Effective André-Oort, differential equations, and

Effective André-Oort, differential equations, and
the isomorphism problem for countable
differentially closed fields
James Freitag
1
University of California, Berkeley
June 2014
1
JF is partially supported by NSF MSPRF 1204510. This material is based
upon work supported by the National Science Foundation under Grant No.
0932078 000 while JF was in residence at the Mathematical Sciences Research
Institute in Berkeley, California, during the Spring 2014 semester.
Freitag
Groups
Figure: Three things
Freitag
Groups
The isomorphism problem - notation
I
L a countable language, XL is the space of structures with
underlying universe N.
I
For σ ∈ Lω1 ,ω , Mod(σ) = {M ∈ XL | M |= σ}.
∼
=σ is the equivalence relation of isomorphism on Mod(σ).
Given X , Y Polish spaces, E , F equivalence relations on X , Y ,
E is Borel reducible to F if there is Borel map f : X → Y
such that
x1 E x2 ↔ f (x1 ) E f (x2 ).
I
I
I
I
∼
=σ is Borel complete if it is Borel bireducible with the
universal S∞ -orbit equivalence relation as complicated as
possible.
In this talk, I will usually let σ be the theory of differentially
closed fields of characteristic zero.
Freitag
Groups
The isomorphism problem - notation
I
L a countable language, XL is the space of structures with
underlying universe N.
I
For σ ∈ Lω1 ,ω , Mod(σ) = {M ∈ XL | M |= σ}.
∼
=σ is the equivalence relation of isomorphism on Mod(σ).
Given X , Y Polish spaces, E , F equivalence relations on X , Y ,
E is Borel reducible to F if there is Borel map f : X → Y
such that
x1 E x2 ↔ f (x1 ) E f (x2 ).
I
I
I
I
∼
=σ is Borel complete if it is Borel bireducible with the
universal S∞ -orbit equivalence relation as complicated as
possible.
In this talk, I will usually let σ be the theory of differentially
closed fields of characteristic zero.
Freitag
Groups
The isomorphism problem - notation
I
L a countable language, XL is the space of structures with
underlying universe N.
I
For σ ∈ Lω1 ,ω , Mod(σ) = {M ∈ XL | M |= σ}.
∼
=σ is the equivalence relation of isomorphism on Mod(σ).
Given X , Y Polish spaces, E , F equivalence relations on X , Y ,
E is Borel reducible to F if there is Borel map f : X → Y
such that
x1 E x2 ↔ f (x1 ) E f (x2 ).
I
I
I
I
∼
=σ is Borel complete if it is Borel bireducible with the
universal S∞ -orbit equivalence relation as complicated as
possible.
In this talk, I will usually let σ be the theory of differentially
closed fields of characteristic zero.
Freitag
Groups
The isomorphism problem - notation
I
L a countable language, XL is the space of structures with
underlying universe N.
I
For σ ∈ Lω1 ,ω , Mod(σ) = {M ∈ XL | M |= σ}.
∼
=σ is the equivalence relation of isomorphism on Mod(σ).
Given X , Y Polish spaces, E , F equivalence relations on X , Y ,
E is Borel reducible to F if there is Borel map f : X → Y
such that
x1 E x2 ↔ f (x1 ) E f (x2 ).
I
I
I
I
∼
=σ is Borel complete if it is Borel bireducible with the
universal S∞ -orbit equivalence relation as complicated as
possible.
In this talk, I will usually let σ be the theory of differentially
closed fields of characteristic zero.
Freitag
Groups
The isomorphism problem - notation
I
L a countable language, XL is the space of structures with
underlying universe N.
I
For σ ∈ Lω1 ,ω , Mod(σ) = {M ∈ XL | M |= σ}.
∼
=σ is the equivalence relation of isomorphism on Mod(σ).
Given X , Y Polish spaces, E , F equivalence relations on X , Y ,
E is Borel reducible to F if there is Borel map f : X → Y
such that
x1 E x2 ↔ f (x1 ) E f (x2 ).
I
I
I
I
∼
=σ is Borel complete if it is Borel bireducible with the
universal S∞ -orbit equivalence relation as complicated as
possible.
