Theorems · Theorem · number theory

WeierstrassCurve.hasAdditiveReduction_iff

∀ (R : Type u_1) [inst : CommRing R] [inst_1 : IsDomain R] [inst_2 : IsDiscreteValuationRing R] {K : Type u_2}
  [inst_3 : Field K] [inst_4 : Algebra R K] [inst_5 : IsFractionRing R K] (W : WeierstrassCurve K),
  WeierstrassCurve.HasAdditiveReduction R W ↔
    WeierstrassCurve.IsMinimal R W ∧
      (IsDedekindDomain.HeightOneSpectrum.valuation K (IsDiscreteValuationRing.maximalIdeal R)) W.Δ < 1 ∧
        (IsDedekindDomain.HeightOneSpectrum.valuation K (IsDiscreteValuationRing.maximalIdeal R)) W.c₄ < 1

Defined in

Mathlib.AlgebraicGeometry.EllipticCurve.Reduction

Cited by

1 results in Mathlib

Foundations

Depth 160 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingIsDomainIsDiscreteValuationRingFieldAlgebraIsFractionRing