Simultaneous torsion points in a Weierstrass
family of elliptic curves.
Myrto Mavraki
University of British Columbia
University of Michigan, December 5, 2015
Motivation
Theorem (Lang, 1965. Ihara-Serre-Tate)
Let C be an irreducible complex plane curve which contains
infinitely many points with both coordinates roots of unity. Then
the curve C is given by an equation of the form x n y m − ω = 0, for
n, m ∈ Z and a root of unity ω.
Motivation
Theorem (Lang, 1965. Ihara-Serre-Tate)
Let C be an irreducible complex plane curve which contains
infinitely many points with both coordinates roots of unity. Then
the curve C is given by an equation of the form x n y m − ω = 0, for
n, m ∈ Z and a root of unity ω.
Let µ denote the set of roots of unity.
Motivation
Theorem (Lang, 1965. Ihara-Serre-Tate)
Let C be an irreducible complex plane curve which contains
infinitely many points with both coordinates roots of unity. Then
the curve C is given by an equation of the form x n y m − ω = 0, for
n, m ∈ Z and a root of unity ω.
Let µ denote the set of roots of unity.
(ξ, ζ) ∈ µ × µ ↔
torsion points of G2m
↔
special
points
Motivation
Theorem (Lang, 1965. Ihara-Serre-Tate)
Let C be an irreducible complex plane curve which contains
infinitely many points with both coordinates roots of unity. Then
the curve C is given by an equation of the form x n y m − ω = 0, for
n, m ∈ Z and a root of unity ω.
Let µ denote the set of roots of unity.
(ξ, ζ) ∈ µ × µ ↔
torsion points of G2m
↔
Curves
↔ translates of algebraic subgroups ↔
x ny m − ω = 0
of G2m by a torsion point
special
points
special
curves
Motivation
Theorem (Lang, 1965. Ihara-Serre-Tate)
Let C be an irreducible complex plane curve which contains
infinitely many points with both coordinates roots of unity. Then
the curve C is given by an equation of the form x n y m − ω = 0, for
n, m ∈ Z and a root of unity ω.
Let µ denote the set of roots of unity.
(ξ, ζ) ∈ µ × µ ↔
torsion points of G2m
↔
Curves
↔ translates of algebraic subgroups ↔
x ny m − ω = 0
of G2m by a torsion point
special
points
special
curves
Philosophical restatement
If a curve has an infinite (Zariski dense) set of special points, then
it is a special curve.
Dynamical reformulation
The map φ : P1 → P1 given by
[x : y ] 7→ [x 2 : y 2 ]
induces a map φ × φ : P1 × P1 → P1 × P1 as follows.
(P, Q) 7→ (φ(P), φ(Q)).
Dynamical reformulation
The map φ : P1 → P1 given by
[x : y ] 7→ [x 2 : y 2 ]
induces a map φ × φ : P1 × P1 → P1 × P1 as follows.
(P, Q) 7→ (φ(P), φ(Q)).
e×µ
e where µ
e := µ ∪ {0, ∞}.
Prepφ×φ (P1 × P1 ) = µ
Dynamical reformulation
The map φ : P1 → P1 given by
[x : y ] 7→ [x 2 : y 2 ]
induces a map φ × φ : P1 × P1 → P1 × P1 as follows.
(P, Q) 7→ (φ(P), φ(Q)).
e×µ
e where µ
e := µ ∪ {0, ∞}.
Prepφ×φ (P1 × P1 ) = µ
Special points
↔
(P, Q) ∈ Prepφ×φ .
Dynamical reformulation
The map φ : P1 → P1 given by
[x : y ] 7→ [x 2 : y 2 ]
induces a map φ × φ : P1 × P1 → P1 × P1 as follows.
(P, Q) 7→ (φ(P), φ(Q)).
e×µ
e where µ
e := µ ∪ {0, ∞}.
Prepφ×φ (P1 × P1 ) = µ
Special points
Special curves
↔
↔
(P, Q) ∈ Prepφ×φ .
preperiodic curves under the action of φ × φ.
