The full BSD formula from a small Selmer group

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Family 002 · Number theory
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-10

1. The question

An elliptic curve can have infinitely many rational points even though its equation is very simple. The curve $y^2=x^3+ax+b$, when nonsingular and completed by its point at infinity, carries a group law. A basic arithmetic question asks how many independent rational points are needed to generate all the others, apart from a finite torsion subgroup.

The Birch–Swinnerton-Dyer conjecture, usually abbreviated BSD, predicts that this arithmetic information is encoded in an analytic function built from counting points modulo primes. It predicts both an order of vanishing and an exact leading coefficient. Knowing how many times a function vanishes does not determine the first nonzero coefficient, just as knowing that a polynomial has a double root does not determine its quadratic term.

Family 002 contains three manuscripts claiming the full formula when a prime-power Selmer group has corank zero or one. M2 supplies the unrestricted converse from this arithmetic condition to analytic rank and finiteness. M3 treats the exact power of two in the coefficient formula. M1 adds the odd-prime comparisons and assembles the exact equality.

The condition is on a Selmer group, not simply the number of independent rational points already found on the curve.

2. Background and essential definitions

Three measures of rank

For an elliptic curve $E/\mathbb Q$, the Mordell–Weil theorem gives

$$E(\mathbb Q)\cong\mathbb Z^r\oplus T,$$

where $T$ is finite. The integer $r$ is the Mordell–Weil rank. Rank zero means only finitely many rational points; positive rank means infinitely many.

At a prime $\ell$ of good reduction, let $N_\ell$ count the points of the reduced curve, including its point at infinity, and set $a_\ell=\ell+1-N_\ell$. These counts contribute factors

$$\left(1-a_\ell\ell^{-s}+\ell^{1-2s}\right)^{-1}$$

to the Hasse–Weil function $L(E,s)$. There are specified factors at bad primes as well. Modularity gives analytic continuation and a functional equation with center $s=1$. The analytic rank is the order of vanishing there, written $\operatorname{ord}_{s=1}L(E,s)$.

A Selmer group packages arithmetic compatibility conditions over $\mathbb Q$, the real numbers, and all prime-adic completions. For a prime $p$, the full $p$-power Selmer group uses all powers $p^n$, with the usual local conditions everywhere. It is larger than the information supplied by rational points alone.

The discrepancy is measured by the Tate–Shafarevich group, written $\operatorname{Sha}(E/\mathbb Q)$. Its elements describe principal homogeneous spaces for $E$: curves with an $E$-action which look like $E$ after choosing a point. The classes in Sha have points over every completion but may have no rational point. Thus they record a failure of passing from local solvability to global solvability.

The standard exact sequence is

$$0\longrightarrow E(\mathbb Q)\otimes(\mathbb Q_p/\mathbb Z_p) \longrightarrow\operatorname{Sel}_{p^\infty}(E/\mathbb Q) \longrightarrow\operatorname{Sha}(E/\mathbb Q)[p^\infty] \longrightarrow0.$$

Here $[p^\infty]$ selects elements killed by some power of $p$. The corank $s_p(E)$ counts the copies of the divisible group $\mathbb Q_p/\mathbb Z_p$ in the Selmer group, ignoring finite parts. It counts the rational-point rank and potentially an additional divisible contribution from Sha. If Sha is finite, then $s_p(E)=r$. Assuming $r\leq1$ alone does not establish the Selmer condition.

The factors in the leading coefficient

Write $\Omega_E$ for the integral of the absolute value of a minimal Néron differential over all real components of $E(\mathbb R)$. A Néron differential is the invariant differential normalized by a minimal integral model. The finite Tamagawa number $c_\ell$ is the index of the subgroup reducing to the identity component inside $E(\mathbb Q_\ell)$; it is one at good primes.

The regulator $\operatorname{Reg}_E$ measures the volume of the full free rational-point lattice using the canonical height pairing. To fix the convention used in these manuscripts, let $h_x(P)=\log\max(|u|,|v|)$ when $x(P)=u/v$ in lowest terms, and put

$$H(P)=\lim_{n\to\infty}4^{-n}h_x([2^n]P), \qquad B(P,Q)=\frac{H(P+Q)-H(P)-H(Q)}2.$$

The regulator is the determinant of $B$ on a basis of $E(\mathbb Q)/T$, and equals one in rank zero. A subgroup of index $m$ has regulator $m^2\operatorname{Reg}_E$. This distinction matters when a construction produces some independent points without proving that they form a full basis.

