1 · The essentials
This family of manuscripts concerns elliptic curves, equations whose rational-number solutions are studied. The Birch–Swinnerton-Dyer formula seeks to connect those solutions with the behavior of a function attached to the curve. The main manuscript claims the complete formula when a particular measure, taken from a group called the Selmer group, is zero or one. It is enough for this condition to hold at a single prime number. Under this hypothesis, the manuscript also states that two measures of rank agree and that another important group, the Tate–Shafarevich group, is finite. The announced formula includes contributions from every prime, without the additional restrictions listed in the abstract. The result is not formalized in Lean and has not been formally verified; the repository history records a correction or revision for this family.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, imagine a financial statement whose two sides describe the same thing in different ways. The announced result says that one particular check, attached to a single prime and returning zero or one, suffices to establish agreement of the entire statement, not merely some of its accounts.
Why it matters
The point is to move from a condition on a Selmer group to an exact formula containing all prime factors. The repository summary says that combining this result with result 006 yields full BSD for a density-one set of quadratic twists of every elliptic curve over the rationals. This consequence remains tied to the manuscripts’ announced results, without formal verification.
2 · Going further
Let be an elliptic curve. The main manuscript assumes that, for some prime , the full -power Selmer group has -corank . It then claims that the analytic and Mordell–Weil ranks both equal , that the Tate–Shafarevich group is finite, and that the complete BSD leading-term formula holds. The abstract specifies that all prime factors are included, with no additional hypotheses on reduction, rational torsion, isogenies, complex multiplication, or residual Galois representations.
The abstracts do not give the steps of the argument: they present its conclusions and the connections between manuscripts. The second manuscript addresses the Selmer converse for every elliptic curve over and every prime , in coranks zero and one. It claims equality of both ranks with that corank and finiteness of the entire Tate–Shafarevich group; these conclusions also appear in the main result, which additionally claims the complete exact formula.
The third manuscript focuses on the part of the BSD formula associated with the prime , assuming that the two-power Selmer group has corank at most one. Combined with quadratic-twist Selmer distribution, it claims this formula for a density-one set of twists indexed by positive and negative squarefree integers, ordered by absolute value. The common rank is zero or one, each with density one half. The repository summary attributes the corresponding full-formula consequence to the main result combined with result 006.
The second manuscript also gives an application at the additive prime : for every prime , the cubic is claimed to have analytic and Mordell–Weil rank one and finite Tate–Shafarevich group. It concludes that each such is a sum of two rational cubes.
Useful notions
- Elliptic curves and rational points
- Primes and prime powers
- Analytic and Mordell–Weil ranks
- Selmer groups and corank
- Quadratic twists and density
What was known before / what is new
Before
The source presents the Birch–Swinnerton-Dyer formula and the Selmer converse as the problems addressed by these manuscripts. It also mentions a distribution of quadratic-twist Selmer groups used for the density consequences. It supplies neither a history of these problems nor earlier results from which to reconstruct a chronology.
New
The main manuscript’s claimed contribution is the complete BSD formula whenever the full prime-power Selmer group has corank zero or one at at least one prime. The announcement includes all prime factors and finiteness of the Tate–Shafarevich group, without the additional hypotheses listed in the abstract. The companion manuscripts address the Selmer converse at every prime and the formula at the prime .
Limits and assumptions
The central hypothesis concerns the full prime-power Selmer group, not a single prime-power level. The main result concerns elliptic curves over for which this corank is zero or one at some prime; higher coranks are outside the announcement’s scope. A density-one conclusion is not a conclusion for every twist. The abstracts do not detail the argument or the nature of the correction or revision recorded in the repository history. The main result is not formalized in Lean and has not been formally verified; the source supplies no further list of open questions.
Mini glossary
- Mordell–Weil rank —
- A measure of the number of independent rational points of infinite order on an elliptic curve.
- Analytic rank —
- The order of vanishing of the curve’s -function at the point considered by the BSD formula.
- Selmer corank —
- A size measure for the full prime-power Selmer group, expressed here over .
- Leading-term formula —
- An equality concerning the first nonzero coefficient in the expansion of the -function at the point under consideration.
- Density one —
- A property of a set whose proportion tends to one in the chosen counting order.
Keywords : Birch–Swinnerton-Dyer · Selmer groups · Elliptic curves
AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.
3 · Technical details and sources
Claim made in the manuscript (source text)
Proves the full Birch–Swinnerton-Dyer leading-term formula for every elliptic curve over ℚ whose full q-power Selmer group has corank zero or one for some prime q, including finiteness of the Tate–Shafarevich group. With result 006, this gives full BSD for a density-one set of quadratic twists of every elliptic curve over ℚ.
To be verified: the proof is written in the manuscript, but no software has checked it and no validation by independent mathematicians is established. It may contain errors.
Manuscripts in this family
Exact Birch–Swinnerton-Dyer Formula from Low Selmer Corank
07/10/2026We prove the full Birch–Swinnerton-Dyer leading-term formula for every elliptic curve over ℚ whose full q-power Selmer group has corank zero or one at some prime q. The analytic and Mordell–Weil ranks equal that corank, and the Tate–Shafarevich group is finite. The formula includes all prime factors and requires no additional hypotheses on reduction, rational torsion, isogenies, complex multiplication, or residual Galois representations.
The Selmer converse for elliptic curves at every prime
07/10/2026We prove the Selmer converse in coranks zero and one for every elliptic curve over ℚ and every prime p: if the full p-power Selmer group has ℤp-corank , then the analytic and Mordell–Weil ranks both equal r, and the entire Tate–Shafarevich group is finite. As an application at the additive prime 3, we prove that for every prime , the cubic has analytic and Mordell–Weil rank one and finite Tate–Shafarevich group. In particular, every such ℓ is a sum of two rational cubes.
The two-primary Birch–Swinnerton-Dyer formula in Selmer corank at most one
06/10/2026We prove the two-primary Birch and Swinnerton-Dyer leading-term formula for every elliptic curve over the rationals whose two-power Selmer group has corank at most one. In this range, the algebraic rank, analytic rank, and Selmer corank are equal, and the Tate–Shafarevich group is finite. Combined with the quadratic-twist Selmer distribution, this gives the exact two-primary formula for a density-one set of signed squarefree twists of each fixed curve, ordered by absolute value; the common rank is zero or one, with each value having density one half.