Dynamical reformulation
The map φ : P1 → P1 given by
[x : y ] 7→ [x 2 : y 2 ]
induces a map φ × φ : P1 × P1 → P1 × P1 as follows.
(P, Q) 7→ (φ(P), φ(Q)).
e×µ
e where µ
e := µ ∪ {0, ∞}.
Prepφ×φ (P1 × P1 ) = µ
Special points
Special curves
↔
↔
(P, Q) ∈ Prepφ×φ .
preperiodic curves under the action of φ × φ.
Reformulation of Lang’s theorem.
A curve C ⊂ P1 × P1 contains infinitely many (a Zariski dense set)
elements in Prepφ×φ if and only if C is a preperiodic curve under
the action of φ × φ.
Questions that arise
Question
What if φ : P1 → P1 is a Lattès map corresponding to an elliptic
curve E ? For example, φ could be the Lattès map, corresponding
to multiplication by 2 on E . When does a curve C ⊂ P1 × P1
contain infinitely many points in Prepφ×φ ?
Questions that arise
Question
What if φ : P1 → P1 is a Lattès map corresponding to an elliptic
curve E ? For example, φ could be the Lattès map, corresponding
to multiplication by 2 on E . When does a curve C ⊂ P1 × P1
contain infinitely many points in Prepφ×φ ?
Special points
↔
points in Prepφ×φ
↔ (P, Q) ∈ (E × E )tors .
Questions that arise
Question
What if φ : P1 → P1 is a Lattès map corresponding to an elliptic
curve E ? For example, φ could be the Lattès map, corresponding
to multiplication by 2 on E . When does a curve C ⊂ P1 × P1
contain infinitely many points in Prepφ×φ ?
Special points
↔
points in Prepφ×φ
↔ (P, Q) ∈ (E × E )tors .
Theorem (Laurent, McQuillan, Raynaud. Manin-Mumford
conjecture)
If an irreducible subvariety of a semiabelian variety contains a
Zariski dense set of torsion points, then it is a translate of an
algebraic subgroup by a torsion point.
Questions that arise
Question
What if φ : P1 → P1 is a Lattès map corresponding to an elliptic
curve E ? For example, φ could be the Lattès map, corresponding
to multiplication by 2 on E . When does a curve C ⊂ P1 × P1
contain infinitely many points in Prepφ×φ ?
Special points
↔
points in Prepφ×φ
↔ (P, Q) ∈ (E × E )tors .
Theorem (Laurent, McQuillan, Raynaud. Manin-Mumford
conjecture)
If an irreducible subvariety of a semiabelian variety contains a
Zariski dense set of torsion points, then it is a translate of an
algebraic subgroup by a torsion point.
Lang’s result is the case A = G2m . Our question is the case
A = E × E.
Pink-Zilber conjecture, an example
Pink-Zilber conjecture, an example
Consider the family of elliptic curves
Eλ : y 2 = x(x − 1)(x − λ), λ ∈ C \ {0, 1}.
and the two sections
Pλ = (2,
p
p
2(2 − λ)), Qλ = (3, 6(3 − λ)).
Pink-Zilber conjecture, an example
Consider the family of elliptic curves
Eλ : y 2 = x(x − 1)(x − λ), λ ∈ C \ {0, 1}.
and the two sections
Pλ = (2,
p
p
2(2 − λ)), Qλ = (3, 6(3 − λ)).
Let T (2) = {λ ∈ C \ {0, 1} | Pλ ∈ (Eλ )tors }; similarly T (3).
Pink-Zilber conjecture, an example
Consider the family of elliptic curves
Eλ : y 2 = x(x − 1)(x − λ), λ ∈ C \ {0, 1}.
and the two sections
Pλ = (2,
p
p
2(2 − λ)), Qλ = (3, 6(3 − λ)).
Let T (2) = {λ ∈ C \ {0, 1} | Pλ ∈ (Eλ )tors }; similarly T (3).
