Wiles · official BSD description
Official Clay problem description; separates rank equality from its refined formula.
The exact passages cited in OpenMath. Each passage identifies how its wording was recorded.
For odd positive squarefree n, BSD for y²=x³−n²x would make a finite ternary-quadratic count criterion equivalent to being a congruent number.
Passage ID: birch-and-swinnerton-dyer-conjecture:bsd-congruent-number-consequence:v1
Source item: §1 Early History, PDF page 3: odd squarefree Tunnell criterion
Source record: clay-wiles-bsd@official-description
Under BSD rank equality, L(E,1) = 0 exactly when E(Q) is infinite.
Passage ID: birch-and-swinnerton-dyer-conjecture:bsd-finite-points-consequence:v1
Source item: PDF pages 1–2: Mordell theorem and “In particular” consequence of rank BSD
Source record: clay-wiles-bsd@official-description
Strong BSD predicts rank equality, finite Sha, and the leading coefficient formula with real period, regulator, Tamagawa factors and torsion.
Passage ID: birch-and-swinnerton-dyer-conjecture:bsd-full-target:v1
Source item: PDF page 2, refinement in Remark 1; standard full-L, total-real-period convention
Source record: clay-wiles-bsd@official-description
The Gross–Zagier height formula links Heegner points to central L-derivatives and supplies a key low-rank input.
Passage ID: birch-and-swinnerton-dyer-conjecture:bsd-gross-zagier-history:v1
Source item: §2, PDF pages 3–4; reference [11], Gross–Zagier (1986)
Source record: clay-wiles-bsd@official-description
General modularity of elliptic curves over Q removes the modularity restriction from the low-analytic-rank rank theorem.
Passage ID: birch-and-swinnerton-dyer-conjecture:bsd-modularity-history:v1
Source item: PDF page 4, low-rank synthesis; references [1], [24], [25]
Source record: clay-wiles-bsd@official-description
E(Q) is isomorphic to Z^r plus a finite torsion group; it is finite exactly when r = 0.
Passage ID: birch-and-swinnerton-dyer-conjecture:bsd-mordell-weil:v1
Source item: PDF page 1: Mordell theorem and rank definition
Source record: clay-wiles-bsd@official-description
Wiles distinguishes the rank statement from the refined leading-coefficient formula and records the analytic-rank-zero-or-one theorem.
Passage ID: birch-and-swinnerton-dyer-conjecture:bsd-official-rank-description:v1
Source item: PDF pages 2 and 4, conjecture/refinement and low-rank theorem
Source record: clay-wiles-bsd@official-description
Computer experiments in the early 1960s motivated the conjecture; Notes on elliptic curves II was published in 1965.
Passage ID: birch-and-swinnerton-dyer-conjecture:bsd-origin-history:v1
Source item: §2 Recent History, PDF page 3; reference [4], Notes on elliptic curves II (1965)
Source record: clay-wiles-bsd@official-description
For every elliptic curve E/Q, rank E(Q) equals ord_{s=1} L(E,s).
Passage ID: birch-and-swinnerton-dyer-conjecture:bsd-rank-target:v1
Source item: PDF page 2: rank conjecture; all finite Euler factors restored for the standard L-function
Source record: clay-wiles-bsd@official-description
Explicit citations in versioned problem publications. A citation records a use; it does not confer proof or independent verification.
Problem publication v17
Source assessment
Proved
Exact scoped statement in cited source
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 assessment
Proved
Exact scoped statement in cited source
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 assessment
Open
Exact scoped statement in cited source
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
Scope
the full BSD formula holds for every E/Q
Source assessment
Proved
Exact scoped statement in cited source
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
Connection
The full BSD scenario explicitly includes finiteness of the whole Tate–Shafarevich group.
Connection
The strong assertion includes the rank statement and additional arithmetic information.
Source assessment
Open
Exact scoped statement in cited source
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
Scope
Sha is finite for every E/Q
Source assessment
Proved
Exact scoped statement in cited source
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 assessment
Proved
Exact scoped statement in cited source
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 assessment
Open
Exact scoped statement in cited source
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
Scope
BSD rank equality holds for every E/Q
Connection
Rank BSD specialized to E_n would supply the missing sufficient direction of Tunnell’s exact counting criterion.
Question
Birch and Swinnerton-Dyer conjecture
History
Heegner points and central derivatives
History
The low-analytic-rank theorem applies to all E/Q
History
Finite generation makes rank a precise invariant
History
The published conjecture grows from computer experiments
Definition
Rank
Further reading
Read the setting and notation before comparing the results.
Linked manuscripts, code and data. External proof reports, statement correspondence and OpenMath execution are separate checks.
Manuscript / exposition · external
External evidence linked; no OpenMath execution or independent reproduction is asserted.
Resource revision: official-description
Recorded authorship
Andrew Wiles
The cited work does not report AI involvement; this is not an assertion that no AI was used.
OpenMath source records preserve exact citations. Observation dates order these records; they do not establish the latest upstream paper or proof.
Wiles · official BSD description
No upstream full text or code is republished by this record.
Source terms ↗