BSD.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

L4DEL explanations are a separate layer. They do not change the Lean source. Intuition is explanatory prose; numbered items are a reading path; sources link the wording back to evidence.

L4DEL: Lean4 Domain Explanation Language

Annotation provenance. This overlay contains 10 curated AI-written annotations and 7190 mechanical source-derived annotations. They have not been human-reviewed as explanations; treat them as unverified and not mathematical authority. Retained generation history: The 71 original curated annotations were produced in a Claude Sonnet 5.5 session at medium effort; Claude Opus 5.5 is recorded in that session as the advisor model, not the direct author. The bulk source-derived expansion was generated mechanically from pinned source comments by tooling developed in a GPT-5.6 Sol session at ultra reasoning; GPT-5.6 Sol did not originate the quoted source prose. On 2026-10-17 the mechanically generated source-comment annotations of the Formal Conjectures, Carleson, FLT, PFR, CompFiles, Mathematics in Lean and Theorem Proving in Lean overlays were regenerated by the same generator after three corrections made in a Claude Sonnet 5.5 session: titles no longer cut a sentence inside mathematics or end on a dangling word, workflow markers are removed from the quoted prose, and declaration names follow the scanner's namespace resolution. The boilerplate step line was dropped, and comments that were only a row of dashes no longer yield annotations. No new prose was written and no human review is claimed.

Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

3 declarations, 2 contain sorry, 0 declared as axioms

The Birch and Swinnerton-Dyer (BSD) Conjecture

References:

The Clay Institute, official problem description by Andrew Wiles: claymath.org
[BSD1965] B. J. Birch and H. P. F. Swinnerton-Dyer. "Notes on elliptic curves. II." Journal fur die reine und angewandte Mathematik 218 (1965), 79-108, doi
[Tate1966] John Tate. "On the conjectures of Birch and Swinnerton-Dyer and a geometric analog." Seminaire Bourbaki, Vol. 9, Exp. No. 306 (1966), 415-440, numdam
[Gross2011] Benedict H. Gross. "Lectures on the conjecture of Birch and Swinnerton-Dyer." Arithmetic of L-functions, IAS/Park City Math. Ser. 18, AMS (2011), 169-209, math.harvard.edu
[Ang2025] David Kurniadi Angdinata. "L-functions of Dirichlet twists of elliptic curves: computations and congruences." PhD thesis, University College London (2025), discovery.ucl.ac.uk
[Ada] Tom Adamczewski. "Autoformalized conjectures", Birch and Swinnerton-Dyer

Imports (2)

def BSD.Weak The weak Birch and Swinnerton-Dyer conjecture for a number field line 60 source
Source documentation
The weak Birch and Swinnerton-Dyer conjecture for a number field $K$: for every elliptic curve $E$ over $K$, a meromorphic continuation of its $L$-series has order $\operatorname{rank}_{\mathbb{Z}} E(K)$ at $s = 1$. [Gross2011], Conjecture 2.10 states the conjecture assuming only a meromorphic continuation near $s = 1$, while HasseWeil.HasMeromorphicContinuation asks for one on all of $\mathbb{C}$.
The rank is AddCommGroup.freeRank, which requires $E(K)$ to be finitely generated. That is the Mordell--Weil theorem, which Mathlib does not have and which this repository states as a sorry in EllipticCurveRank.mordell_weil, so it appears here as a hypothesis.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, NumberField, DecidableEq
Proof uses: WeierstrassCurve, WeierstrassCurve.IsElliptic, AddGroup.FG, WeierstrassCurve.Affine.Point, WeierstrassCurve.toAffine, Complex, HasseWeil.HasMeromorphicContinuation, Eq, WithTop, Int, meromorphicOrderAt, OfNat.ofNat, AddCommGroup.freeRank
15 instances, coercions and auxiliary constants left out.
```
def Weak (K : Type*) [Field K] [NumberField K] [DecidableEq K] : Prop :=
…
```
theorem BSD.weak_birch_swinnerton_dyer_conjecture Weak Birch and Swinnerton-Dyer conjecture ([Tate1966], Conjecture (A)) sorry line 67 source
Source documentation
Weak Birch and Swinnerton-Dyer conjecture ([Tate1966], Conjecture (A)).

depends on sorry what this means
Statement uses: Field, NumberField, DecidableEq, BSD.Weak
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
1 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem weak_birch_swinnerton_dyer_conjecture (K : Type*) [Field K] [NumberField K]
    [DecidableEq K] : Weak K := by
…
```
Mentions: BSD.Weak
theorem BSD.weak_birch_swinnerton_dyer_conjecture_rat The weak Birch and Swinnerton-Dyer conjecture over $\mathbb{Q}$, a Clay Millennium Prize Problem sorry line 74 source
Source documentation
The weak Birch and Swinnerton-Dyer conjecture over $\mathbb{Q}$, a Clay Millennium Prize Problem.

depends on sorry what this means
Statement uses: BSD.Weak, Rat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
7 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem weak_birch_swinnerton_dyer_conjecture_rat : Weak ℚ := by
…
```
Mentions: BSD.Weak