Remark
1 There are infinitely many λ ∈ T (2) (similarly for T (3)).
Pink-Zilber conjecture, an example
Consider the family of elliptic curves
Eλ : y 2 = x(x − 1)(x − λ), λ ∈ C \ {0, 1}.
and the two sections
Pλ = (2,
p
p
2(2 − λ)), Qλ = (3, 6(3 − λ)).
Let T (2) = {λ ∈ C \ {0, 1} | Pλ ∈ (Eλ )tors }; similarly T (3).
Remark
1 There are infinitely many λ ∈ T (2) (similarly for T (3)).
2
Pλ ∈ Eλ is not identically torsion. Same for Qλ .
Pink-Zilber conjecture, an example
Consider the family of elliptic curves
Eλ : y 2 = x(x − 1)(x − λ), λ ∈ C \ {0, 1}.
and the two sections
Pλ = (2,
p
p
2(2 − λ)), Qλ = (3, 6(3 − λ)).
Let T (2) = {λ ∈ C \ {0, 1} | Pλ ∈ (Eλ )tors }; similarly T (3).
Remark
1 There are infinitely many λ ∈ T (2) (similarly for T (3)).
2
Pλ ∈ Eλ is not identically torsion. Same for Qλ .
3
{Pλ }{λ∈C} and {Qλ }{λ∈C} are linearly independent over Z:
there are no (n, m) ∈ Z \ {(0, 0)} such that
[n]Pλ + [m]Qλ = O.
Masser and Zannier’s theorem
Question (Masser)
Can T (2) ∩ T (3) be infinite?
Masser and Zannier’s theorem
Question (Masser)
Can T (2) ∩ T (3) be infinite?
Theorem (Masser-Zannier, 2010)
There are at most finitely many λ ∈ C such that Pλ and Qλ are
simultaneously torsion points for Eλ .
Masser and Zannier’s theorem
Question (Masser)
Can T (2) ∩ T (3) be infinite?
Theorem (Masser-Zannier, 2010)
There are at most finitely many λ ∈ C such that Pλ and Qλ are
simultaneously torsion points for Eλ .
T (2) (respectively T (3)) is infinite
Masser and Zannier’s theorem
Question (Masser)
Can T (2) ∩ T (3) be infinite?
Theorem (Masser-Zannier, 2010)
There are at most finitely many λ ∈ C such that Pλ and Qλ are
simultaneously torsion points for Eλ .
T (2) (respectively T (3)) is infinite but sparse;
Masser and Zannier’s theorem
Question (Masser)
Can T (2) ∩ T (3) be infinite?
Theorem (Masser-Zannier, 2010)
There are at most finitely many λ ∈ C such that Pλ and Qλ are
simultaneously torsion points for Eλ .
T (2) (respectively T (3)) is infinite but sparse; think bounded Weil
height.
Masser and Zannier’s theorem
Question (Masser)
Can T (2) ∩ T (3) be infinite?
Theorem (Masser-Zannier, 2010)
There are at most finitely many λ ∈ C such that Pλ and Qλ are
simultaneously torsion points for Eλ .
T (2) (respectively T (3)) is infinite but sparse; think bounded Weil
height.
Double sparseness =⇒ finiteness.
General theorem
In fact a much more general statement is true.
General theorem
In fact a much more general statement is true. There is only one
special case when there are infinitely many torsion parameters in
the intersection.
General theorem
In fact a much more general statement is true. There is only one
special case when there are infinitely many torsion parameters in
the intersection.
Theorem (Masser-Zannier, 2012)
Let Pt = (a(t), ∗), Qt = (b(t), ∗) ∈ Et (C(t)). There are infinitely
many t = λ ∈ C \ {0, 1} such that (Pλ , Qλ ) ∈ (Eλ × Eλ )tors if and
only if there are (n, m) ∈ Z \ {(0, 0)}, such that
[n]Pt + [m]Qt = O.
General theorem
In fact a much more general statement is true. There is only one
special case when there are infinitely many torsion parameters in
the intersection.
