Post-critically finite maps
Galois representations and finite ramification
A finiteness theorem for PCF maps

Post-critically finite rational functions overnumber fields
Rafe Jones
Carleton college

September 26, 2013
Laboratoire de mathématiques de BesançonRafe Jones

Post-critically finite rational functions over number fields

Post-critically finite maps
Galois representations and finite ramification
A finiteness theorem for PCF maps

OutlineI. Post-critically finite (PCF) maps: definitions and examples
II. Galois representations and finite ramification
III. A finiteness theorem for PCF maps over number fields:

Theorem (R. Benedetto, P. Ingram, RJ, A. Levy, 2013)
Let d, B ∈ Z with d ≥ 2 and B ≥ 1. Up to conjugacy, there are
only finitely many PCF rational functions of degree d defined overa number field of degree at most B, except for flexible Lattès maps.

Rafe Jones

Post-critically finite rational functions over number fields

Post-critically finite maps
Galois representations and finite ramification
A finiteness theorem for PCF maps

Setup
ExamplesSetup

Let φ ∈ C(z) be a rational function of degree d ≥ 2.
Denote by φn the n-fold composition of φ with itself.
We say φ and ψ are conjugate if there is a Möbius transformationf ∈ PGL2 (C) with f ◦ φ ◦ f −1 = ψ.

Rafe Jones

Post-critically finite rational functions over number fields

Post-critically finite maps
Galois representations and finite ramification
A finiteness theorem for PCF maps

Setup
ExamplesSetup, continued

Riemann-Hurwitz: counting multiplicity, φ has 2d − 2 criticalpoints in P1 (C).

Definition
The orbit of α ∈ C under φ is the set
Oφ (α) = {α, φ(α), φ2 (α), . . .}.
We say that φ is post-critically finite (PCF) if for every criticalpoint γ of φ, the orbit Oφ (γ) is finite.
Every conjugate of a PCF map is PCF.

Rafe Jones

Post-critically finite rational functions over number fields

Post-critically finite maps
Galois representations and finite ramification
A finiteness theorem for PCF maps

Setup
ExamplesExamplesI φ(z) = z d . Critφ = {0, ∞}. 0 7→ 0, ∞ 7→ ∞I φ(z) = 1/z 2 . Critφ = {0, ∞}. 0 7→ ∞ 7→ 0

I φ(z) = z 2 − 2. Critφ = {0, ∞}. 0 7→ −2 7→ 2 7→ 2, ∞ 7→ ∞I φ(z) = z 2 − 1. Critφ = {0, ∞}. 0 7→ −1 7→ 0, ∞ 7→ ∞2I φ(z) = 6z +16z+16. Critφ = {0, −2}.
−3z 2 −4z−4

0 7→ −4 7→ −4/3 7→ −4/3, −2 7→ −1 7→ −2
42+1I φ(z) = x +2x. #Critφ = 6.
4x 3 −4x

Rafe Jones

Post-critically finite rational functions over number fields

Post-critically finite maps
Galois representations and finite ramification
A finiteness theorem for PCF maps

Setup
ExamplesLattès maps
We say φ ∈ C(z) is a Lattès map if there is an elliptic curve E , amorphism α : E → E , and a finite separable map π such that thefollowing diagram commutes:
E

α

π

/E
π



P1

Rafe Jones

φ


/ P1

Post-critically finite rational functions over number fields

Post-critically finite maps
Galois representations and finite ramification
A finiteness theorem for PCF maps

E

Setup
Examples

α

π

/E
π



P1

φ


/ P1

Natural choices: let π be the double cover given by π(P) = x(P),and let α = [m].
The resulting maps are called flexible Lattès maps.
If α = [m] for fixed m, and E varies, we obtain a family of
non-conjugate maps.

Rafe Jones

Post-critically finite rational functions over number fields

Post-critically finite maps
Galois representations and finite ramification
A finiteness theorem for PCF maps

Setup
Examples

[m]

E
x

/E
x



P1

φE ,m


/ P1

Claim: φE ,m is PCF.
[m] is unramified, so critical points of φE ,m come from P 6= Q with
x(P) = x(Q)and[m]P = [m]Q.

x(P) = x(Q) ⇒ Q = −P, so want P 6= −P and [m]P = [−m]Pi. e., want P ∈ E [2m] \ E [2]Rafe Jones

Post-critically finite rational functions over number fields

Post-critically finite maps
Galois representations and finite ramification
A finiteness theorem for PCF maps

Setup
ExamplesThus CritφE ,m = {x(P) : P ∈ E [2m] \ E [2]}.
For n ≥ 1,

φnE ,m (x(P)) = x([mn ]P).

