Good reduction outside finitely many primes

An elliptic Weierstrass equation over the fraction field of a Dedekind domain has integral coefficients and unit discriminant in almost every completed valuation ring. The equation itself therefore gives an elliptic integral model at all but finitely many primes; no global minimal equation is needed.

For a nonzero multiplier m, one can also exclude the primes dividing m. The resulting finite exceptional set supplies the good-reduction and residue-characteristic conditions used to show that Kummer classes, and hence Selmer classes, are unramified outside a finite set.

The statements use the ring of integers 𝒪[K_v] of the canonical valuative relation, so that they apply directly to local Galois cohomology. An elliptic integral model means that its discriminant is a unit in that ring, a stronger condition than just ellipticity over K_v.

References

• J. Silverman, The Arithmetic of Elliptic Curves, VII.5.1 and X.4.2.

theorem TauCeti.eventually_isElliptic_integralModel_adicCompletion (R : Type u_1) [CommRing R] [IsDedekindDomain R] {K : Type u_2} [Field K] [Algebra R K] [IsFractionRing R K] (W : WeierstrassCurve K) [W.IsElliptic] : ∀ᶠ (v : IsDedekindDomain.HeightOneSpectrum R) in Filter.cofinite, ∃ (x : WeierstrassCurve.IsIntegral (↥(ValuativeRel.valuation (IsDedekindDomain.HeightOneSpectrum.adicCompletion K v)).integer) (W.baseChange (IsDedekindDomain.HeightOneSpectrum.adicCompletion K v))), (WeierstrassCurve.integralModel (↥(ValuativeRel.valuation (IsDedekindDomain.HeightOneSpectrum.adicCompletion K v)).integer) (W.baseChange (IsDedekindDomain.HeightOneSpectrum.adicCompletion K v))).IsElliptic

An elliptic equation has an elliptic integral model over almost every completed valuation ring. The model base changes to the given equation itself, without a change of variables.

theorem TauCeti.exists_finset_isElliptic_integralModel_adicCompletion (R : Type u_1) [CommRing R] [IsDedekindDomain R] {K : Type u_2} [Field K] [Algebra R K] [IsFractionRing R K] (W : WeierstrassCurve K) [W.IsElliptic] {m : ℕ} (hm : ↑m ≠ 0) (S : Finset (IsDedekindDomain.HeightOneSpectrum R)) : ∃ (T : Finset (IsDedekindDomain.HeightOneSpectrum R)), S ⊆ T ∧ (∀ v ∉ T, ∃ (x : WeierstrassCurve.IsIntegral (↥(ValuativeRel.valuation (IsDedekindDomain.HeightOneSpectrum.adicCompletion K v)).integer) (W.baseChange (IsDedekindDomain.HeightOneSpectrum.adicCompletion K v))), (WeierstrassCurve.integralModel (↥(ValuativeRel.valuation (IsDedekindDomain.HeightOneSpectrum.adicCompletion K v)).integer) (W.baseChange (IsDedekindDomain.HeightOneSpectrum.adicCompletion K v))).IsElliptic) ∧ ∀ v ∉ T, ↑m ∉ v.asIdeal

Outside a finite set containing any prescribed finite set of primes, an elliptic equation has an elliptic integral model and a nonzero multiplier is prime to the residue characteristic. This supplies the exceptional set for the unramified local Kummer conditions.