Theorem (Masser-Zannier, 2012)
Let Pt = (a(t), ∗), Qt = (b(t), ∗) ∈ Et (C(t)). There are infinitely
many t = λ ∈ C \ {0, 1} such that (Pλ , Qλ ) ∈ (Eλ × Eλ )tors if and
only if there are (n, m) ∈ Z \ {(0, 0)}, such that
[n]Pt + [m]Qt = O.
Masser-Zannier’s proof involves:
• Model theory, results of Pila, Pila and Wilkie.
• The geometry of an elliptic curve, uniformization map.
Connection
If Et = E is a constant family of elliptic curves, then Masser and
Zannier’s theorem says
The curve C = {(Pt , Qt ) : t ∈ C} contains infinitely many
torsion points of E × E if and only if there are
(n, m) ∈ Z \ {(0, 0)} such that [n]Pt + [m]Qt = O.
Connection
If Et = E is a constant family of elliptic curves, then Masser and
Zannier’s theorem says
The curve C = {(Pt , Qt ) : t ∈ C} contains infinitely many
torsion points of E × E if and only if there are
(n, m) ∈ Z \ {(0, 0)} such that [n]Pt + [m]Qt = O.
This is a special case of the Manin-Mumford conjecture.
Connection
If Et = E is a constant family of elliptic curves, then Masser and
Zannier’s theorem says
The curve C = {(Pt , Qt ) : t ∈ C} contains infinitely many
torsion points of E × E if and only if there are
(n, m) ∈ Z \ {(0, 0)} such that [n]Pt + [m]Qt = O.
This is a special case of the Manin-Mumford conjecture.
Lang’s theorem also implied.
Let a(t), b(t) ∈ C(t). The curve C = {(a(t), b(t)) : t ∈ C}
contains infinitely many torsion points of G2m if and only if there
are (n, m) ∈ Z \ {(0, 0)} such that a(t)n b(t)m = 1.
Connection
If Et = E is a constant family of elliptic curves, then Masser and
Zannier’s theorem says
The curve C = {(Pt , Qt ) : t ∈ C} contains infinitely many
torsion points of E × E if and only if there are
(n, m) ∈ Z \ {(0, 0)} such that [n]Pt + [m]Qt = O.
This is a special case of the Manin-Mumford conjecture.
Lang’s theorem also implied.
Let a(t), b(t) ∈ C(t). The curve C = {(a(t), b(t)) : t ∈ C}
contains infinitely many torsion points of G2m if and only if there
are (n, m) ∈ Z \ {(0, 0)} such that a(t)n b(t)m = 1.
Hence, Masser’s question can be viewed as a varying
Manin-Mumford.
Stoll’s result
Stoll’s result
2 and 3 are different modulo 2.
Stoll’s result
2 and 3 are different modulo 2.
Fix an embedding ι : Q ,→ Q2 , and consider the reduction map
ρ : P1 (Q2 ) → P1 (F2 ).
Stoll’s result
2 and 3 are different modulo 2.
Fix an embedding ι : Q ,→ Q2 , and consider the reduction map
ρ : P1 (Q2 ) → P1 (F2 ).
For a ∈ Q \ {0, 1}, write
T (a) = {λ ∈ C \ {0, 1} : (a,
p
a(a − 1)(a − λ)) ∈ (Eλ )tors }.
Stoll’s result
2 and 3 are different modulo 2.
Fix an embedding ι : Q ,→ Q2 , and consider the reduction map
ρ : P1 (Q2 ) → P1 (F2 ).
For a ∈ Q \ {0, 1}, write
T (a) = {λ ∈ C \ {0, 1} : (a,
p
a(a − 1)(a − λ)) ∈ (Eλ )tors }.
Theorem (Stoll, 2014)
Let α, β ∈ Q \ {0, 1} such that ρ(α) 6= ρ(β). Then
T (α) ∩ T (β) ⊂ {α, β}.
Stoll’s approach
Examples
1
T (2) ∩ T (3) ⊂ {2, 3} and one can check T (2) ∩ T (3) = ∅.
2
For a primitive 3rd root of unity ω, T (ω) ∩ T (ω 2 ) ⊂ {ω, ω 2 }
and one can check T (ω) ∩ T (ω 2 ) = {ω, ω 2 }.
