The full BSD formula from low Selmer corank

The full Birch–Swinnerton-Dyer formula is claimed for every elliptic curve with a small Selmer group at one prime.

3 papers October 6, 2026 Explained by AI

An elliptic curve has an L-function, built by counting its points modulo every prime. The Birch–Swinnerton-Dyer formula predicts that function's first nonzero coefficient at its centre exactly. Since 1990, analytic ranks 0 and 1 gave the rank and a finite hidden group, but not the exact formula in general. These three papers claim the full formula whenever one Selmer group is small. They need no extra conditions on the curve or the prime.

First step

1965

Birch and Swinnerton-Dyer, the conjecture from machine computations

Significance

Landmark

If it holds, it gives the exact Birch–Swinnerton-Dyer formula for every rank 0 or 1 curve meeting one Selmer test.

Trust

Claimed, not checked

Field

Number theory

3 papers

The explanation

Take the curve y² + y = x³ − x² − 10x − 20. It is an elliptic curve, a cubic curve whose points can be added. Its rational points form a group with five elements. One is a point at infinity. The others are (5, 5), (5, −6), (16, 60) and (16, −61). Since the group is finite, the curve has rank 0.

Now count solutions modulo each prime p. Modulo 2, 3 and 5 there are 5 points, counting infinity. Modulo 7 there are 10. Write a_p for p + 1 minus the count. So a_2 = −2, a_3 = −1, a_5 = 1 and a_7 = −2. The L-function packs all these numbers into one function of a variable s. Its centre is s = 1. There its value is about 0.253842.

The Birch–Swinnerton-Dyer formula predicts that value exactly. For rank 0 it reads L(1) ÷ Ω = (#Sha × c) ÷ (#torsion)². Here Ω is the real period, an integral over the curve's real points, about 1.269209. The ratio L(1) ÷ Ω agrees with 1/5 to 16 digits. The torsion group has 5 points, so its square is 25. The local factor c comes from the bad prime 11, and it equals 5. So the formula says 1/5 = #Sha × 5/25. That forces Sha, the hidden group, to have exactly one element. For this curve, all this was already known.

Sha is the Tate–Shafarevich group. It collects curves that look solvable at every prime, and over the real numbers, but have no rational point. It is conjectured to be finite. The rank is the number of independent points that generate infinitely many others. The analytic rank is the order of vanishing of the L-function at the centre. For rank r, the formula uses the r-th derivative and a regulator, a determinant of point sizes.

The conjecture dates from 1965. Modularity, completed in 2001, gave every such L-function a value at the centre. Gross and Zagier in 1986, and Kolyvagin in 1990, handled analytic ranks 0 and 1. They proved the rank matches and Sha is finite. That still left the exact formula, prime by prime. Earlier exact results needed conditions on how the curve behaves modulo its bad primes, on the prime, or on the curve's symmetries.

The Selmer group is the computable side. For a prime q, the q-power Selmer group combines the rational points with the q-part of Sha. Its corank counts the rank plus any infinitely divisible part of Sha. It can be bounded by finite calculations.

The three papers claim the formula whenever the Selmer corank is 0 or 1 at some prime qSource 1. The Selmer converse paper claims rank equals analytic rank equals that corank, and Sha is finiteSource 2. A second paper claims the formula's 2-part is exactSource 3. The main paper claims every odd prime part is exactSource 4. Together they give the whole formula. There are no extra conditions on reduction, torsion, isogenies or complex multiplicationSource 5.

The converse paper also claims an old cube problem in special cases. For every prime ℓ leaving remainder 4, 7 or 8 after division by 9, ℓ is a sum of two rational cubesSource 6. For example, 13 = (7/3)³ + (2/3)³, since 343/27 + 8/27 = 351/27. Other recent work has treated these cases by Heegner-point methods.

The obstacle is that rank and finiteness do not fix the coefficient's prime factors. The main paper measures the mismatch at each odd prime p. For curves without complex multiplication, it shows the mismatch is never negative. It then pairs the curve with a twisted partner, so that one has analytic rank 0 and the other rank 1. The two mismatches add to zero, so both vanishSource 7. Curves with complex multiplication use a known rank-zero theorem of Burungale and FlachSource 8.

The papers lean on companions in the openai/math release. The Selmer converse uses the 2-converse and twist densities from the Goldfeld density paperSource 9. The main paper uses both other papers here. None has a published specialist review.

How sure is this? No Lean formalization is listed for these papers in the openai/math release. So no computer proof checker has checked them. No specialist review has been published.

What it doesn't show. This does not cover curves of rank 2 or more, or curves whose Selmer coranks are at least 2 at every prime.

