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Math-2346 · mathematics · Number theory

Parity of 2-Primary Tate–Shafarevich Groups

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Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
For every elliptic curve $E/K$ over a number field, the $\mathbb{F}_2$-dimension of $\operatorname{Sha}(E/K)[2]$ is even.

Formulation references: [S1], [S2].

Short English statement

For every elliptic curve $E/K$ over a number field, the $\mathbb{F}_2$-dimension of $\operatorname{Sha}(E/K)[2]$ is even.

Sources

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

Math-2346-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2346

The arithmetic conclusions require a rational elliptic curve with Selmer corank zero or one at the specified prime; the rational-two-torsion companion has that additional hypothesis. Rank equality, finite Tate--Shafarevich group, or the stated leading-coefficient identities in those cases do not extend to all ranks and number fields in this problem. Average conclusions for quadratic twists also leave individual exceptional curves untreated.

Parity of 2-Primary Tate–Shafarevich Groups

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 002; 2026-09-24), the asserted result is that for any prime, a rational elliptic curve of Selmer corank zero or one has the corresponding analytic and rational-point ranks, together with finite Tate--Shafarevich group; the application includes primes congruent to $4$, $7$, or $8$ modulo $9$ as sums of two rational cubes.

Mathematical statement of the claimed result

Manuscript 1.
For every elliptic curve $E/\mathbb Q$, every prime $p$ and $r\in\{0,1\}$, let $s_p(E)$ be the $\mathbb Z_p$-corank of its $p^{\infty}$ Selmer group. The claimed converse is \[ s_p(E)=r\ \Longrightarrow\ \operatorname{ord}_{s=1}L(E,s)=\operatorname{rank}E(\mathbb Q)=r, \qquad\#\operatorname{Sha}(E/\mathbb Q)<\infty. \] No additional reduction, CM, residual-representation, rational-torsion or isogeny hypothesis is imposed. Coranks greater than one and an exact BSD leading coefficient are not concluded by this theorem.

Outline of the argument

For manuscript 1 (family 002; 2026-09-24), a tame deformation gives an upper bound on an Artin-module length. Integral theta-series coefficients and Heegner logarithms produce an incompatible lower bound unless the low-corank converse holds; auxiliary quadratic fields separate the two root-number cases.

Formal verification

No Lean formalization is listed at this snapshot for manuscript 1 (family 002; 2026-09-24).

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.

Sources

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