GoodReduction.lean`

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1 declarations, 1 contain sorry, 0 declared as axioms

Let E be an elliptic curve over the field of fractions k of a DVR, with good reduction. Let n be a positive integer which is nonzero in k. Then the Galois representation on the n-torsion points over k^sep is unramified.

This is the easy direction of the criterion of Néron–Ogg–Shafarevich; see for example [Silverman, The Arithmetic of Elliptic Curves, VII.7.1] or [Serre–Tate, Good reduction of abelian varieties, Theorem 1 for the general abelian variety case].

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Imported by 1 file

FLT.lean

theorem WeierstrassCurve.torsion_unramified_of_good_reduction Documented declaration: torsion_unramified_of_good_reduction sorry line 57 source
Source documentation
If E is an elliptic curve over k (given by a minimal Weierstrass equation) with good reduction over R, and if 𝒪 is a valuation subring of kˢᵉᵖ lying above R, then the inertia subgroup of Gal(kˢᵉᵖ/k) at 𝒪 acts trivially on the n-torsion of E(kˢᵉᵖ). In other words, the Galois representation on the n-torsion points is unramified.

depends on sorry what this means
Statement uses: CommRing, IsDomain, IsDiscreteValuationRing, Field, Algebra, IsFractionRing, WeierstrassCurve, WeierstrassCurve.IsElliptic, WeierstrassCurve.HasGoodReduction, Nat, NeZero, IsLocalRing.ResidueField, IsSepClosure, DecidableEq, ValuationSubring, Eq, Subring, ValuationSubring.toSubring, ValuationSubring.comap, Algebra.algebraMap and 14 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
79 instances, coercions and auxiliary constants left out.
```
theorem WeierstrassCurve.torsion_unramified_of_good_reduction
    -- Assume 𝒪 lies above R, i.e. 𝒪 ∩ k = R
    (h𝒪 : (𝒪.comap (algebraMap k ksep)).toSubring = (algebraMap R k).range) :
    -- Then every element of the inertia subgroup at 𝒪 fixes every n-torsion point of E(ksep)
    ∀ σ ∈ 𝒪.inertiaSubgroup k, ∀ P ∈ AddSubgroup.torsionBy (E⁄ksep).Point (n : ℤ),
      Affine.Point.map (σ : ksep ≃ₐ[k] ksep).toAlgHom P = P :=
…
```