3. Development history and previous partial results

Birch and Swinnerton-Dyer formulated their conjectures in the 1960s from computations relating rational points to point counts over finite fields. Their Notes on elliptic curves II is a primary source for this development. Tate placed the proposed rank and leading-term identities in a wider arithmetic framework, while Cassels explained how the combined expression behaves under isogeny.

The modularity work of Wiles, Taylor–Wiles, and Breuil–Conrad–Diamond–Taylor established the analytic setting for all elliptic curves over $\mathbb Q$. The Gross–Zagier height formula relates derivatives of suitable L-functions to heights of Heegner points. Kolyvagin’s Euler systems then yield equality of analytic and arithmetic ranks, and finiteness of Sha, when analytic rank is zero or one.

Those results run from analytic information to arithmetic conclusions. A Selmer converse reverses the direction: small arithmetic Selmer corank must force a nonzero central value or derivative. That reversal is a separate problem, particularly when the curve has difficult reduction or a reducible representation on its torsion points.

Iwasawa theory, which studies arithmetic across towers of number fields, supplied increasingly strong converse and coefficient results. Kato’s zeta elements provide one direction of comparison; Skinner–Urban, Wei Zhang, and Jetchev–Skinner–Wan obtained major low-rank results under their respective local and residual hypotheses. Later work treated additional ordinary, supersingular, Eisenstein, and multiplicative cases. These labels distinguish how the curve and its torsion behave at the chosen prime; the manuscript’s historical sections explain the precise earlier restrictions.

Complex multiplication, meaning extra endomorphisms of the elliptic curve, provides another substantial theory. Burungale–Flach establish full rank-zero BSD for their class of CM curves, with consequences over $\mathbb Q$. Exact two-primary formulas were also known in special CM and quadratic-twist settings. The family claims to remove the remaining curve and prime restrictions under the stated low-Selmer-corank hypothesis, while retaining every numerical factor in BSD.

4. What the new manuscripts claim

The complete formula

M1’s Theorem 1.1 states that, for every elliptic curve $E/\mathbb Q$, if some prime $q$ satisfies $s_q(E)\in\{0,1\}$, then

$$r=\operatorname{rank}E(\mathbb Q) =\operatorname{ord}_{s=1}L(E,s)=s_q(E), \qquad \#\operatorname{Sha}(E/\mathbb Q)<\infty,$$

and

$$\frac{L^{(r)}(E,1)}{r!} =\frac{\Omega_E\operatorname{Reg}_E\, \#\operatorname{Sha}(E/\mathbb Q)\prod_{\ell\text{ finite}}c_\ell} {(\#T)^2}.$$

Here $L^{(r)}$ is the $r$-th derivative, so the left side is the first nonzero Taylor coefficient. The uncompleted L-function includes all finite Euler factors. The total real period already includes every real component; no extra real-component factor is inserted.

No additional assumption is imposed on reduction, rational torsion, rational isogenies, complex multiplication, or residual Galois representations. The prime $q$ supplying the input condition need not equal a prime whose valuation is being checked in the output formula.

The roles of the three manuscripts

Manuscript	Principal contribution within the family
M1	The exact full leading-term formula, completing the odd-prime comparisons and assembling all primes.
M2	The low-corank Selmer converse at every prime, with finiteness of the entire Sha group.
M3	The exact two-primary leading-term formula, with no restriction at two.

To state M3 precisely, define the normalized positive quantity

$$Q_E=\frac{L^{(r)}(E,1)(\#T)^2} {r!\,\Omega_E\operatorname{Reg}_E\prod_\ell c_\ell}.$$

It claims that $s_2(E)\leq1$ gives the rank and finiteness conclusions, $Q_E\in\mathbb Q_{>0}$, and

$$v_2(Q_E)=v_2\!\left(\#\operatorname{Sha}(E/\mathbb Q)\right).$$

The valuation $v_p$ counts the exponent of $p$ in a nonzero rational number. Matching the valuation at two is one part of the full equality, not the whole equality.

M2 also derives the positive prime cube-sum cases $\ell\equiv4,7,8\pmod9$: such primes are sums of two rational cubes, and the corresponding cubic has rank one and finite Sha. The paper acknowledges earlier direct treatments of these cases; this application illustrates its unrestricted converse at three.

Combining the family with the twist-density results discussed in family 006 gives the formula for a density-one set of signed squarefree twists of each fixed curve. That additional conclusion uses the distribution input.

