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Math-2252 · mathematics · Graph theory

Finiteness of a Shafarevich–Tate group needed by the lifting method

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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.
Prove finiteness of the Shafarevich–Tate group of the elliptic-curve lift required by the Huang–Raskind method, in the general cases where it is not known.

Formulation references: [S1], [S2].

Short English statement

Prove finiteness of the Shafarevich–Tate group of the elliptic-curve lift required by the Huang–Raskind method, in the general cases where it is not known.

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2252-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2252

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.

Finiteness of a Shafarevich–Tate group needed by the lifting method

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 004; 2026-09-24), the asserted result is that a rational elliptic curve with a nonidentity rational point of order two satisfies the low-corank two-adic Selmer converse even at arbitrary reduction at two.

Mathematical statement of the claimed result

Manuscript 1.
For an elliptic curve $E/\mathbb Q$, put $s_2(E)=\operatorname{corank}_{\mathbb Z_2}\operatorname{Sel}_{2^{\infty}}(E/\mathbb Q)$. If $E(\mathbb Q)[2]\ne0$ and $s_2(E)\in\{0,1\}$, the manuscript claims \[ \operatorname{ord}_{s=1}L(E,s)=\operatorname{rank}_{\mathbb Z}E(\mathbb Q)=s_2(E),\qquad \#\operatorname{Sha}(E/\mathbb Q)<\infty. \] Both rational two-torsion and the low-corank condition belong to this theorem; it does not cover arbitrary elliptic curves or arbitrary Selmer corank.

Outline of the argument

For manuscript 1 (family 004; 2026-09-24), binary coefficient systems detect central values or Heegner points. Integral interpolation and arithmetic contraction identities isolate one address in a network of twists, transferring a nonvanishing witness back to the original curve.

Formal verification

No Lean formalization is listed at this snapshot for manuscript 1 (family 004; 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.

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