Stoll’s approach
Examples
1
T (2) ∩ T (3) ⊂ {2, 3} and one can check T (2) ∩ T (3) = ∅.
2
For a primitive 3rd root of unity ω, T (ω) ∩ T (ω 2 ) ⊂ {ω, ω 2 }
and one can check T (ω) ∩ T (ω 2 ) = {ω, ω 2 }.
ψn (λ, x) := the n−th reduced division polynomial of Eλ
Stoll’s approach
Examples
1
T (2) ∩ T (3) ⊂ {2, 3} and one can check T (2) ∩ T (3) = ∅.
2
For a primitive 3rd root of unity ω, T (ω) ∩ T (ω 2 ) ⊂ {ω, ω 2 }
and one can check T (ω) ∩ T (ω 2 ) = {ω, ω 2 }.
ψn (λ, x) := the n−th reduced division polynomial of Eλ
Its roots are the x coordinates of the points of Eλ whose order
divides n and is > 2.
Stoll’s approach
Examples
1
T (2) ∩ T (3) ⊂ {2, 3} and one can check T (2) ∩ T (3) = ∅.
2
For a primitive 3rd root of unity ω, T (ω) ∩ T (ω 2 ) ⊂ {ω, ω 2 }
and one can check T (ω) ∩ T (ω 2 ) = {ω, ω 2 }.
ψn (λ, x) := the n−th reduced division polynomial of Eλ
Its roots are the x coordinates of the points of Eλ whose order
divides n and is > 2.
Inductively, Stoll shows that
ψn (λ, x) ≡ 2e(n) (λ − x 2 )
mod 2e(n)+1 Z[λ, x].
Stoll’s approach
Examples
1
T (2) ∩ T (3) ⊂ {2, 3} and one can check T (2) ∩ T (3) = ∅.
2
For a primitive 3rd root of unity ω, T (ω) ∩ T (ω 2 ) ⊂ {ω, ω 2 }
and one can check T (ω) ∩ T (ω 2 ) = {ω, ω 2 }.
ψn (λ, x) := the n−th reduced division polynomial of Eλ
Its roots are the x coordinates of the points of Eλ whose order
divides n and is > 2.
Inductively, Stoll shows that
ψn (λ, x) ≡ 2e(n) (λ − x 2 )
mod 2e(n)+1 Z[λ, x].
λ ∈ T (α) \ {α} =⇒ ψn (λ, α) = 0 =⇒ ρ(λ) = ρ(α2 ).
Dynamical reformulation of Masser’s question
Consider the Lattès map induced by multiplication by 2 on Eλ
(x 2 − λ)2
.
4x(x − 1)(x − λ)
p
Here fλ (a) is the x−coordinate of [2](a, a(a − 1)(a − λ)) ∈ Eλ .
fλ (x) =
Dynamical reformulation of Masser’s question
Consider the Lattès map induced by multiplication by 2 on Eλ
(x 2 − λ)2
.
4x(x − 1)(x − λ)
p
Here fλ (a) is the x−coordinate of [2](a, a(a − 1)(a − λ)) ∈ Eλ .
fλ (x) =
(a,
p
a(a − 1)(a − λ)) ∈ (Eλ )tors ⇐⇒ a ∈ Prepfλ
Dynamical reformulation of Masser’s question
Consider the Lattès map induced by multiplication by 2 on Eλ
(x 2 − λ)2
.
4x(x − 1)(x − λ)
p
Here fλ (a) is the x−coordinate of [2](a, a(a − 1)(a − λ)) ∈ Eλ .
fλ (x) =
(a,
p
a(a − 1)(a − λ)) ∈ (Eλ )tors ⇐⇒ a ∈ Prepfλ
Question (Reformulation)
Are there infinitely many λ ∈ C such that 2, 3 ∈ Prepfλ ?
Another approach and more questions
DeMarco, Wang and Ye re-proved Masser and Zannier’s result
using the equidistribution theorem of Baker-Rumely.
Another approach and more questions
DeMarco, Wang and Ye re-proved Masser and Zannier’s result
using the equidistribution theorem of Baker-Rumely.
