Finiteness of the Tate–Shafarevich group Ш(E/ℚ) for an elliptic curve of analytic rank 2

Open. Gross–Zagier (1986) and Kolyvagin (1988–1990), with modularity, prove that analytic rank 0 or 1 implies equality of ranks and finiteness of Ш; the method needs a non-torsion Heegner point, which exists only in analytic rank 1. Skinner–Urban and others proved Iwasawa main conjecture divisibilities for GL₂, giving p-parts of the BSD formula in many rank 0/1 cases. Bhargava–Skinner (2014) showed a positive proportion of curves have rank one, and Bhargava–Skinner–Zhang (2014) that more than 66% of curves, ordered by height, satisfy the rank part of BSD; all such curves have rank 0 or 1. For analytic rank at least 2 the rank equality is checked only curve by curve, and finiteness of Ш is unknown for every such curve. Chan-Ho Kim (arXiv:2609.15328, preprint) shows how Heegner points detect finiteness of the p-primary part of Ш in arbitrary rank under Kolyvagin-type hypotheses, a criterion rather than an unconditional proof. OpenAI's 6–7 October 2026 release (github.com/openai/math, families 002 and 006, unrefereed) claims the full BSD formula whenever some p^∞-Selmer group has corank 0 or 1, and Goldfeld's conjecture for quadratic twists; these claims stay inside corank ≤ 1 and do not address rank 2.

Primarily conceptual. Higher-rank Euler systems would extend Iwasawa-theoretic methods that are also used for Galois representations and for explicit determination of rational points on curves. Cryptographic uses of curves rely on point counts, not on BSD.

A proof, for some explicit E/ℚ or an infinite family with analytic rank 2, that rank E(ℚ) = 2 and Ш(E/ℚ) is finite, accepted by experts. Verification of the full BSD formula for such a curve would be a stronger result.