Goldfeld’s conjecture: half the twists have rank zero, half rank one

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Family 006 · Number theory
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-10

1. The question

An elliptic curve can have finitely many rational points or infinitely many generated from a finite collection. Its rank measures the number of independent generators of infinite order. If one varies a curve in a natural one-parameter family, how often should extra generators appear?

Goldfeld’s prediction is strikingly simple: in a quadratic-twist family, half the curves should have rank zero, half rank one, and larger ranks should be exceptional. The analytic version concerns the order of vanishing of an L-function. A further prediction says that the average analytic rank should tend to one-half.

Family 006 contains two manuscripts addressing these assertions. M1 claims the half-and-half analytic distribution for every elliptic curve over $\mathbb Q$, using a new pointwise implication from Selmer groups to L-functions. M2 claims the mean is also one-half by separately controlling rare twists of large analytic rank. That second step is necessary: knowing what happens almost everywhere need not determine an average.

2. Background and essential definitions

An elliptic curve over $\mathbb Q$ can be represented by a nonsingular cubic

$$E:\quad y^2=x^3+Ax+B,$$

with rational coefficients, together with a point at infinity serving as the identity for its group law. Its rational points form a finitely generated abelian group,

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

where $T$ is finite. The nonnegative integer $r(E)$ is the algebraic, or Mordell–Weil, rank.

For a nonzero squarefree integer $d$, the quadratic twist can be written as

$$E^{(d)}:\quad dy^2=x^3+Ax+B.$$

Over the field obtained by adjoining $\sqrt d$, a change of variable identifies it with $E$. Over $\mathbb Q$, its rational points can behave differently. Squarefree means that no prime square divides $d$; using these parameters avoids repeating the same squareclass.

The L-function $L(E,s)$ packages counts of points modulo primes. Modularity supplies its analytic continuation and functional equation. Its analytic rank is

$$a(E)=\operatorname{ord}_{s=1}L(E,s),$$

the multiplicity of its central zero; a nonzero value gives rank zero. The functional equation has a sign, called the root number. A positive sign permits even analytic rank, while a negative sign forces odd analytic rank. Ranks zero and one are the smallest possibilities in the two cases. The BSD conjecture predicts $a(E)=r(E)$.

The full 2-power Selmer group is an arithmetic enlargement of the rational-point information obtained from repeated doubling and local solvability. Its corank satisfies

$$c_2(E)=r(E)+ \operatorname{corank}_{\mathbb Z_2}\operatorname{Sha}(E/\mathbb Q)[2^\infty].$$

Here $\operatorname{Sha}$ is the Shafarevich–Tate group, measuring certain obstructions to passing from local to global solutions, and $[2^\infty]$ denotes its subgroup killed by powers of two. Thus low rational-point rank alone does not ensure low Selmer corank.

Both manuscripts count the same set:

$$\mathcal D(X)=\{d\in\mathbb Z:0<|d|\le X, \ d\text{ squarefree}\}.$$

Positive and negative parameters are included together and ordered by absolute value. Density is a limiting proportion within this set.

3. Development history and previous partial results

Goldfeld’s 1979 paper proposed an average analytic-rank prediction using quadratic-field discriminants. The present manuscripts formulate their conclusions in the signed squarefree convention above. Different orderings and restrictions need to be kept explicit when comparing theorems.

Analytic work established many twists with nonzero central values or nonzero first derivatives. Results of Bump–Friedberg–Hoffstein, Murty–Murty, Iwaniec, and Perelli–Pomykała developed this approach. Positive proportions were obtained in special families, including Vatsal’s work on a particular modular curve and Kriz–Li’s results for curves admitting a rational 3-isogeny. These advances did not yield the exact two half-densities for every curve.

The arithmetic side developed through Selmer statistics. Alexander Smith’s unrestricted theorem establishes that full 2-power Selmer coranks zero and one each occur half the time for every rational elliptic curve’s quadratic twists. Its analytic consequence assumes BSD. M1 uses this Selmer distribution as an input and claims to supply the missing pointwise analytic implication.

The direction of that implication matters. Gross–Zagier and Kolyvagin established that analytic rank zero or one gives the corresponding algebraic rank and finiteness of $\operatorname{Sha}$. A converse starts from Selmer information and must produce analytic nonvanishing. Earlier converse results covered substantial ranges at odd primes, with reduction or residual-representation conditions, and important complex-multiplication cases at two. M1 claims to remove these restrictions at the prime two.