Where it came from

This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026.

Hype check

Landmark only if all three papers and the Goldfeld density paper they use hold. No Lean formalization is listed and no specialist review is published.

Words used

Elliptic curve: A cubic curve, such as y² = x³ + ax + b, whose points can be added like numbers.
Rank: The number of independent rational points needed, with finitely many others, to generate all rational points.
L-function: A function built from the curve's point counts modulo every prime.
Analytic rank: How many times the L-function vanishes at its centre, s = 1.
Tate–Shafarevich group (Sha): Curves that have points modulo every prime and over the reals, but no rational point.
Selmer group: A computable group combining the rational points with part of Sha, for one prime q.
Corank: The number of independent infinitely divisible directions in such a group.
Complex multiplication (CM): Extra symmetries of a curve beyond multiplying points by whole numbers.

The original papers (3)

Exact Birch–Swinnerton-Dyer Formula from Low Selmer Corank (PDF, opens in a new tab)
October 7, 2026
The Selmer converse for elliptic curves at every prime (PDF, opens in a new tab)
October 7, 2026
The two-primary Birch–Swinnerton-Dyer formula in Selmer corank at most one (PDF, opens in a new tab)
October 6, 2026

All quoted lines (9)

If $s_q(E)\in\{0,1\}$ , then $r(E)=a(E)=s_q(E),\qquad \#\Sha(E/\Q)<\infty,$ and, with $r=r(E)$ and the preceding normalizations, $\frac{L^{(r)}(E,1)}{r!} =\frac{\Omega_E\Reg_E\,\#\Sha(E/\Q) \prod_{\ell\text{ finite}}c_\ell(E)} {\#E(\Q)_{\rm tors}^{2}}.$
Exact Birch–Swinnerton-Dyer Formula from Low Selmer Corank Back to text
We 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 $\Z_p$ -corank $r\in\{0,1\}$ , then the analytic and Mordell–Weil ranks both equal $r$ , and the entire Tate–Shafarevich group is finite.
The Selmer converse for elliptic curves at every prime Back to text
We establish the two-primary formula for every elliptic curve over $\mathbb Q$ whose usual two-power Selmer group has corank zero or one.
The two-primary Birch–Swinnerton-Dyer formula in Selmer corank at most one Back to text
The new assertion proved here is the exact valuation of the leading-term formula at every odd prime, independently of the prime $q$ in the hypothesis.
Exact Birch–Swinnerton-Dyer Formula from Low Selmer Corank Back to text
There are no additional hypotheses on reduction, rational torsion, isogenies, complex multiplication, or residual Galois representations.
Exact Birch–Swinnerton-Dyer Formula from Low Selmer Corank Back to text
In particular, every such $\ell$ is a sum of two rational cubes.
The Selmer converse for elliptic curves at every prime Back to text
Applying the first inequality to both curves in the second identity gives $X_p(E)=0$ in the non-CM case.
Exact Birch–Swinnerton-Dyer Formula from Low Selmer Corank Back to text
That theorem supplies the rank-zero CM input in our final step.
Exact Birch–Swinnerton-Dyer Formula from Low Selmer Corank Back to text
The present proof uses two distinct results of: the unrestricted converse at $2$ settles that prime, while the analytic twist-density theorem supplies the auxiliary fields for the odd-prime argument.
The Selmer converse for elliptic curves at every prime Back to text

Other sources

OpenAI, “Exact Birch–Swinnerton-Dyer Formula from Low Selmer Corank”, openai/math release preprint (October 7, 2026) The new assertion proved here is the exact valuation of the leading-term formula at every odd prime, independently of the prime $q$ in the hypothesis.
OpenAI, “Goldfeld's analytic density conjecture and the 2-converse for elliptic curves”, openai/math release preprint (October 7, 2026)
openai/math release notes (2026): model provenance and review status manuscripts and supporting proof artifacts produced by an internal OpenAI model
B. J. Birch and H. P. F. Swinnerton-Dyer, “Notes on elliptic curves. II”, Journal für die reine und angewandte Mathematik 218 (1965), 79–108
B. H. Gross and D. B. Zagier, “Heegner points and derivatives of L-series”, Inventiones Mathematicae 84 (1986), 225–320
V. A. Kolyvagin, “Euler systems”, in The Grothendieck Festschrift, Vol. II, Progress in Mathematics 87, Birkhäuser (1990), 435–483
C. Breuil, B. Conrad, F. Diamond and R. Taylor, “On the modularity of elliptic curves over ℚ: wild 3-adic exercises”, Journal of the AMS 14 (2001), 843–939