Elliptic curves

Elliptic curves given by Weierstrass equations, their points over finite fields and number fields, torsion, reduction and Galois representations.

• Galois action on torsion points
• Isogeny volcanoes
• Point counts and Frobenius traces
• Supersingular elliptic curves
• Weierstrass curves
• Hasse bound

TheoremNéron–Ogg–Shafarevich criterion, easy direction#✓fully proved

Let \(R\) be a discrete valuation ring with fraction field \(k\), and let \(E\) be an elliptic curve over \(k\) given by a Weierstrass equation with good reduction at \(R\). Let \(n\) be a natural number whose image in the residue field of \(R\) is nonzero, let \(k^{\mathrm{sep}}\) be a separable closure of \(k\), and let \(\mathcal{O}\) be a valuation subring of \(k^{\mathrm{sep}}\) whose intersection with \(k\) is \(R\). Then every element \(\sigma\) of the inertia subgroup of \(\mathcal{O}\) in \(\mathrm{Gal}(k^{\mathrm{sep}}/k)\) fixes every point \(P \in E(k^{\mathrm{sep}})\) with \(nP = 0\).

Proof: We use Silverman's elementary valuation filtration at the origin in \((z, w)\) coordinates, together with reduction to the nonsingular residue curve; see The Arithmetic of Elliptic Curves, IV.1, VII.3, and VII.7.

theorem good_reduction_torsion_unramified (R : Type*)
    [CommRing R] [IsDomain R] [IsDiscreteValuationRing R]
    (k : Type*) [Field k] [Algebra R k] [IsFractionRing R k]
    (E : WeierstrassCurve k) [E.IsElliptic] [E.HasGoodReduction R]
    (n : ℕ) [NeZero (n : IsLocalRing.ResidueField R)]
    (ksep : Type*) [Field ksep] [Algebra k ksep]
    [IsSepClosure k ksep] [DecidableEq ksep]
    (𝒪 : ValuationSubring ksep)
    (h𝒪 : (𝒪.comap (algebraMap k ksep)).toSubring = (algebraMap R k).range) :
    ∀ σ ∈ 𝒪.inertiaSubgroup k, ∀ P : (E⁄ksep).Point, (n : ℤ) • P = 0 →
      WeierstrassCurve.Affine.Point.map (σ : ksep ≃ₐ[k] ksep).toAlgHom P = P

Sources

• J. H. Silverman, The Arithmetic of Elliptic Curves, 2nd ed., GTM 106, Springer, Theorem VII.7.1.

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