In this talk, I will usually let σ be the theory of differentially
closed fields of characteristic zero.
Freitag
Groups
The isomorphism problem - notation
I
L a countable language, XL is the space of structures with
underlying universe N.
I
For σ ∈ Lω1 ,ω , Mod(σ) = {M ∈ XL | M |= σ}.
∼
=σ is the equivalence relation of isomorphism on Mod(σ).
Given X , Y Polish spaces, E , F equivalence relations on X , Y ,
E is Borel reducible to F if there is Borel map f : X → Y
such that
x1 E x2 ↔ f (x1 ) E f (x2 ).
I
I
I
I
∼
=σ is Borel complete if it is Borel bireducible with the
universal S∞ -orbit equivalence relation as complicated as
possible.
In this talk, I will usually let σ be the theory of differentially
closed fields of characteristic zero.
Freitag
Groups
How complicated is the isomorphism problem for DCF0 ?
I
I
DCF0 is the model companion of differential fields of
characteristic zero - if you can solve a δ-equation in some
δ-field extension, you can solve it in your model.
Old question (Poizat/Lascar): To determine the isomorphism
class of Mod(DCF0 ) is it enough to specify
1. |{x | δ(x) = 0}|. Constants
2. |{x | x satisfies no δ-equation}|. δ-transcendentals
I
Vaught for ω-stable theories I (DCF0 , ℵ0 ) = ℵ0 or 2ℵ0
Freitag
Groups
How complicated is the isomorphism problem for DCF0 ?
I
I
DCF0 is the model companion of differential fields of
characteristic zero - if you can solve a δ-equation in some
δ-field extension, you can solve it in your model.
Old question (Poizat/Lascar): To determine the isomorphism
class of Mod(DCF0 ) is it enough to specify
1. |{x | δ(x) = 0}|. Constants
2. |{x | x satisfies no δ-equation}|. δ-transcendentals
I
Vaught for ω-stable theories I (DCF0 , ℵ0 ) = ℵ0 or 2ℵ0
Freitag
Groups
How complicated is the isomorphism problem for DCF0 ?
I
I
DCF0 is the model companion of differential fields of
characteristic zero - if you can solve a δ-equation in some
δ-field extension, you can solve it in your model.
Old question (Poizat/Lascar): To determine the isomorphism
class of Mod(DCF0 ) is it enough to specify
1. |{x | δ(x) = 0}|. Constants
2. |{x | x satisfies no δ-equation}|. δ-transcendentals
I
Vaught for ω-stable theories I (DCF0 , ℵ0 ) = ℵ0 or 2ℵ0
Freitag
Groups
Figure: The universe
Freitag
Groups
How complicated is the isomorphism problem for DCF0 ?
I
Theorem (Hrushovski/Sokolovic): There are 2ℵ0 countable
models of DCF0 .
I
I
Pf. Idea: use certain strongly minimal definable groups called
Manin kernels to code finite graphs (∼
=DCF0 is Borel complete).
∼
strongly minimal sets ↔= -class of M ∈ Mod(DCF0 )
I
Manin Kernels = {Tor (A) | A an abelian variety}cl
I
The closure is taken in the Kolchin topology
Freitag
Groups
How complicated is the isomorphism problem for DCF0 ?
I
Theorem (Hrushovski/Sokolovic): There are 2ℵ0 countable
models of DCF0 .
I
I
Pf. Idea: use certain strongly minimal definable groups called
Manin kernels to code finite graphs (∼
=DCF0 is Borel complete).
∼
strongly minimal sets ↔= -class of M ∈ Mod(DCF0 )
I
Manin Kernels = {Tor (A) | A an abelian variety}cl
I
The closure is taken in the Kolchin topology
Freitag
Groups
How complicated is the isomorphism problem for DCF0 ?
I
Theorem (Hrushovski/Sokolovic): There are 2ℵ0 countable
models of DCF0 .
I
I
Pf. Idea: use certain strongly minimal definable groups called
Manin kernels to code finite graphs (∼
=DCF0 is Borel complete).
∼
strongly minimal sets ↔= -class of M ∈ Mod(DCF0 )
I
Manin Kernels = {Tor (A) | A an abelian variety}cl
I
The closure is taken in the Kolchin topology
Freitag
Groups
How complicated is the isomorphism problem for DCF0 ?