5. Why the advance matters

Importance: 5/5, provisional. BSD connects rational solutions, complex analysis, local arithmetic, and the failure of the local-to-global principle. An exact leading-term formula under an unrestricted low-Selmer-corank criterion would be a major advance on a central arithmetic problem.

The word “exact” has substantial content. Equality up to an unknown power of two, or up to factors at bad primes, leaves part of the conjecture unresolved. Likewise, proving that Sha is finite does not compute its order. The family claims to retain precisely the finite information that distinguishes these assertions.

Its reach is large but qualified: the initial Selmer condition remains essential. The density-one twist consequence means that exceptions have asymptotic proportion zero in the specified family; it does not eliminate every exceptional twist or prove BSD for arbitrary rank.

6. The main idea

Force analytic nonvanishing from arithmetic smallness

For the converse, M2 reduces an odd-prime problem to showing that a certain Heegner point is not torsion. Heegner points arise from elliptic curves with complex multiplication mapped into the curve under study. The appropriate height formula then connects this point to an L-value or derivative.

Assuming the point is torsion, the paper obtains two incompatible measurements of the same auxiliary extension space. The low Selmer corank bounds how many Galois extensions satisfying specified local conditions can exist. A theta-series construction, transferred through congruences of automorphic forms, produces more such extensions than that bound permits.

A theta series is a structured generating function built from arithmetic data. Here the construction retains a truncated deformation parameter, so it records successive layers of extension information rather than only isolated character values. The stated contradiction forces the Heegner point to be non-torsion. The argument uses the separate two-converse and twist-density inputs identified in the manuscript.

Measure the error prime by prime

Once rank and finiteness are available, define

$$X_p(E)=v_p(Q_E)-v_p\!\left(\#\operatorname{Sha}(E/\mathbb Q)\right).$$

For odd $p$, M1 proves a nonnegative discrepancy for non-CM curves of analytic rank at most one. A second comparison pairs $E$ with a suitable quadratic twist $E^D$ of complementary analytic rank and gives

$$X_p(E)+X_p(E^D)=0.$$

Two nonnegative numbers with sum zero must both vanish. CM curves are handled using the appropriate CM rank-zero input and a paired comparison. The same numerical idea appears in M3’s treatment at two, with additional machinery suited to that prime.

The difficult step is obtaining exact comparisons. The papers encode arithmetic in integral lattices and determinant lines, which track finite indices as well as vector-space dimensions. Their quotients must be units, elements with no unwanted divisibility, rather than merely integral elements. Theta comparisons and auxiliary deformation directions exclude possible extra factors. M3 treats the distinct torsion-representation cases and the real place explicitly.

Finally, M3 supplies positivity, rationality, and the correct valuation at two; M1 supplies every odd valuation. The positive rational number $Q_E/\#\operatorname{Sha}$ then has exponent zero at every prime. Its prime factorization is empty, so it equals one. That last elementary observation turns the local comparisons into the full real-valued BSD formula.

7. Scope, limitations, and open questions

These statements concern elliptic curves over $\mathbb Q$. They do not assert unrestricted BSD over all number fields or in higher rank. In particular, a known Mordell–Weil rank of zero or one is not substituted for the full Selmer-corank assumption.

The precise normalizations of the period, height pairing, regulator, Euler factors, and local conditions are part of the result. Changing a height by a factor of two or using a sublattice of rational points changes the numerical formula unless the corresponding correction is made.

The family does not supply an efficient general algorithm to find a basis of rational points or compute Sha. Nor does its twist-density consequence imply a bound on every exceptional curve. Higher-corank cases and broader base fields remain outside the stated coverage.

8. Sources and further reading

The three Theorems 1.1 contain the principal assertions. M1’s introductory discrepancy calculation explains the final assembly; M2’s introduction describes the extension-length contradiction; M3’s first section distinguishes integrality from the stronger unit comparison needed at two.

The historical sources linked above explain the development from point-counting experiments to Heegner points and Iwasawa theory. Family 006 treats the complementary distribution question: how often the relevant small coranks occur among quadratic twists.

Source manuscripts

• M1. OpenAI, Exact Birch–Swinnerton-Dyer Formula from Low Selmer Corank (October 7, 2026).
• M2. OpenAI, The Selmer converse for elliptic curves at every prime (October 7, 2026).
• M3. OpenAI, The two-primary Birch–Swinnerton-Dyer formula in Selmer corank at most one (October 6, 2026).

Sources consulted on 10 October 2026 at repository revision fd4aeeb. Download the manuscript citations.