Theorem · version 2
For every elliptic curve E/Q, E(Q) ≅ Z^r ⊕ E(Q)_tors with r ≥ 0 and finite torsion. Consequently E(Q) is finite if and only if r = 0.
An unconditional foundation. Knowing finite generation does not compute the rank of an arbitrary curve.
Proved
Only the cited assertion with its exact assumptions. This report does not resolve the canonical Problem or establish independent OpenMath mathematical review.
Wiles · official BSD description · PDF page 1: Mordell theorem and rank definition →
Observed Oct 10, 2026
In this problem · publication v12 · Selected context
Supporting result · Proved
Finite generation defines the rank used in BSD without determining it for every curve.
Relationship not independently reviewed. This concerns its use in this problem.
In this problem · publication v17
Supporting result · Proved
Finite generation defines the rank used in BSD without determining it for every curve.
Relationship not independently reviewed. This concerns its use in this problem.
E(Q) is isomorphic to Z^r plus a finite torsion group; it is finite exactly when r = 0.
paraphrase · No upstream full text or code is republished by this record.
Andrew Wiles · Published
Official Clay problem description; separates rank equality from its refined formula.
Source revision: official-description. Observed Oct 10, 2026.
The cited work does not report AI involvement; this is not an assertion that no AI was used.
Official description PDF: official-description. External evidence linked; no OpenMath execution or independent reproduction is asserted.
Statement versions and problem publication versions are separate. A new publication can update context or sources while retaining the same statement.
Assessments retained in these publication records. Earlier statement versions retain their own source histories.
Proved
Only the cited assertion with its exact assumptions. This report does not resolve the canonical Problem or establish independent OpenMath mathematical review.
Observed Oct 10, 2026 · Source passage →