noncomputable def Lec17AbelJacobi.SpecC : AlgebraicGeometry.Scheme

The scheme Spec ℂ, used as the base in the smooth-complex-curve setup of Lec 17.

structure Lec17AbelJacobi.SmoothCompactComplexCurve : Type 1

A smooth compact complex curve: an integral scheme over Spec ℂ that is proper and smooth of relative dimension 1.

• toScheme : AlgebraicGeometry.Scheme
• structureMorphism : self.toScheme ⟶ SpecC
• isIntegral : AlgebraicGeometry.IsIntegral self.toScheme
• isProper : AlgebraicGeometry.IsProper self.structureMorphism
• isSmoothOfRelDim1 : AlgebraicGeometry.SmoothOfRelativeDimension 1 self.structureMorphism

noncomputable def Lec17AbelJacobi.periodPairToWeierstrassCurve (L : PeriodPair) : WeierstrassCurve ℂ

Converts a complex period lattice into the Weierstrass cubic y² = 4x³ - g₂(L) x - g₃(L), with aᵢ coefficients adjusted to standard short form.

def Lec17AbelJacobi.LatticePairHomothetic (L₁ L₂ : PeriodPair) : Prop

Two period pairs are homothetic if their lattices are scalar multiples of each other by some nonzero complex α.

structure Lec17AbelJacobi.PeriodLatticeData : Type

Period lattice data attached to a curve of genus g: a full-rank ℤ-submodule of ℂ^g, representing the periods of holomorphic 1-forms.

• genus : ℕ
• lattice : Submodule ℤ (Fin self.genus → ℂ)

def Lec17AbelJacobi.PeriodLatticeData.Equivalent (D₁ D₂ : PeriodLatticeData) : Prop

Two period lattice data are equivalent if there is a ℂ-linear isomorphism between the ambient spaces carrying one lattice onto the other.

def Lec17AbelJacobi.SmoothCompactComplexCurve.IsIsomorphicOver (X₁ X₂ : SmoothCompactComplexCurve) : Prop

Two smooth compact complex curves are isomorphic over ℂ if there is an isomorphism of schemes compatible with their structure morphisms to Spec ℂ.

noncomputable def Lec17AbelJacobi.periodLatticeOf (X : SmoothCompactComplexCurve) : PeriodLatticeData

Assigns to a smooth compact complex curve its period lattice in ℂ^{genus}.

theorem Lec17AbelJacobi.torelli_reconstruction (X₁ X₂ : SmoothCompactComplexCurve) : X₁.IsIsomorphicOver X₂ ↔ (periodLatticeOf X₁).Equivalent (periodLatticeOf X₂)

Torelli's theorem: a smooth compact complex curve is determined up to isomorphism by its period lattice.

structure Lec17AbelJacobi.AbelianGroupScheme : Type 1

A (proper smooth) abelian group scheme over Spec ℂ: a proper smooth scheme over ℂ with a chosen identity section.

• toScheme : AlgebraicGeometry.Scheme
• structureMorphism : self.toScheme ⟶ SpecC
• isProper : AlgebraicGeometry.IsProper self.structureMorphism
• isSmooth : AlgebraicGeometry.Smooth self.structureMorphism
• identitySection : SpecC ⟶ self.toScheme
• identitySection_comp : CategoryTheory.CategoryStruct.comp self.identitySection self.structureMorphism = CategoryTheory.CategoryStruct.id SpecC

theorem Lec17AbelJacobi.jacobian_variety_prop26 (X : SmoothCompactComplexCurve) : ∃ (JacX : AbelianGroupScheme) (AJmap : X.toScheme ⟶ JacX.toScheme), CategoryTheory.CategoryStruct.comp AJmap JacX.structureMorphism = X.structureMorphism

Proposition 26 (Lec 17): existence of the Jacobian variety and the Abel–Jacobi map X → Jac(X) over ℂ, compatible with structure morphisms to Spec ℂ.

structure Lec17AbelJacobi.IrreducibleCompleteCurve (k : Type u_1) [Field k] : Type 1

An irreducible complete curve over k: an integral Dedekind domain regarded as the coordinate ring of a 1-dimensional smooth curve.

• coordinateRing : Type
• instCommRing : CommRing self.coordinateRing
• instIsDomain : IsDomain self.coordinateRing
• instIsDedekind : IsDedekindDomain self.coordinateRing

structure Lec17AbelJacobi.CurveMorphism {k : Type u_1} [Field k] (X Y : IrreducibleCompleteCurve k) : Type

A morphism X → Y of irreducible complete curves, contravariantly described by a ring homomorphism on coordinate rings Y.coordinateRing → X.coordinateRing.

• ringHom : Y.coordinateRing →+* X.coordinateRing

def Lec17AbelJacobi.CurveMorphism.comp {k : Type u_1} [Field k] {X Y Z : IrreducibleCompleteCurve k} (g : CurveMorphism Y Z) (f : CurveMorphism X Y) : CurveMorphism X Z

Composition of curve morphisms, defined contravariantly via composition of the underlying ring maps.

def Lec17AbelJacobi.CurveMorphism.IsConstant {k : Type u_1} [Field k] {X Y : IrreducibleCompleteCurve k} (f : CurveMorphism X Y) : Prop

A curve morphism is constant iff the underlying ring map is not injective (i.e. collapses some function on the target curve).

def Lec17AbelJacobi.CurveMorphism.IsFinite {k : Type u_1} [Field k] {X Y : IrreducibleCompleteCurve k} (f : CurveMorphism X Y) : Prop

A curve morphism is finite iff X.coordinateRing is module-finite over Y.coordinateRing via f.ringHom.