For means, conditional results already illustrated the extra difficulty. Heath-Brown bounded analytic-rank averages under Riemann-hypothesis assumptions for twists. Fiorilli obtained an exact conditional mean, using both a Riemann-hypothesis assumption and an additional averaged cancellation hypothesis. M2 claims an unconditional analytic tail estimate instead. Its separate algebraic-moment consequence uses exponential rank bounds of Koymans and Smith.

4. What the new manuscripts claim

Pointwise converse and half-densities

M1, Theorem 1.1, states that for every elliptic curve $E/\mathbb Q$,

$$c_2(E)\in\{0,1\} \quad\Longrightarrow\quad a(E)=r(E)=c_2(E),\qquad \operatorname{Sha}(E/\mathbb Q)\text{ is finite}.$$

There are no restrictions on reduction type, complex multiplication, rational torsion, or rational isogenies. It concerns the full 2-power Selmer corank, not the dimension of a single finite Selmer group.

Combining this with Smith’s statistics, M1, Theorem 1.2, claims that for $j=0,1$,

$$\lim_{X\to\infty} \frac{\#\{d\in\mathcal D(X):a(E^{(d)})=j\}} {\#\mathcal D(X)}=\frac12.$$

Consequently, analytic rank at least two has density zero. On a density-one set, analytic and algebraic ranks agree and the whole Shafarevich–Tate group is finite.

M1 also gives a finite-descent criterion. Put

$$d_2(E)=\dim_{\mathbb F_2}\operatorname{Sel}_2(E/\mathbb Q) -\dim_{\mathbb F_2}E(\mathbb Q)[2],$$

where $E(\mathbb Q)[2]$ consists of rational points killed by doubling. If $d_2(E)$ is zero or one, Corollary 1.3 concludes that all three ranks equal $d_2(E)$, $\operatorname{Sha}$ is finite, and its entire 2-primary subgroup is trivial.

Mean and tail control

M2, Theorem 1.1, claims

$$\lim_{X\to\infty}\frac1{\#\mathcal D(X)} \sum_{d\in\mathcal D(X)}a(E^{(d)})=\frac12.$$

The additional input is its Theorem 1.2: for constants $C_E>0$ and an integer threshold $R_E$, every integer $R\ge R_E$ satisfies

$$\limsup_{X\to\infty}\frac1X \sum_{\substack{d\in\mathcal D(X)\\a(E^{(d)})>R}} a(E^{(d)})\le\frac{C_E}{R}.$$

The height limit comes first; the rank threshold is then allowed to grow. This is a bound on total rank contribution, not merely on how many exceptional twists occur.

M2 also gives algebraic-rank moment limits. For fixed real $t$ and fixed positive integer $m$,

$$\begin{aligned} \lim_{X\to\infty}\frac1{\#\mathcal D(X)} \sum_{d\in\mathcal D(X)} e^{t r(E^{(d)})} &=\frac{1+e^t}{2},\\ \lim_{X\to\infty}\frac1{\#\mathcal D(X)} \sum_{d\in\mathcal D(X)} r(E^{(d)})^m &=\frac12. \end{aligned}$$

These stronger moment statements concern algebraic ranks. They are not higher analytic-rank moment theorems.

5. Why the advance matters

Provisional importance: 5/5. Goldfeld’s conjecture is a central prediction about the distribution of rational points in families. Resolving both its analytic density and mean formulations for every fixed rational elliptic curve would connect detailed arithmetic statistics to analytic behavior with exceptional generality.

The pointwise converse also matters independently. Statistical statements allow exceptions; a pointwise theorem applies to a prescribed curve once its Selmer hypothesis is known. That makes the result useful beyond the distribution problem and explains its role in the BSD claims of family 002.

The two manuscripts also clarify a general probabilistic issue. Among $N$ objects, one exceptional object of value $N$ has proportion $1/N$, tending to zero, but contributes one to the average. Density-zero high ranks therefore cannot simply be ignored. M2’s tail estimate is the part that rules out this obstruction for analytic rank.

