Abelian Integrals and Categoricity Martin Bays April 2, 2012 - PowerPoint PPT Presentation

Abelian Integrals and Categoricity Martin Bays April 2, 2012

Abelian integrals � w dz where w ∈ acl ( C ( z )) . i.e. � ω where ω is a meromorphic differential form on a Riemann surface C . e.g. � dz dz � √ = z 3 + az + b w on E := { w 2 = z 3 + az + b } .

Abelian integrals � w dz where w ∈ acl ( C ( z )) . i.e. � ω where ω is a meromorphic differential form on a Riemann surface C . e.g. � dz dz � √ = z 3 + az + b w on E := { w 2 = z 3 + az + b } .

Abelian integrals � w dz where w ∈ acl ( C ( z )) . i.e. � ω where ω is a meromorphic differential form on a Riemann surface C . e.g. � dz dz � √ = z 3 + az + b w on E := { w 2 = z 3 + az + b } .

Abelian integrals Multifunction I ω : C ( C ) 2 → C � Q I ω ( P , Q ) = ω, P value depends on path from P to Q on C . Status of C � ω := < C ; + , · , I ω > ? Example: ω = dz z on C = P 1 ◮ � b dz z = exp − 1 ( b ) = ln ( b ) + 2 π i Z 1 ◮ � b dz z = exp − 1 ( b ) − exp − 1 ( a ) = ln ( b ) − ln ( a ) + 2 π i Z a ◮ So C � ω interdefinable with C exp = < C ; + , · , exp > . ◮ Zilber: conjectural categorical description of C exp . ◮ Involves transcendence conjectures - e.g. e e ∈ Q ?

Abelian integrals Multifunction I ω : C ( C ) 2 → C � Q I ω ( P , Q ) = ω, P value depends on path from P to Q on C . Status of C � ω := < C ; + , · , I ω > ? Example: ω = dz z on C = P 1 ◮ � b dz z = exp − 1 ( b ) = ln ( b ) + 2 π i Z 1 ◮ � b dz z = exp − 1 ( b ) − exp − 1 ( a ) = ln ( b ) − ln ( a ) + 2 π i Z a ◮ So C � ω interdefinable with C exp = < C ; + , · , exp > . ◮ Zilber: conjectural categorical description of C exp . ◮ Involves transcendence conjectures - e.g. e e ∈ Q ?

Abelian integrals Multifunction I ω : C ( C ) 2 → C � Q I ω ( P , Q ) = ω, P value depends on path from P to Q on C . Status of C � ω := < C ; + , · , I ω > ? Example: ω = dz z on C = P 1 ◮ � b dz z = exp − 1 ( b ) = ln ( b ) + 2 π i Z 1 ◮ � b dz z = exp − 1 ( b ) − exp − 1 ( a ) = ln ( b ) − ln ( a ) + 2 π i Z a ◮ So C � ω interdefinable with C exp = < C ; + , · , exp > . ◮ Zilber: conjectural categorical description of C exp . ◮ Involves transcendence conjectures - e.g. e e ∈ Q ?

Abelian integrals of the first kind Suppose ω ∈ Ω := space of holomorphic differential forms on a Riemann surface C . Say ω = ω 1 , . . . , ω g basis for Ω , where g = genus ( C ) . Fix P 0 ∈ C ( C ) . Fact (Abel, Jacobi) C embeds in its Jacobian J = Pic 0 ( C ) such that �� Q � Q � = π − 1 ( Q ) ω 1 , . . . , ω g P 0 P 0 where π : C g ։ J ( C ) is a homomorphism with kernel a lattice (the periods). Status of < C ; + , · , π > ?

Abelian integrals of the first kind Suppose ω ∈ Ω := space of holomorphic differential forms on a Riemann surface C . Say ω = ω 1 , . . . , ω g basis for Ω , where g = genus ( C ) . Fix P 0 ∈ C ( C ) . Fact (Abel, Jacobi) C embeds in its Jacobian J = Pic 0 ( C ) such that �� Q � Q � = π − 1 ( Q ) ω 1 , . . . , ω g P 0 P 0 where π : C g ։ J ( C ) is a homomorphism with kernel a lattice (the periods). Status of < C ; + , · , π > ?

Abelian integrals of the first kind Suppose ω ∈ Ω := space of holomorphic differential forms on a Riemann surface C . Say ω = ω 1 , . . . , ω g basis for Ω , where g = genus ( C ) . Fix P 0 ∈ C ( C ) . Fact (Abel, Jacobi) C embeds in its Jacobian J = Pic 0 ( C ) such that �� Q � Q � = π − 1 ( Q ) ω 1 , . . . , ω g P 0 P 0 where π : C g ։ J ( C ) is a homomorphism with kernel a lattice (the periods). Status of < C ; + , · , π > ?

Linear reduct Let O := { η ∈ Mat g ( C ) | η ( ker ( π )) ≤ ker ( π ) } ∼ = End ( J ) . Let O 0 := Q ⊗ Z O . Consider C g as a new sort with just the O 0 -module structure: C g ; + , ( η ) η ∈O 0 � � � � π : C g → J ( C ) Cov ( J ) := � C ; + , ·� Lemma T J := Th ( Cov ( J )) has quantifier elimination and axiomatisation: � C g + , ( η ) η ∈O 0 � is a O 0 -module; π is a surjective O -homomorphism; � C ; + , ·� | = ACF 0 .