I
Theorem (Hrushovski/Sokolovic): There are 2ℵ0 countable
models of DCF0 .
I
I
Pf. Idea: use certain strongly minimal definable groups called
Manin kernels to code finite graphs (∼
=DCF0 is Borel complete).
∼
strongly minimal sets ↔= -class of M ∈ Mod(DCF0 )
I
Manin Kernels = {Tor (A) | A an abelian variety}cl
I
The closure is taken in the Kolchin topology
Freitag
Groups
Figure: The orthogonality classes of strongly minimal sets
Conjecture (Hrushovski) Trivial sets are ℵ0 -categorical.
The rest fo the talk will be devoted to describing a counterexample.
Freitag
Groups
Figure: The orthogonality classes of strongly minimal sets
Conjecture (Hrushovski) Trivial sets are ℵ0 -categorical.
The rest fo the talk will be devoted to describing a counterexample.
Freitag
Groups
Figure: Three related things
Freitag
Groups
Figure: Three related things and three problems to solve
Freitag
Groups
Mazur’s Question
I
The André-Oort conjecture sometimes tells us that in
products of Shimura varieties, the intersection
Y
(non-weakly-special subvarieties)∩ (generalized Hecke orbits)
is finite. Explicit bound?
I
e.g let Eτ denote the elliptic curve with j-invariant τ . Fix
τ ∈ C and σ ∈ Aut(P1 (C).
|{ν ∈ C | Eν ∼ Eτ and Eσ(ν) ∼ Eσ(τ ) }| < ℵ0
I
In the remainder of the talk, I will only talk about the
particular case of elliptic curves, but we know things about
moduli of abelian varieties as well.
Freitag
Groups
Mazur’s Question
I
The André-Oort conjecture sometimes tells us that in
products of Shimura varieties, the intersection
Y
(non-weakly-special subvarieties)∩ (generalized Hecke orbits)
is finite. Explicit bound?
I
e.g let Eτ denote the elliptic curve with j-invariant τ . Fix
τ ∈ C and σ ∈ Aut(P1 (C).
|{ν ∈ C | Eν ∼ Eτ and Eσ(ν) ∼ Eσ(τ ) }| < ℵ0
I
In the remainder of the talk, I will only talk about the
particular case of elliptic curves, but we know things about
moduli of abelian varieties as well.
Freitag
Groups
Mazur’s Question
I
The André-Oort conjecture sometimes tells us that in
products of Shimura varieties, the intersection
Y
(non-weakly-special subvarieties)∩ (generalized Hecke orbits)
is finite. Explicit bound?
I
e.g let Eτ denote the elliptic curve with j-invariant τ . Fix
τ ∈ C and σ ∈ Aut(P1 (C).
|{ν ∈ C | Eν ∼ Eτ and Eσ(ν) ∼ Eσ(τ ) }| < ℵ0
I
In the remainder of the talk, I will only talk about the
particular case of elliptic curves, but we know things about
moduli of abelian varieties as well.
Freitag
Groups
Mazur’s Question
I
The André-Oort conjecture sometimes tells us that in
products of Shimura varieties, the intersection
Y
(non-weakly-special subvarieties)∩ (generalized Hecke orbits)
is finite. Explicit bound?
I
e.g let Eτ denote the elliptic curve with j-invariant τ . Fix
τ ∈ C and σ ∈ Aut(P1 (C).
|{ν ∈ C | Eν ∼ Eτ and Eσ(ν) ∼ Eσ(τ ) }| < ℵ0
I
In the remainder of the talk, I will only talk about the
particular case of elliptic curves, but we know things about
moduli of abelian varieties as well.
Freitag
Groups
Mazur’s Question
I
The André-Oort conjecture sometimes tells us that in
products of Shimura varieties, the intersection
Y
(non-weakly-special subvarieties)∩ (generalized Hecke orbits)
is finite. Explicit bound?