If P ∈ E [2m] \ E [2], then [mn ]P ∈ E [2] for n ≥ 1.
Therefore φE ,m is PCF, as desired.
4

2+1
Example: E : y 2 = x 3 − x, m = 2. φE ,m = x 4x+2x
3 −4x .

Rafe Jones

Post-critically finite rational functions over number fields

Post-critically finite maps
Galois representations and finite ramification
A finiteness theorem for PCF maps

PCF maps in dynamics
Galois representations from pre-image trees
Finite ramification

PCF maps in dynamics

Rafe Jones

Post-critically finite rational functions over number fields

Post-critically finite maps
Galois representations and finite ramification
A finiteness theorem for PCF maps

PCF maps in dynamics
Galois representations from pre-image trees
Finite ramification

Arboreal Galois representations
Let K be a number field, φ ∈ K (z), and b ∈ P1 (K ).
The preimage tree of φ with root b has vertex set
G
φ−i (b),i≥0with two elements connected iff φ maps one to the other.
Denote this tree by T∞ , and its truncation to the nth level by Tn .
For simplicity, assume that T∞ contains no critical points, so that
it is a complete d-ary rooted tree.

Rafe Jones

Post-critically finite rational functions over number fields

Post-critically finite maps
Galois representations and finite ramification
A finiteness theorem for PCF maps!
√
− −3−2 5""!

PCF maps in dynamics
Galois representations from pre-image trees
Finite ramification!
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First two levels of preimage tree of f (x) = x x+1 , b = 0.

Rafe Jones

Post-critically finite rational functions over number fields

Post-critically finite maps
Galois representations and finite ramification
A finiteness theorem for PCF maps

PCF maps in dynamics
Galois representations from pre-image trees
Finite ramification

Let Kn = K (φ−n (b)), and note Kn+1 ⊇ Kn . Let K∞ =

S

Kn .

Let Gn = Gal(Kn /K ), and G∞ = lim Gn . All these objects depend←−on φ and b, but to ease notation we don’t make explicit reference
to this dependence.
We have injections
Gn ,→ Aut(Tn ) ∼= (Sd )wr(n)G∞ ,→ Aut(T∞ ).

The latter is the arboreal Galois representation associated to φ, b.
When d = 2, Gn is a 2-group, and G∞ is a pro-2 group.

Rafe Jones

Post-critically finite rational functions over number fields

Post-critically finite maps
Galois representations and finite ramification
A finiteness theorem for PCF maps

PCF maps in dynamics
Galois representations from pre-image trees
Finite ramification

Lattès maps, once again

If φE ,m is a flexible Lattès map and b = ∞, then
Kn = K (x(E [mn ])).
When m = ` is prime, thenG∞ ,→ GL2 (Z` ),and G∞ is a subgroup of index at most 2 of the image of the`-adic representation attached to E .

Rafe Jones

Post-critically finite rational functions over number fields

Post-critically finite maps
Galois representations and finite ramification
A finiteness theorem for PCF maps

PCF maps in dynamics
Galois representations from pre-image trees
Finite ramification

Finitely ramified representations

Theorem (Aitken-Hajir-Maire 2005, Hajir-Cullinan 2012)
Let K be a number field and φ ∈ K (z). If φ is PCF, then theextension K∞ /K is ramified over only finitely many primes of K .
Already known for Lattès maps φE ,` with b = ∞: K∞ can ramify
only at ` and the primes of bad reduction for E .