Linear reduct Let O := { η ∈ Mat g ( C ) | η ( ker ( π )) ≤ ker ( π ) } ∼ = End ( J ) . Let O 0 := Q ⊗ Z O . Consider C g as a new sort with just the O 0 -module structure: C g ; + , ( η ) η ∈O 0 � � � � π : C g → J ( C ) Cov ( J ) := � C ; + , ·� Lemma T J := Th ( Cov ( J )) has quantifier elimination and axiomatisation: � C g + , ( η ) η ∈O 0 � is a O 0 -module; π is a surjective O -homomorphism; � C ; + , ·� | = ACF 0 .

Linear reduct Let O := { η ∈ Mat g ( C ) | η ( ker ( π )) ≤ ker ( π ) } ∼ = End ( J ) . Let O 0 := Q ⊗ Z O . Consider C g as a new sort with just the O 0 -module structure: C g ; + , ( η ) η ∈O 0 � � � � π : C g → J ( C ) Cov ( J ) := � C ; + , ·� Lemma T J := Th ( Cov ( J )) has quantifier elimination and axiomatisation: � C g + , ( η ) η ∈O 0 � is a O 0 -module; π is a surjective O -homomorphism; � C ; + , ·� | = ACF 0 .

Categoricity Theorem (Categoricity over ker ( π ) ) Suppose J is defined over a number field. Then Cov ( J ) is specified up to isomorphism by: its first order theory T J ; its cardinality; the isomorphism type of ker ( π ) . ◮ Zilber: analogous statement for G m . ◮ Gavrilovich: similar statement, but assuming 2 ℵ 0 = ℵ 1 .

Categoricity Theorem (Categoricity over ker ( π ) ) Suppose J is defined over a number field. Then Cov ( J ) is specified up to isomorphism by: its first order theory T J ; its cardinality; the isomorphism type of ker ( π ) . ◮ Zilber: analogous statement for G m . ◮ Gavrilovich: similar statement, but assuming 2 ℵ 0 = ℵ 1 .

Atomicity B ⊆ C finite algebraically independent M B := π − 1 ( J ( acl ( B ))) � Cov ( J ) . Lemma (Atomicity) M B is atomic, hence unique, over � B ′ � B M B ′ . Categoricity theorem follows: Cov ( J ) is built uniquely over a transcendence basis of C .

Atomicity B ⊆ C finite algebraically independent M B := π − 1 ( J ( acl ( B ))) � Cov ( J ) . Lemma (Atomicity) M B is atomic, hence unique, over � B ′ � B M B ′ . Categoricity theorem follows: Cov ( J ) is built uniquely over a transcendence basis of C .

Proof Equivalent by QE: Lemma (Atomicity) ◮ a ∈ J ( acl ( B )) ; ◮ a i in simple subgroups, no O -linear relations; ◮ k ∂ := � B ′ � B acl ( B ′ ) ; Then exist only finitely many types tp ACF (( a n ) n / k ∂ ) .

Proof Lemma (Atomicity) ◮ k ∂ := � B ′ � B acl ( B ′ ) ; Then exist only finitely many types tp ACF (( a n ) n / k ∂ ) . Proof. ◮ k := k ∂ ( a ) Step I (“Mordell-Weil”): Bound n such that a n ∈ J ( k ) ;

Proof Lemma (Atomicity) Exist only finitely many types tp ACF (( a n ) n / k ∂ ) . Proof. ◮ k := k ∂ ( a ) Step I (“Mordell-Weil”): Bound n such that a n ∈ J ( k ) ; Step II (“Kummer”): More generally, bound k -rational imaginaries a n + Z n for subgroups Z n ≤ Tor n ( J ) - i.e. bound index [ Tor n ( J ) : Z n ] .

Proof Lemma (Atomicity) Exist only finitely many types tp ACF (( a n ) n / k ∂ ) . Proof. ◮ k := k ∂ ( a ) Step I (“Mordell-Weil”): Bound n such that a n ∈ J ( k ) ; Step II (“Kummer”): More generally, bound k -rational imaginaries a n + Z n for subgroups Z n ≤ Tor n ( J ) - i.e. bound index [ Tor n ( J ) : Z n ] . Step IIa Find number field k 0 such that J and all Z n may be taken over k 0 ; Step IIb By Faltings, the isogenous quotients J / Z n fall into finitely many isomorphism classes; hence reduce to bounding rational points in J as in Step I.

Proof Lemma (Atomicity) Exist only finitely many types tp ACF (( a n ) n / k ∂ ) . Proof. ◮ k := k ∂ ( a ) Step I (“Mordell-Weil”): Bound n such that a n ∈ J ( k ) ; Step II (“Kummer”): More generally, bound k -rational imaginaries a n + Z n for subgroups Z n ≤ Tor n ( J ) - i.e. bound index [ Tor n ( J ) : Z n ] . Step IIa Find number field k 0 such that J and all Z n may be taken over k 0 ; Step IIb By Faltings, the isogenous quotients J / Z n fall into finitely many isomorphism classes; hence reduce to bounding rational points in J as in Step I.

Proof Lemma (Atomicity) Exist only finitely many types tp ACF (( a n ) n / k ∂ ) . Proof. ◮ k := k ∂ ( a ) Step I (“Mordell-Weil”): Bound n such that a n ∈ J ( k ) ; and moreover such that η a n ∈ J ( k ) for η ∈ O . Step II (“Kummer”): More generally, bound k -rational imaginaries a n + Z n for subgroups Z n ≤ Tor n ( J ) - i.e. bound index [ Tor n ( J ) : Z n ] . Step IIa Find number field k 0 such that J and all Z n may be taken over k 0 ; Step IIb By Faltings, the isogenous quotients J / Z n fall into finitely many isomorphism classes; hence reduce to bounding rational points in J as in Step I.