I
e.g let Eτ denote the elliptic curve with j-invariant τ . Fix
τ ∈ C and σ ∈ Aut(P1 (C).
|{ν ∈ C | Eν ∼ Eτ and Eσ(ν) ∼ Eσ(τ ) }| < ℵ0
I
In the remainder of the talk, I will only talk about the
particular case of elliptic curves, but we know things about
moduli of abelian varieties as well.
Freitag
Groups
Pila’s Question
I
Theorem: (Ax-Lindemann-Weierstrass) W ⊆ C. Given
a1 , . . . , an ∈ C(W ) linearly dependent over Q modulo C, then
e a1 , . . . , e an are algebraically independent over C.
Goal: Formulate suitable generalizations of the theorem to
the case j : H → C.
I
If ai = gaj for some g ∈ GL+
2 (Q) then j(ai ) and j(aj ) are
algebraically dependent via a modular polynomial.
I
Defn z1 , . . . , zn ∈ C(W ) are modularly independent if no pair
satisfies a GL+
2 (Q)-relation and no zi is constant.
I
Note: The notion is essentially binary.
Freitag
Groups
Pila’s Question
I
Theorem: (Ax-Lindemann-Weierstrass) W ⊆ C. Given
a1 , . . . , an ∈ C(W ) linearly dependent over Q modulo C, then
e a1 , . . . , e an are algebraically independent over C.
Goal: Formulate suitable generalizations of the theorem to
the case j : H → C.
I
If ai = gaj for some g ∈ GL+
2 (Q) then j(ai ) and j(aj ) are
algebraically dependent via a modular polynomial.
I
Defn z1 , . . . , zn ∈ C(W ) are modularly independent if no pair
satisfies a GL+
2 (Q)-relation and no zi is constant.
I
Note: The notion is essentially binary.
Freitag
Groups
Pila’s Question
I
Theorem: (Ax-Lindemann-Weierstrass) W ⊆ C. Given
a1 , . . . , an ∈ C(W ) linearly dependent over Q modulo C, then
e a1 , . . . , e an are algebraically independent over C.
Goal: Formulate suitable generalizations of the theorem to
the case j : H → C.
I
If ai = gaj for some g ∈ GL+
2 (Q) then j(ai ) and j(aj ) are
algebraically dependent via a modular polynomial.
I
Defn z1 , . . . , zn ∈ C(W ) are modularly independent if no pair
satisfies a GL+
2 (Q)-relation and no zi is constant.
I
Note: The notion is essentially binary.
Freitag
Groups
Pila’s Question
I
Theorem: (Ax-Lindemann-Weierstrass) W ⊆ C. Given
a1 , . . . , an ∈ C(W ) linearly dependent over Q modulo C, then
e a1 , . . . , e an are algebraically independent over C.
Goal: Formulate suitable generalizations of the theorem to
the case j : H → C.
I
If ai = gaj for some g ∈ GL+
2 (Q) then j(ai ) and j(aj ) are
algebraically dependent via a modular polynomial.
I
Defn z1 , . . . , zn ∈ C(W ) are modularly independent if no pair
satisfies a GL+
2 (Q)-relation and no zi is constant.
I
Note: The notion is essentially binary.
Freitag
Groups
I
Defn z1 , . . . , zn ∈ C(W ) are modularly independent if no pair
satisfies a GL+
2 (Q)-relation and no zi is constant.
I
Theorem: (Pila) W ⊆ C. Given a1 , . . . , an ∈ C(W )
modularly independent, the 3n functions
{j(ai ), j 0 (ai ), j 00 (ai )}ni=1
are algebraically independent over C(W ).
I
This is one of three essential steps in Pila’s proof of the
Andre-Oort conjecture for Cn .
I
Main tool of proof: The essential tool is the Pila-Wilkie
Counting Theorem: if in an o-minimal expansion of a real
closed field,
limsupT
|{q̄ ∈ X (Qalg ) | H(q̄) ≤ T }|
=∞
T
then X contains a connected semi-algebraic set /Q.
Freitag
Groups
I
Defn z1 , . . . , zn ∈ C(W ) are modularly independent if no pair
satisfies a GL+
2 (Q)-relation and no zi is constant.
I
Theorem: (Pila) W ⊆ C. Given a1 , . . . , an ∈ C(W )
modularly independent, the 3n functions
{j(ai ), j 0 (ai ), j 00 (ai )}ni=1
are algebraically independent over C(W ).
