–Weil rank r and finiteness of the whole Tate–Shafarevich group, without reduction, CM, residual-representation, torsion or isogeny restrictions.
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