def Lec17AbelJacobi.CurveMorphism.IsIsomorphism {k : Type u_1} [Field k] {X Y : IrreducibleCompleteCurve k} (f : CurveMorphism X Y) : Prop

A curve morphism is an isomorphism iff its underlying ring map is bijective.

def Lec17AbelJacobi.CurveMorphism.IsBirational {k : Type u_1} [Field k] {X Y : IrreducibleCompleteCurve k} (f : CurveMorphism X Y) : Prop

A curve morphism is birational iff its ring map is injective and every element of X.coordinateRing is a fraction of elements pulled back from Y.coordinateRing.

def Lec17AbelJacobi.IrreducibleCompleteCurve.IsNormal {k : Type u_1} [Field k] (X : IrreducibleCompleteCurve k) : Prop

A curve is normal iff its coordinate ring is integrally closed in its field of fractions.

def Lec17AbelJacobi.IrreducibleCompleteCurve.Pic0Group {k : Type u_1} [Field k] (X : IrreducibleCompleteCurve k) : Type

The Picard group Pic⁰(X) of a curve, modeled as the ideal class group of its Dedekind coordinate ring.

@[implicit_reducible]

noncomputable instance Lec17AbelJacobi.instCommGroupPic0Group {k : Type u_1} [Field k] (X : IrreducibleCompleteCurve k) : CommGroup X.Pic0Group

Pic⁰(X) inherits a commutative group structure from the class group.

theorem Lec17AbelJacobi.normalization_factorization {k : Type u_1} [Field k] {X Y : IrreducibleCompleteCurve k} (f : CurveMorphism X Y) (hf : ¬f.IsConstant) : ∃ (NorY : IrreducibleCompleteCurve k) (g : CurveMorphism X NorY) (normMap : CurveMorphism NorY Y), g.IsBirational ∧ X.IsNormal ∧ NorY.IsNormal ∧ normMap.IsFinite ∧ f = normMap.comp g

Normalization factorization: any non-constant morphism f : X → Y factors as a birational map X → NorY followed by a finite normalization NorY → Y, with both X and NorY normal.

theorem Lec17AbelJacobi.birational_to_normal_is_iso {k : Type u_1} [Field k] {X Y : IrreducibleCompleteCurve k} (f : CurveMorphism X Y) (hf : f.IsBirational) (hY : Y.IsNormal) : f.IsIsomorphism

A birational morphism into a normal curve is automatically an isomorphism (Zariski's main theorem for smooth complete curves).

theorem Lec17AbelJacobi.finite_comp_of_iso_finite {k : Type u_1} [Field k] {X Y Z : IrreducibleCompleteCurve k} (g : CurveMorphism X Y) (h : CurveMorphism Y Z) (hg : g.IsIsomorphism) (hh : h.IsFinite) : (h.comp g).IsFinite

Composition of an isomorphism g : X → Y and a finite map h : Y → Z is again finite.

theorem Lec17AbelJacobi.toClass_surjective {F : Type u_1} [Field F] [DecidableEq F] (W : WeierstrassCurve.Affine F) [WeierstrassCurve.IsElliptic W] : Function.Surjective ⇑WeierstrassCurve.Affine.Point.toClass

Abel–Jacobi surjectivity for an elliptic Weierstrass curve: the map from affine points to the class group of the coordinate ring is surjective.

theorem Lec17AbelJacobi.genus_one_isomorphic_to_pic0 {F : Type u_1} [Field F] [DecidableEq F] (W : WeierstrassCurve.Affine F) [WeierstrassCurve.IsElliptic W] : Function.Bijective ⇑WeierstrassCurve.Affine.Point.toClass

Corollary 21 (Lec 17): for an elliptic curve W, the Abel–Jacobi map W.Point → Pic⁰(W) is a bijection (in fact a group isomorphism).

@[implicit_reducible]

instance Lec17AbelJacobi.elliptic_curve_addCommGroup {F : Type u_1} [Field F] [DecidableEq F] (W : WeierstrassCurve.Affine F) [WeierstrassCurve.IsElliptic W] : AddCommGroup W.Point

The affine points of an elliptic Weierstrass curve form an abelian group.

theorem Lec17AbelJacobi.normalization_is_finite_lec17 (k : Type u_1) [Field k] (B : Type u_2) [CommRing B] [IsDomain B] [Algebra k B] [Algebra.FiniteType k B] (K : Type u_3) [Field K] [Algebra B K] [IsFractionRing B K] (L : Type u_4) [Field L] [Algebra K L] [FiniteDimensional K L] [Algebra B L] [IsScalarTower B K L] : Module.Finite B ↥(integralClosure B L)

Normalization is finite (Lec 17): for B a finite-type k-domain with fraction field K and L/K a finite extension, the integral closure of B in L is module- finite over B.

theorem Lec17AbelJacobi.normalization_existence (k : Type u_1) [Field k] (B : Type u_2) [CommRing B] [IsDomain B] [Algebra k B] [Algebra.FiniteType k B] (K : Type u_3) [Field K] [Algebra B K] [IsFractionRing B K] [Algebra k K] [IsScalarTower k B K] (L : Type u_4) [Field L] [Algebra K L] [Algebra B L] [IsScalarTower B K L] [Algebra k L] [IsScalarTower k B L] [FiniteDimensional K L] : Module.Finite B ↥(integralClosure B L) ∧ Algebra.FiniteType k ↥(integralClosure B L) ∧ IsIntegrallyClosed ↥(integralClosure B L) ∧ Function.Injective ⇑(algebraMap B ↥(integralClosure B L))

Existence of the normalization with all relevant properties (Lec 17): the integral closure of B in L is B-module finite, of finite type over k, integrally closed, and the structure map B → integralClosure B L is injective.