I
This is one of three essential steps in Pila’s proof of the
Andre-Oort conjecture for Cn .
I
Main tool of proof: The essential tool is the Pila-Wilkie
Counting Theorem: if in an o-minimal expansion of a real
closed field,
limsupT
|{q̄ ∈ X (Qalg ) | H(q̄) ≤ T }|
=∞
T
then X contains a connected semi-algebraic set /Q.
Freitag
Groups
I
Defn z1 , . . . , zn ∈ C(W ) are modularly independent if no pair
satisfies a GL+
2 (Q)-relation and no zi is constant.
I
Theorem: (Pila) W ⊆ C. Given a1 , . . . , an ∈ C(W )
modularly independent, the 3n functions
{j(ai ), j 0 (ai ), j 00 (ai )}ni=1
are algebraically independent over C(W ).
I
This is one of three essential steps in Pila’s proof of the
Andre-Oort conjecture for Cn .
I
Main tool of proof: The essential tool is the Pila-Wilkie
Counting Theorem: if in an o-minimal expansion of a real
closed field,
limsupT
|{q̄ ∈ X (Qalg ) | H(q̄) ≤ T }|
=∞
T
then X contains a connected semi-algebraic set /Q.
Freitag
Groups
I
Defn z1 , . . . , zn ∈ C(W ) are modularly independent if no pair
satisfies a GL+
2 (Q)-relation and no zi is constant.
I
Theorem: (Pila) W ⊆ C. Given a1 , . . . , an ∈ C(W )
modularly independent, the 3n functions
{j(ai ), j 0 (ai ), j 00 (ai )}ni=1
are algebraically independent over C(W ).
I
This is one of three essential steps in Pila’s proof of the
Andre-Oort conjecture for Cn .
I
Main tool of proof: The essential tool is the Pila-Wilkie
Counting Theorem: if in an o-minimal expansion of a real
closed field,
limsupT
|{q̄ ∈ X (Qalg ) | H(q̄) ≤ T }|
=∞
T
then X contains a connected semi-algebraic set /Q.
Freitag
Groups
I
Defn z1 , . . . , zn ∈ C(W ) are modularly independent if no pair
satisfies a GL+
2 (Q)-relation and no zi is constant.
I
Theorem: (Pila) W ⊆ C. Given a1 , . . . , an ∈ C(W )
modularly independent, the 3n functions
{j(ai ), j 0 (ai ), j 00 (ai )}ni=1
are algebraically independent over C(W ).
I
This is one of three essential steps in Pila’s proof of the
Andre-Oort conjecture for Cn .
I
Main tool of proof: The essential tool is the Pila-Wilkie
Counting Theorem: if in an o-minimal expansion of a real
closed field,
limsupT
|{q̄ ∈ X (Qalg ) | H(q̄) ≤ T }|
=∞
T
then X contains a connected semi-algebraic set /Q.
Freitag
Groups
I
Pila-Wilkie Counting Theorem: if in an o-minimal expansion
of a real closed field,
limsupT
|{q̄ ∈ X (Qalg ) | H(q̄) ≤ T }|
=∞
T
then X contains a connected semi-algebraic set.
I
e.g. the graph X of a polynomial f ∈ Z[x1 , . . . , xn−1 ] then
the graph of f : Rn−1 → R then N(X , t) > ct 2(n−1)/d .
I
The point Counting theorem detects algebraic dependence
via point counting.
Freitag
Groups
I
Pila-Wilkie Counting Theorem: if in an o-minimal expansion
of a real closed field,
limsupT
|{q̄ ∈ X (Qalg ) | H(q̄) ≤ T }|
=∞
T
then X contains a connected semi-algebraic set.
I
e.g. the graph X of a polynomial f ∈ Z[x1 , . . . , xn−1 ] then
the graph of f : Rn−1 → R then N(X , t) > ct 2(n−1)/d .
I
The point Counting theorem detects algebraic dependence
via point counting.
Freitag
Groups
I
Pila-Wilkie Counting Theorem: if in an o-minimal expansion
of a real closed field,
limsupT
|{q̄ ∈ X (Qalg ) | H(q̄) ≤ T }|
=∞
T
then X contains a connected semi-algebraic set.
