Selected research history · publication v10
Follow the ideas, results and limitations over time. This is a selected history, not a complete literature survey or a measure of how close the problem is to being solved.
Current state & remaining question →Viewing the timeline from pinned publication v10.
11 selected milestones · Source coverage checked 2026-10-10 · Historical annotation and mathematical review remain separate.
Foundation
Mordell’s theorem describes rational points as a finite-rank free group plus finite torsion.
Scope & review: Finite generation does not determine the rank of each elliptic curve.
Historical annotation review: pending. The date refers to the publication, with the precision supported by the source.
Foundation
Birch–Swinnerton-Dyer’s Notes II records the rank and L-function prediction motivated by early computations.
Scope & review: This is a cited publication year. Agreement in finite data did not establish the general formula.
Historical annotation review: pending. The date refers to the publication, with the precision supported by the source.
Foundation
The published Gross–Zagier formula provides the height/L-derivative connection later combined with Euler systems.
Scope & review: The current analytic-rank-≤1 synthesis also uses Kolyvagin and modularity; the 1986 paper alone is not that full synthesis.
Historical annotation review: pending. The date refers to the publication, with the precision supported by the source.
Special case proved
General modularity, published in 2001, completes the classical scope: analytic rank 0 or 1 gives rank equality and finite Sha for every E/Q.
Scope & review: This is a synthesis of earlier Gross–Zagier/Kolyvagin inputs with modularity, not a proof of every leading coefficient or higher-rank case.
Historical annotation review: pending. The date refers to the publication, with the precision supported by the source.
Special case proved
Miller’s published proof covers 16,714 of 16,725 enumerated analytic-rank-≤1 curves of conductor <5000.
Scope & review: The 11 excluded curve/prime pairs describe that paper’s boundary, not today’s list of unsolved cases.
Historical annotation review: pending. The date refers to the publication, with the precision supported by the source.
Special case proved
Jetchev–Skinner–Wan’s author preprint gives the p-part for semistable analytic-rank-one curves at good p ≥5 with irreducible E[p].
Scope & review: The theorem is prime-specific and hypothesis-specific; its later journal publication is a separate bibliographic event.
Historical annotation review: pending. The date refers to the preprint, with the precision supported by the source.
Conditional result
Skinner’s published theorem obtains analytic rank one from algebraic rank one, finite Sha and the stated semistable reduction pattern.
Scope & review: Finite Sha is assumed here; this converse cannot be used to prove its own premise.
Historical annotation review: pending. The date refers to the publication, with the precision supported by the source.
Special case proved
Burungale–Skinner–Tian–Wan v2 gives selected supersingular p-parts and full BSD for the explicitly constrained 46a1 twist family.
Scope & review: These are two different scopes. The selection omits p=3’s extra conditions and does not cover every quadratic twist.
Historical annotation review: pending. The date refers to the preprint, with the precision supported by the source.
Algorithmic advance
Banwait–Huang v3 supplies finite algorithms for subfamilies satisfying cited BSD criteria and clarifies the base-curve restrictions.
Scope & review: The four-base list in the family result is not an unconditional full-BSD assertion for every listed curve; there is no enumeration of all BSD cases.
Historical annotation review: pending. The date refers to the preprint, with the precision supported by the source.
Proposal
A separate provider manuscript proposes Goldfeld’s 0/1 half-densities for every E/Q in the signed squarefree ordering.
Scope & review: An asymptotic family statement allows exceptional twists; mean rank and full leading terms require additional results.
Historical annotation review: pending. The date refers to the preprint, with the precision supported by the source.
Proposal
The provider proposes the complete leading-term formula and finite Sha when a full q-power Selmer corank is 0 or 1.
Scope & review: The manuscript and its companion dependencies remain unassessed here. Mordell–Weil rank ≤1 is not the stated hypothesis.
Historical annotation review: pending. The date refers to the preprint, with the precision supported by the source.