Rafe Jones

Post-critically finite rational functions over number fields

Post-critically finite maps
Galois representations and finite ramification
A finiteness theorem for PCF maps

PCF maps in dynamics
Galois representations from pre-image trees
Finite ramification

Let φ(x) = p(x)/q(x) with p, q relatively prime.
Assume for simplicity that ∞ is not a critical point of φ and b = 0.
Then K∞ can ramify only over primes of K dividing one of thefollowing:Q
1. γ φi (γ), where the product is over the critical points γ of φ,and over i with 1 ≤ i ≤ n.
2. The leading coefficient of pq 0 − qp 0 .
3. The leading coefficients of p and q.
4. The resultant of p and q.
When φ is a monic polynomial, (3) and (4) in the above list areboth 1, and (2) is just the degree of φ.

Rafe Jones

Post-critically finite rational functions over number fields

Post-critically finite maps
Galois representations and finite ramification
A finiteness theorem for PCF maps

PCF maps in dynamics
Galois representations from pre-image trees
Finite ramification

Example Let K = Q and φ(x) = x 2 − 2. 0 7→ −2 7→ 2 7→ 2
K∞ is ramified over Q only at the prime 2.
n∼∼
Kn = Q(ζ2n+2 + ζ2−1
n+2 ). So Gn = Z/2 Z, G∞ = Z2 .

Example Let K = Q and φ(x) = (x + 1)2 − 2. −1 7→ −2 7→ −1
K∞ is ramified over Q only at 2 and ∞.
#G2 = 23
#G3 = 26
#G4 = 211
#G5 = 222
#G6 = 243 (J. Klüners)#G7 = 286 (M. Watkins - MAGMA)#G8 = 2171 ?

Rafe Jones

Post-critically finite rational functions over number fields

Post-critically finite maps
Galois representations and finite ramification
A finiteness theorem for PCF maps

PCF maps in dynamics
Galois representations from pre-image trees
Finite ramification

Question: Does there exists a number field K and a PCF map
φ ∈ K (x) of degree 2 such that K∞ is unramified at 2?

Rafe Jones

Post-critically finite rational functions over number fields

Post-critically finite maps
Galois representations and finite ramification
A finiteness theorem for PCF maps

PCF maps in dynamics
Galois representations from pre-image trees
Finite ramification

An overgroup for G∞
Return to an arbitrary number field K . Instead of b = 0, takeb = t, where t is transcendental over K , and work over K (t).
Let φ ∈ K (x) be post-critically finite, and put
Kn,t := K (t)(φ−n (t))

K (t)

Gn

:= Gal(Kn,t /K (t)).

K (t)The isomorphism class of G∞ is invariant under conjugation of
φ. It contains G∞ as a subgroup (specialization t = b).
We have an exact sequence
C(t)
K (t)
1 → G∞→ G∞→ Gal(L/K ) → 1,

where L = K ∩ K∞,t .
Rafe Jones

Post-critically finite rational functions over number fields

Post-critically finite maps
Galois representations and finite ramification
A finiteness theorem for PCF maps

C(t)The group G∞
f (x).

PCF maps in dynamics
Galois representations from pre-image trees
Finite ramificationis the profinite iterated monodromy group of

It is a (topologically) finitely generated group satisfying a property
known as self-similarity.
The action of its generators on T∞ is given by an explicitly
computable finite automaton, which can be calculated via abeautiful theory involving lifts of loops in C. (V. Nekrashevych)

Rafe Jones

Post-critically finite rational functions over number fields

Post-critically finite maps
Galois representations and finite ramification
A finiteness theorem for PCF maps

PCF maps in dynamics
Galois representations from pre-image trees
Finite ramification

Example: K = Q, φ(x) = x 2 − 2. Then G∞ is the pro-2
completion of the infinite Dihedral group D∞ .
C(t)Example: K = Q, φ(x) = x 2 − 1. Then G∞ is the pro-2
completion of the Basilica group B. Note that (x + 1)2 − 2 isconjugate to x 2 − 1 by x 7→ x + 1.
C(t)

K (t)

Recent work of R. Pink shows that G∞
group.

is also a self-similar

Moreover, if φ is a (PCF) quadratic polynomial whose finite criticalC(t)point does not have a periodic orbit, then G∞ is isomorphic to a
K (t)subgroup of G∞ of index at most 4.

