TateShafarevich.lean`

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The Tate–-Shafarevich group of an elliptic curve over a number field

The non-singular points over an algebraic closure of a field K carry a discrete Galois representation. For a number field, the Tate–Shafarevich group is the intersection of the kernels of restriction to all infinite and finite completions, where finite places are indexed by the non-zero prime ideals of the ring of integers of K. The coefficient module remains the geometric points over the separable closure of K, with its action restricted to each local Galois group.

References

[Silverman2009] Joseph H. Silverman, The Arithmetic of Elliptic Curves, 2nd ed., Graduate Texts in Mathematics 106, Springer (2009), doi. Chapter X, §4, pp. 331–333, especially the definition on p. 332, gives the classical Tate–-Shafarevich group over a number field, using local geometric points.
J. S. Milne, Arithmetic Duality Theorems
J. S. Milne, Elliptic Curves, 2nd ed., Chapter IV, Remark 1.11, p. 111, explains the use of a separable closure over imperfect fields.

Imports (4)

Imported by 1 file

FormalConjecturesForMathlib.lean

lemma Field.absoluteGaloisGroup.one_toAlgHom line 46 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, Eq, AlgHom, AlgebraicClosure, OfNat.ofNat, Field.absoluteGaloisGroup, AlgHom.id
Proof uses: rfl
28 instances, coercions and auxiliary constants left out.
lemma Field.absoluteGaloisGroup.mul_toAlgHom line 50 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, Field.absoluteGaloisGroup, Eq, AlgHom, AlgebraicClosure, HMul.hMul, AlgHom.comp
Proof uses: rfl
30 instances, coercions and auxiliary constants left out.
lemma WeierstrassCurve.Affine.Point.map_id' line 55 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, Field, WeierstrassCurve.Affine, DecidableEq, Algebra, IsScalarTower, WeierstrassCurve.Affine.Point, WeierstrassCurve.Affine.baseChange, Eq, AddMonoidHom, WeierstrassCurve.Affine.Point.map, AlgHom.id
Proof uses: WeierstrassCurve.Affine.Point.zero, Eq.refl, Eq.symm, WeierstrassCurve.Affine.Nonsingular, WeierstrassCurve.Affine.Point.some
24 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma ContinuousLinearMap.mk_apply line 61 source
no axioms what this means
Statement uses: Semiring, RingHom, TopologicalSpace, AddCommMonoid, Module, LinearMap, Continuous, AddHom.toFun, LinearMap.toAddHom, Eq, ContinuousLinearMap, ContinuousLinearMap.mk
Proof uses: rfl
13 instances, coercions and auxiliary constants left out.
def WeierstrassCurve.Affine.galoisRepresentation The natural Galois action on geometric non-singular points with their discrete topology line 89 source
Source documentation
The natural Galois action on geometric non-singular points with their discrete topology.
This uses AlgebraicClosure K to match Field.absoluteGaloisGroup. For number fields, which are perfect, an algebraic closure is also a separable closure. Over an imperfect field, the coefficient module for the usual Galois cohomology of an elliptic curve should use SeparableClosure K instead.
TODO: use SeparableClosure K for the coefficient field when the API for absolute Galois groups supports this directly. The automorphism group of the algebraic closure itself gives the usual absolute Galois group by restriction to the separable closure.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, WeierstrassCurve.Affine, TopRep, Int, Field.absoluteGaloisGroup
Proof uses: TopRep.of, WeierstrassCurve.Affine.Point, AlgebraicClosure, WeierstrassCurve.Affine.baseChange, Classical.propDecidable, Eq, WeierstrassCurve.Affine.instTopologicalSpacePointAlgebraicClosureBaseChange, ContRepresentation.ofMonoidHom, MonoidHom.mk, ContinuousLinearMap, RingHom.id, OneHom.mk, LinearMap, AddMonoidHom.toIntLinearMap, WeierstrassCurve.Affine.Point.map, ContinuousLinearMap.mk
39 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
```
noncomputable def galoisRepresentation : TopRep ℤ <| Field.absoluteGaloisGroup K := .of <|
…
```
def WeierstrassCurve.Affine.tateShafarevich The subgroup of the first continuous Galois cohomology group consisting of classes whose restrictions vanish at every infinite and finite place line 99 source
Source documentation
The subgroup of the first continuous Galois cohomology group consisting of classes whose restrictions vanish at every infinite and finite place. When W.IsElliptic, this is the classical Tate--Shafarevich group of W over K. The construction also applies to the nonsingular locus of a singular Weierstrass curve.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, WeierstrassCurve.Affine, NumberField, AddSubgroup, Int, TopModuleCat.toModuleCat, continuousCohomology, Field.absoluteGaloisGroup, OfNat.ofNat, Nat, WeierstrassCurve.Affine.galoisRepresentation
Proof uses: Min.min, ContinuousCohomology.tateShafarevich, NumberField.InfinitePlace, NumberField.InfinitePlace.Completion, Real, IsDedekindDomain.HeightOneSpectrum, NumberField.RingOfIntegers, IsDedekindDomain.HeightOneSpectrum.adicCompletion
37 instances, coercions and auxiliary constants left out.
```
noncomputable def tateShafarevich [NumberField K] :
    AddSubgroup <| continuousCohomology 1 W.galoisRepresentation :=
…
```