FormalConjecturesForMathlib.AlgebraicGeometry.EllipticCurve.MinimalDiscriminant

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module

public import Mathlib.Algebra.BigOperators.Finprod
public import Mathlib.AlgebraicGeometry.EllipticCurve.Reduction
public import Mathlib.NumberTheory.NumberField.Completion.FinitePlace

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.

@[expose] public sectionnamespace WeierstrassCurveopen NumberField IsDedekindDomainvariable {K : Type*} [Field K] [NumberField K]

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.

noncomputable def minimalDiscriminantExponent (W : WeierstrassCurve K)
    (v : HeightOneSpectrum (𝓞 K)) : ℕ :=
  (-WithZero.log ((IsDiscreteValuationRing.maximalIdeal (v.adicCompletionIntegers K)).valuation
    (v.adicCompletion K) ((W⁄(v.adicCompletion K)).minimal (v.adicCompletionIntegers K)).Δ)).toNat

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.

noncomputable def minimalDiscriminant (W : WeierstrassCurve K) : Ideal (𝓞 K) :=
  ∏ᶠ v : HeightOneSpectrum (𝓞 K), v.asIdeal ^ W.minimalDiscriminantExponent v

end WeierstrassCurve