Tamagawa.lean`

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Tamagawa numbers of elliptic curves

For a non-archimedean local field K, the Tamagawa number is [E(K) : E₀(K)], where E₀(K) is the points with non-singular reduction on a minimal Weierstrass model. For a global field K, the Tamagawa product is the finitely supported product over all non-archimedean completions.

We describe E₀(K) as a set using integral projective representatives, because its group structure has not been proven, and count its distinct additive translates in E(K). The local definitions take in an integral model, which gives the usual local definitions when it is minimal.

References

[Silverman2009] Joseph H. Silverman, The Arithmetic of Elliptic Curves, 2nd ed., Graduate Texts in Mathematics 106, Springer (2009), doi. Chapter VII, §§1–2 and §6: minimal equations, reduction of points, and E(K) / E₀(K); see pp. 187–188 for the reduction map and E₀(K), Proposition VII.2.1 for the subgroup and homomorphism properties, and Theorem VII.6.1 and Corollary VII.6.2 on p. 200.
LMFDB, Tamagawa number

Imports (5)

Imported by 1 file

FormalConjecturesForMathlib.lean

def WeierstrassCurve.Projective.nonsingularReduction The points with an integral projective representative whose reduction is non-singular line 52 source
Source documentation
The points with an integral projective representative whose reduction is non-singular.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, IsLocalRing, Algebra, WeierstrassCurve.Projective, Set, WeierstrassCurve.Projective.Point, WeierstrassCurve.Projective.baseChange
Proof uses: Set.ofPred, Exists, Fin, OfNat.ofNat, Nat, And, Eq, WeierstrassCurve.Projective.PointClass, WeierstrassCurve.Projective.Point.point, Quotient.mk, MulAction.orbitRel, Units, RingHom, Algebra.algebraMap, WeierstrassCurve.Projective.Nonsingular, IsLocalRing.ResidueField, WeierstrassCurve.Projective.map, IsLocalRing.residue
18 instances, coercions and auxiliary constants left out.
```
def nonsingularReduction {R : Type*} [CommRing R] [IsLocalRing R] (K : Type*) [CommRing K]
    [Algebra R K] (W : Projective R) : Set (W⁄K).Point :=
…
```
def WeierstrassCurve.Projective.tamagawaNumber The local Tamagawa number attached to an integral model line 59 source
Source documentation
The local Tamagawa number attached to an integral model. For a minimal model this is the usual Tamagawa number. This is 0 if there are infinitely many distinct translates.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, IsDomain, IsDiscreteValuationRing, Field, Algebra, IsFractionRing, WeierstrassCurve.Projective, Nat
Proof uses: Nat.card, Set, WeierstrassCurve.Projective.Point, WeierstrassCurve.Projective.baseChange, Set.range, Set.image, HAdd.hAdd, WeierstrassCurve.Projective.nonsingularReduction
17 instances, coercions and auxiliary constants left out.
```
def tamagawaNumber {R : Type*} [CommRing R] [IsDomain R] [IsDiscreteValuationRing R] (K : Type*)
    [Field K] [Algebra R K] [IsFractionRing R K] (W : Projective R) : ℕ :=
…
```
def WeierstrassCurve.Projective.tamagawaProduct The product of local Tamagawa numbers for minimal integral models at all the non-archimedean completions line 65 source
Source documentation
The product of local Tamagawa numbers for minimal integral models at all the non-archimedean completions. This is 1 if infinitely many local factors differ from 1.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, NumberField, WeierstrassCurve.Projective, Nat
Proof uses: finprod, IsDedekindDomain.HeightOneSpectrum, NumberField.RingOfIntegers, WeierstrassCurve.Projective.tamagawaNumber, Subtype, IsDedekindDomain.HeightOneSpectrum.adicCompletion, Membership.mem, ValuationSubring, IsDedekindDomain.HeightOneSpectrum.adicCompletionIntegers, WeierstrassCurve.integralModel, WeierstrassCurve.minimal, WeierstrassCurve.Projective.baseChange
18 instances, coercions and auxiliary constants left out.
```
def tamagawaProduct {K : Type*} [Field K] [NumberField K] (W : Projective K) : ℕ :=
…
```