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Math-2783 · mathematics · Number theory

Goldfeld analytic rank density over Q

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Fix any elliptic curve E over Q, and order its quadratic twists by signed squarefree integer parameters d with $0<|d|<=X$. Must twists of analytic rank zero and twists of analytic rank one each have natural proportion tending to one half, with all higher analytic ranks having density zero?

Short English statement

Fix any elliptic curve E over Q, and order its quadratic twists by signed squarefree integer parameters d with $0<|d|<=X$. Must twists of analytic rank zero and twists of analytic rank one each have natural proportion tending to one half, with all higher analytic ranks having density zero?

Sources

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Math-2783-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2783

The result supplies the conclusion in this problem with its displayed hypotheses and quantifiers. The broader statements and applications in the manuscript are kept separate from the particular question above. The supporting result is manuscript 1 (family 006; 2026-09-23).

Goldfeld analytic rank density over Q

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 006; 2026-09-23), the asserted result is that among signed squarefree twists of each rational elliptic curve, analytic ranks zero and one each occur with density one half; the paper also claims the low-corank two-Selmer converse.

Mathematical statement of the claimed result

Manuscript 1.
For $E/\mathbb Q$, write $a(E)=\operatorname{ord}_{s=1}L(E,s)$, $r(E)=\operatorname{rank}E(\mathbb Q)$ and $c_2(E)=\operatorname{corank}_{\mathbb Z_2}\operatorname{Sel}_{2^{\infty}}(E/\mathbb Q)$. The first claimed theorem is \[ c_2(E)\in\{0,1\}\Rightarrow a(E)=r(E)=c_2(E),\quad\#\operatorname{Sha}(E/\mathbb Q)<\infty. \] Let $\mathcal D(X)$ be the nonzero signed squarefree integers with $|d|\le X$ and $E^{(d)}$ the quadratic twist. The analytic Goldfeld assertion is, for each $j=0,1$, \[ \lim_{X\to\infty}\frac{|\{d\in\mathcal D(X):a(E^{(d)})=j\}|}{|\mathcal D(X)|}=\tfrac12. \] It further claims that a density-one set of twists has equal analytic and Mordell--Weil rank and finite whole Tate--Shafarevich group. This density statement does not assert those properties for every twist.

Outline of the argument

For manuscript 1 (family 006; 2026-09-23), finite cubes of twists preserve the required local classes. A central determinant or Heegner coordinate is nonzero at the noncentral cube vertices, and bounded integral interpolation recovers the missing vertex; the Selmer distribution then supplies the density assertion.

Formal verification

No Lean formalization is listed at this snapshot for manuscript 1 (family 006; 2026-09-23).

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

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