6. The main idea

Recover one curve from many auxiliary twists

M1 fixes the target curve and constructs finite families of twists indexed by binary vectors

$$x\in\mathbb F_2^b.$$

The zero vector represents the original curve. Auxiliary primes enter each twist according to linear functions of $x$, while prescribed local behavior at the original bad primes is preserved.

The other vertices are arranged to have nonzero central values or simple central zeros, as appropriate, with uniformly bounded normalized powers of two. An integral interpolation argument then recovers nonvanishing at the missing zero vertex. The uniformity is crucial: the family can grow, but the bound needed for interpolation must not deteriorate with its dimension.

For corank zero, the arithmetic object being interpolated is related to Kato’s classes and detects $L(E,1)$. For corank one, it comes from Heegner points, special points constructed using an imaginary quadratic field. Their heights relate to a central derivative. A fixed companion twist with nonzero central value turns the relevant product formula into a detector for the target derivative.

The finite construction has two branches. With rational two-torsion, graphs encode prescribed quadratic residue signs between auxiliary primes and enforce simultaneous nonvanishing of coefficient tests. Without rational two-torsion, matrices over $\mathbb F_2$ encode local Selmer constraints. Their kernel dimensions interact with identities among coefficient tests to construct the required families. The manuscript does not identify those tests with matrix determinants merely because they satisfy related identities.

Once the unrestricted pointwise converse is obtained, Smith’s half-densities for Selmer coranks transfer to analytic ranks. The arithmetic statistics and the analytic implication have distinct roles.

Control the rank carried by rare exceptions

M2 starts from a basic fact: if an analytic function has a zero of order greater than $k$, its $k$-th derivative vanishes there. Applied to the completed L-function, this produces an exact identity for a weighted series.

The manuscript designs smooth weights and short mollifiers, finite approximations to reciprocal Euler products. Their purpose is to make the first portion of the series close to a positive main contribution while keeping later portions smaller on most twists. Mean-square estimates control the exceptional parameters. Consecutive derivative orders handle both functional-equation signs.

The asserted counting estimate is of order $X/k^2$ for ranks exceeding $k$, uniformly up to

$$k=(\log X)^{3/5}.$$

A separate pointwise bound of order $\log X$ controls ranks beyond that range. Combining the count at the largest admissible $k$ with the pointwise bound makes the far tail negligible. Summing the remaining tail counts yields the displayed rank-weighted estimate.

Finally, split the average into ranks zero and one, ranks two through a fixed $R$, and ranks greater than $R$. M1 supplies the contribution one-half from rank one and makes the middle part vanish. M2 controls the last part by a quantity tending to zero as $R$ grows. This is the passage from densities to the mean.

7. Scope, limitations, and open questions

Every theorem fixes an elliptic curve before the twist parameter tends to infinity. The conclusions do not state uniform rates across all curves. The half-and-half assertion uses both signs of squarefree parameters together; an arbitrary residue class can fix the functional-equation sign and need not have that distribution.

Density zero does not mean finitely many higher-rank twists. The manuscripts do not settle all questions about the existence, frequency, or maximal size of high ranks. M2’s higher moment formulas are algebraic only, with moment order fixed before the limit; no corresponding assertion is made for growing moment orders or thin polynomial subfamilies.

Rank equality and finiteness of $\operatorname{Sha}$ are distinct from the exact BSD leading-coefficient formula. That formula requires the additional manuscripts discussed in family 002. Neither a general BSD theorem for every rank nor a rule deciding every curve’s rank follows from the statements here.

8. Sources and further reading

M1’s Theorems 1.1–1.2 and Corollary 1.3 state the pointwise converse, densities, and finite-descent criterion. Its introduction explains the finite-twist interpolation strategy. M2’s Theorems 1.1–1.2 separate the mean from the tail estimate; Corollary 1.3 gives the algebraic moments.

Goldfeld’s original paper and Smith’s Selmer-distribution theorem are linked in Section 3. Silverman’s The Arithmetic of Elliptic Curves supplies a systematic background treatment. For optional definitions, see quadratic twists and the Birch–Swinnerton-Dyer conjecture.

Source manuscripts

• M1. OpenAI, Goldfeld’s analytic density conjecture and the 2-converse for elliptic curves (October 7, 2026).
• M2. OpenAI, The mean analytic rank of quadratic twists of elliptic curves (October 6, 2026).

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