The approach adopted by them was introduced by Baker and
DeMarco for studying simultaneous preperiodic points for other
1-parameter families of rational maps, such as t 7→ z d + t.
Recovering Stoll’s result (M.,2015)
Let | · | := | · |2 be the 2−adic absolute value such that |2| = 12 .
Recovering Stoll’s result (M.,2015)
Let | · | := | · |2 be the 2−adic absolute value such that |2| = 12 .
Consider a ∈ C2 \ {0, 1} with |a| ≤ 1, e.g. a = 2, 3.
Recovering Stoll’s result (M.,2015)
Let | · | := | · |2 be the 2−adic absolute value such that |2| = 12 .
Consider a ∈ C2 \ {0, 1} with |a| ≤ 1, e.g. a = 2, 3.
Analyse |fλn (a)|.
Recovering Stoll’s result (M.,2015)
Let | · | := | · |2 be the 2−adic absolute value such that |2| = 12 .
Consider a ∈ C2 \ {0, 1} with |a| ≤ 1, e.g. a = 2, 3.
Analyse |fλn (a)|.
• If λ ∈ T (a) \ {a}, then |λ| ≤ 1.
Recovering Stoll’s result (M.,2015)
Let | · | := | · |2 be the 2−adic absolute value such that |2| = 12 .
Consider a ∈ C2 \ {0, 1} with |a| ≤ 1, e.g. a = 2, 3.
Analyse |fλn (a)|.
• If λ ∈ T (a) \ {a}, then |λ| ≤ 1.
Assume to the contrary that |λ| > 1.
Recovering Stoll’s result (M.,2015)
Let | · | := | · |2 be the 2−adic absolute value such that |2| = 12 .
Consider a ∈ C2 \ {0, 1} with |a| ≤ 1, e.g. a = 2, 3.
Analyse |fλn (a)|.
• If λ ∈ T (a) \ {a}, then |λ| ≤ 1.
Assume to the contrary that |λ| > 1.
Then |fλ (a)| =
4|λ|2
|a||a−1|||λ|
≥ 4|λ|.
Recovering Stoll’s result (M.,2015)
Let | · | := | · |2 be the 2−adic absolute value such that |2| = 12 .
Consider a ∈ C2 \ {0, 1} with |a| ≤ 1, e.g. a = 2, 3.
Analyse |fλn (a)|.
• If λ ∈ T (a) \ {a}, then |λ| ≤ 1.
Assume to the contrary that |λ| > 1.
4|λ|2
|a||a−1|||λ| ≥ 4|λ|.
|fλn+1 (a)| = 4|fλn (a)|,
Then |fλ (a)| =
Inductively,
for n ≥ 1.
Recovering Stoll’s result (M.,2015)
Let | · | := | · |2 be the 2−adic absolute value such that |2| = 12 .
Consider a ∈ C2 \ {0, 1} with |a| ≤ 1, e.g. a = 2, 3.
Analyse |fλn (a)|.
• If λ ∈ T (a) \ {a}, then |λ| ≤ 1.
Assume to the contrary that |λ| > 1.
4|λ|2
|a||a−1|||λ| ≥ 4|λ|.
|fλn+1 (a)| = 4|fλn (a)|,
Then |fλ (a)| =
Inductively,
for n ≥ 1.
This contradicts the fact that λ ∈ T (a) and fλ (a) 6= ∞.
Recovering Stoll’s result (M.,2015)
• If λ ∈ T (a) \ {a}, then |λ − a2 | < 1.
Recovering Stoll’s result (M.,2015)
• If λ ∈ T (a) \ {a}, then |λ − a2 | < 1.
Assume to the contrary that |λ − a2 | ≥ 1.
Recovering Stoll’s result (M.,2015)
• If λ ∈ T (a) \ {a}, then |λ − a2 | < 1.
Assume to the contrary that |λ − a2 | ≥ 1.
Since |λ| ≤ 1, we get |fλ (a)| =
4|a2 −λ|2
|a||a−1||a−λ|
≥ 4.