I
e.g. the graph X of a polynomial f ∈ Z[x1 , . . . , xn−1 ] then
the graph of f : Rn−1 → R then N(X , t) > ct 2(n−1)/d .
I
The point Counting theorem detects algebraic dependence
via point counting.
Freitag
Groups
Pila’s question
I
The j-function satisfies a thrid order differential equation over
C:
∂
χ(y ) := S ∂ (y ) + R(y )( (y ))2 = 0
∂t
∂t
where
00 0
y
1 y 00 2
S ∂ (y ) =
−
.
∂t
y0
2 y0
I
Pila’s question: Is this differential equation strongly minimal?
I
Theorem (F., Scanlon) χ(y ) = 0 is strongly minimal and the
algebraic closure relation on the variety is geometrically trivial.
“Algebraic closure = isogeny=modular relations”
I
This set is not ℵ0 -categorical.
I
(think about Ryll-Nardzewski; there are many orbits on X 2 ,
since there are many modular relations).
Freitag
Groups
Pila’s question
I
The j-function satisfies a thrid order differential equation over
C:
∂
χ(y ) := S ∂ (y ) + R(y )( (y ))2 = 0
∂t
∂t
where
00 0
y
1 y 00 2
S ∂ (y ) =
−
.
∂t
y0
2 y0
I
Pila’s question: Is this differential equation strongly minimal?
I
Theorem (F., Scanlon) χ(y ) = 0 is strongly minimal and the
algebraic closure relation on the variety is geometrically trivial.
“Algebraic closure = isogeny=modular relations”
I
This set is not ℵ0 -categorical.
I
(think about Ryll-Nardzewski; there are many orbits on X 2 ,
since there are many modular relations).
Freitag
Groups
Pila’s question
I
The j-function satisfies a thrid order differential equation over
C:
∂
χ(y ) := S ∂ (y ) + R(y )( (y ))2 = 0
∂t
∂t
where
00 0
y
1 y 00 2
S ∂ (y ) =
−
.
∂t
y0
2 y0
I
Pila’s question: Is this differential equation strongly minimal?
I
Theorem (F., Scanlon) χ(y ) = 0 is strongly minimal and the
algebraic closure relation on the variety is geometrically trivial.
“Algebraic closure = isogeny=modular relations”
I
This set is not ℵ0 -categorical.
I
(think about Ryll-Nardzewski; there are many orbits on X 2 ,
since there are many modular relations).
Freitag
Groups
Pila’s question
I
The j-function satisfies a thrid order differential equation over
C:
∂
χ(y ) := S ∂ (y ) + R(y )( (y ))2 = 0
∂t
∂t
where
00 0
y
1 y 00 2
S ∂ (y ) =
−
.
∂t
y0
2 y0
I
Pila’s question: Is this differential equation strongly minimal?
I
Theorem (F., Scanlon) χ(y ) = 0 is strongly minimal and the
algebraic closure relation on the variety is geometrically trivial.
“Algebraic closure = isogeny=modular relations”
I
This set is not ℵ0 -categorical.
I
(think about Ryll-Nardzewski; there are many orbits on X 2 ,
since there are many modular relations).
Freitag
Groups
Figure: Three related things and three problems to solve
Freitag
Groups
Mazur’s Question
I
e.g let Eτ denote the elliptic curve with j-invariant τ . Fix
τ ∈ C and σ ∈ Aut(P1 (C).
|{ν ∈ C | Eν ∼ Eτ and Eσ(ν) ∼ Eσ(τ ) }| < ℵ0
I
Main idea: replace isogeny classes by solutions to a
differential equation.
I
The effective finiteness comes from strong minimality and
intersection theory for differential algebraic geometry.
I
Mazur’s question:
|{ν ∈ C | Eν ∼ Eτ and Eσ(ν) ∼ Eσ(τ ) }| < 367 .
I
Bounds for the more general cases of André-Oort are doubly
exponential in the dimension and degree of certain varieties.
