MinimalDiscriminant.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

2 declarations, 0 contain sorry, 0 declared as axioms

Minimal discriminants of elliptic curves over number fields

The minimal discriminant ideal of an elliptic curve over a number field, together with the local exponents cutting it out.

References

[J. Silverman, The Arithmetic of Elliptic Curves][silverman2009], Section VIII.8.

Imports (3)

Imported by 1 file

FormalConjecturesForMathlib.lean

def WeierstrassCurve.minimalDiscriminantExponent Documented declaration: minimalDiscriminantExponent line 45 source
Source documentation
The exponent of v in the minimal discriminant ideal of W: the natural number n such that the discriminant of a minimal model over the v-adic completion has valuation exp (-n). A vanishing discriminant has valuation 0, whose log is 0, so this is only meaningful for elliptic W.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, NumberField, WeierstrassCurve, IsDedekindDomain.HeightOneSpectrum, NumberField.RingOfIntegers, Nat
Proof uses: Int.toNat, Neg.neg, Int, WithZero.log, Valuation, IsDedekindDomain.HeightOneSpectrum.adicCompletion, WithZero, Multiplicative, IsDedekindDomain.HeightOneSpectrum.valuation, Subtype, Membership.mem, ValuationSubring, IsDedekindDomain.HeightOneSpectrum.adicCompletionIntegers, IsDiscreteValuationRing.maximalIdeal, WeierstrassCurve.Δ, WeierstrassCurve.minimal, WeierstrassCurve.baseChange
30 instances, coercions and auxiliary constants left out.
```
noncomputable def minimalDiscriminantExponent (W : WeierstrassCurve K)
    (v : HeightOneSpectrum (𝓞 K)) : ℕ :=
…
```
def WeierstrassCurve.minimalDiscriminant Documented declaration: minimalDiscriminant line 54 source
Source documentation
The minimal discriminant ideal of an elliptic curve over a number field is the product of the local minimal discriminant ideals v.asIdeal ^ W.minimalDiscriminantExponent v over all nonzero prime ideals v of its ring of integers. Only finitely many exponents are nonzero. See LMFDB.

L4DEL explanation
The minimal discriminant ideal of an elliptic curve over a number field is the product of the local minimal discriminant ideals v.asIdeal ^ W.minimalDiscriminantExponent v over all nonzero prime ideals v of its ring of integers. Only finitely many exponents are nonzero. See LMFDB.
Sources and attribution: Pinned Lean source (1646ca16afd6)

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, NumberField, WeierstrassCurve, Ideal, NumberField.RingOfIntegers
Proof uses: finprod, IsDedekindDomain.HeightOneSpectrum, HPow.hPow, Nat, IsDedekindDomain.HeightOneSpectrum.asIdeal, WeierstrassCurve.minimalDiscriminantExponent
13 instances, coercions and auxiliary constants left out.
```
noncomputable def minimalDiscriminant (W : WeierstrassCurve K) : Ideal (𝓞 K) :=
…
```