Recovering Stoll’s result (M.,2015)
• If λ ∈ T (a) \ {a}, then |λ − a2 | < 1.
Assume to the contrary that |λ − a2 | ≥ 1.
Since |λ| ≤ 1, we get |fλ (a)| =
4|a2 −λ|2
|a||a−1||a−λ|
Inductively, |fλn+1 (a)| = 4|fλn (a)|.
≥ 4.
Recovering Stoll’s result (M.,2015)
• If λ ∈ T (a) \ {a}, then |λ − a2 | < 1.
Assume to the contrary that |λ − a2 | ≥ 1.
Since |λ| ≤ 1, we get |fλ (a)| =
4|a2 −λ|2
|a||a−1||a−λ|
Inductively, |fλn+1 (a)| = 4|fλn (a)|.
Contradiction.
≥ 4.
A Weierstrass family
Eλ : y 2 = x 3 + λ, λ ∈ C \ {0}
A Weierstrass family
Eλ : y 2 = x 3 + λ, λ ∈ C \ {0}
p
T (a) = {λ ∈ C \ {0} : (a, a3 + λ) ∈ (Eλ )tors }
A Weierstrass family
Eλ : y 2 = x 3 + λ, λ ∈ C \ {0}
p
T (a) = {λ ∈ C \ {0} : (a, a3 + λ) ∈ (Eλ )tors }
The Lattès map induced by multiplication by 2 in Eλ is
fλ (x) =
x 4 − 8λx
.
4(x 3 + λ)
A Weierstrass family
Eλ : y 2 = x 3 + λ, λ ∈ C \ {0}
p
T (a) = {λ ∈ C \ {0} : (a, a3 + λ) ∈ (Eλ )tors }
The Lattès map induced by multiplication by 2 in Eλ is
fλ (x) =
x 4 − 8λx
.
4(x 3 + λ)
Let a ∈ C2 . We study the 2−adic absolute value of elements in
T (a), by looking at {|fλn (a)|2 }n∈N .
More impossible intersections
Theorem (M.,2015)
If α, β ∈ Q \ {0} are such that
α
β
∈
/ {−2, − 12 }, then
T (α) ∩ T (β) = ∅.
More impossible intersections
Theorem (M.,2015)
If α, β ∈ Q \ {0} are such that
α
β
∈
/ {−2, − 12 }, then
T (α) ∩ T (β) = ∅.
Moreover, for all a ∈ Q \ {0} we have
T (a) ∩ T (−2a) = {−a3 }.
More impossible intersections
Theorem (M.,2015)
If α, β ∈ Q \ {0} are such that
α
β
∈
/ {−2, − 12 }, then
T (α) ∩ T (β) = ∅.
Moreover, for all a ∈ Q \ {0} we have
T (a) ∩ T (−2a) = {−a3 }.
Theorem (M.,2015)
If α, β ∈ Q2 are such that gcd(6, e(Q2 (α, β)|Q2 )) = 1, |α| =
6 |β|
α3
1
and β 3 ∈
/ {−8, − 8 }, then we have that T (α) ∩ T (β) = ∅.
Moreover, for all a ∈ C2 \ {0} we have T (a) ∩ T (−2a) = {−a3 }.
A trichotomy and more impossible intersections
Theorem (M.,2015)
n
o
α3
3
Let λ ∈ T (α). Then either λ ∈ −α , 8 or
m
m
|λ| ∈ {4|α|3 , 41−(1/4) |α|3 , 22+(1/4) |α|3 : m ∈ N≥1 }.
Moreover, exactly one of the following is true.
1
|λ| = 4|α|3 ⇐⇒ 0, ∞ ∈
/ Ofλ (α).
2
|λ| = 41−(1/4) |α|3 for some m ∈ N≥1 , or λ = −α3 ⇐⇒
∞ ∈ Ofλ (α).
3
|λ| = 22+(1/4) |α|3 for some m ∈ N≥1 , or λ =
0 ∈ Ofλ (α).
m
m
α3
8
⇐⇒
The end
Thank you for your attention!