Freitag
Groups
Mazur’s Question
I
e.g let Eτ denote the elliptic curve with j-invariant τ . Fix
τ ∈ C and σ ∈ Aut(P1 (C).
|{ν ∈ C | Eν ∼ Eτ and Eσ(ν) ∼ Eσ(τ ) }| < ℵ0
I
Main idea: replace isogeny classes by solutions to a
differential equation.
I
The effective finiteness comes from strong minimality and
intersection theory for differential algebraic geometry.
I
Mazur’s question:
|{ν ∈ C | Eν ∼ Eτ and Eσ(ν) ∼ Eσ(τ ) }| < 367 .
I
Bounds for the more general cases of André-Oort are doubly
exponential in the dimension and degree of certain varieties.
Freitag
Groups
Mazur’s Question
I
e.g let Eτ denote the elliptic curve with j-invariant τ . Fix
τ ∈ C and σ ∈ Aut(P1 (C).
|{ν ∈ C | Eν ∼ Eτ and Eσ(ν) ∼ Eσ(τ ) }| < ℵ0
I
Main idea: replace isogeny classes by solutions to a
differential equation.
I
The effective finiteness comes from strong minimality and
intersection theory for differential algebraic geometry.
I
Mazur’s question:
|{ν ∈ C | Eν ∼ Eτ and Eσ(ν) ∼ Eσ(τ ) }| < 367 .
I
Bounds for the more general cases of André-Oort are doubly
exponential in the dimension and degree of certain varieties.
Freitag
Groups
Mazur’s Question
I
e.g let Eτ denote the elliptic curve with j-invariant τ . Fix
τ ∈ C and σ ∈ Aut(P1 (C).
|{ν ∈ C | Eν ∼ Eτ and Eσ(ν) ∼ Eσ(τ ) }| < ℵ0
I
Main idea: replace isogeny classes by solutions to a
differential equation.
I
The effective finiteness comes from strong minimality and
intersection theory for differential algebraic geometry.
I
Mazur’s question:
|{ν ∈ C | Eν ∼ Eτ and Eσ(ν) ∼ Eσ(τ ) }| < 367 .
I
Bounds for the more general cases of André-Oort are doubly
exponential in the dimension and degree of certain varieties.
Freitag
Groups
Mazur’s Question
I
e.g let Eτ denote the elliptic curve with j-invariant τ . Fix
τ ∈ C and σ ∈ Aut(P1 (C).
|{ν ∈ C | Eν ∼ Eτ and Eσ(ν) ∼ Eσ(τ ) }| < ℵ0
I
Main idea: replace isogeny classes by solutions to a
differential equation.
I
The effective finiteness comes from strong minimality and
intersection theory for differential algebraic geometry.
I
Mazur’s question:
|{ν ∈ C | Eν ∼ Eτ and Eσ(ν) ∼ Eσ(τ ) }| < 367 .
I
Bounds for the more general cases of André-Oort are doubly
exponential in the dimension and degree of certain varieties.
Freitag
Groups
Mazur’s Question
I
e.g let Eτ denote the elliptic curve with j-invariant τ . Fix
τ ∈ C and σ ∈ Aut(P1 (C).
|{ν ∈ C | Eν ∼ Eτ and Eσ(ν) ∼ Eσ(τ ) }| < ℵ0
I
Main idea: replace isogeny classes by solutions to a
differential equation.
I
The effective finiteness comes from strong minimality and
intersection theory for differential algebraic geometry.
I
Mazur’s question:
|{ν ∈ C | Eν ∼ Eτ and Eσ(ν) ∼ Eσ(τ ) }| < 367 .
I
Bounds for the more general cases of André-Oort are doubly
exponential in the dimension and degree of certain varieties.
Freitag
Groups
Thanks for listening
I
The majority of this work is joint with Tom Scanlon.
I
Parts of the intersection theory which was briefly mentioned
are joint with Omar Sanchez.
I
Thanks to Barry Mazur, Jonathan Pila, and Ronnie Nagloo
for essential suggestions, complaints, and questions.
I
Thanks to David Masser, Anand Pillay, Omar Sanchez, Dave
Marker, and Rahim Moosa for useful conversations and
comments on early drafts.
I
Thanks very much for listening.
Freitag
Groups