Rafe Jones

Post-critically finite rational functions over number fields

Post-critically finite maps
Galois representations and finite ramification
A finiteness theorem for PCF maps

Multipliers and McMullen’s theorem
Heights
A non-archimedean version of a theorem of Fatou

Theorem (R. Benedetto, P. Ingram, RJ, A. Levy, 2013)
Let d, B ∈ Z with d ≥ 2 and B ≥ 1. Up to conjugacy, there are
only finitely many PCF rational functions of degree d defined overa number field of degree at most B, except for flexible Lattès maps.

Rafe Jones

Post-critically finite rational functions over number fields

Post-critically finite maps
Galois representations and finite ramification
A finiteness theorem for PCF maps

Multipliers and McMullen’s theorem
Heights
A non-archimedean version of a theorem of Fatou

Corollary (M. Manes, D. Yap, 2013)
Suppose that φ ∈ Q(z) is quadratic and PCF. Then φ is conjugate
to one of the following:z2

z2 − 2z2 − 1

1/z 2

1(z−1)212(z−1)22(z−1)2−1
4z 2 −4z−4
9z 2 −12z2z+1
4z−2z 2−2z
2z 2 −4z+1

3z 2 −4z+1
1−4zMoreover, none of these twelve is conjugate to any of the others.

Rafe Jones

Post-critically finite rational functions over number fields

Post-critically finite maps
Galois representations and finite ramification
A finiteness theorem for PCF maps

Multipliers and McMullen’s theorem
Heights
A non-archimedean version of a theorem of Fatou

Multipliers
Let K be a field, φ ∈ K (z).
Let γ ∈ P1 (K ) be a fixed point of φ. The multiplier of γ is
λ := φ0 (γ) ∈ K . The set of multipliers of the fixed points of φ isinvariant under conjugation of φ.
Changing coordinates so γ = 0, we haveφ(z) = λz + higher order terms
in some neighborhood of zero.
We say γ is attracting with respect to an absolute value | · | on K
if |λ| < 1.
Extend these definitions to n-periodic points of φ by considering
them as fixed points of φn .
Rafe Jones

Post-critically finite rational functions over number fields

Post-critically finite maps
Galois representations and finite ramification
A finiteness theorem for PCF maps

Multipliers and McMullen’s theorem
Heights
A non-archimedean version of a theorem of Fatou

McMullen’s theorem
For φ ∈ C(z), denote by Mn (φ) the unordered set of multipliers of
all n-periodic points of φ.

Theorem (McMullen, 1987)
For fixed d ≥ 2, there exists Nd ≥ 1 such that the set
M(Nd ) := M1 (φ) t M2 (φ) t · · · t MNd (φ)
determines the conjugacy class of φ up to finitely many choices,
unless φ is a flexible Lattès map.
Special bonus (Milnor): N2 = 1, and M(1) uniquely determines
the conjugacy class of φ when φ is quadratic.
Rafe Jones

Post-critically finite rational functions over number fields

Post-critically finite maps
Galois representations and finite ramification
A finiteness theorem for PCF maps

Multipliers and McMullen’s theorem
Heights
A non-archimedean version of a theorem of Fatou

Proof strategy
By Thurston rigidity, the multipliers of a PCF map φ ∈ C(z) all lie
in Q. Thus if φ is PCF of degree d, then M(Nd ) ⊂ (Q)k for somepositive integer k.
I If φ is PCF of degree d, show that M(Nd ) belongs to a set ofbounded height.

I Observe that If φ is defined over a number field of degree at

most B, then M(Nd ) is defined over a number field of degree
at most B 0 .
I By standard properties of height, there are only finitely many

possibilities for M(Nd ).

I From McMullen’s theorem, conclude that φ belongs to a finite

collection of conjugacy classes.
Rafe Jones

Post-critically finite rational functions over number fields

Post-critically finite maps
Galois representations and finite ramification
A finiteness theorem for PCF maps

Multipliers and McMullen’s theorem
Heights
A non-archimedean version of a theorem of Fatou

Heights
Definition
The height of α ∈ K , where K is a finite extension of Q, is definedby
X [Kv : Qv ]log max{1, |α|v },h(α) =[K : Q]v ∈MK

where MK denotes the set of absolute values of K , and Kv denotes
the completion of K with respect to the absolute value v .
I h(α) is invariant under finite extensions of K , so h : Q → R is

well-defined.

I for A, B ∈ Z≥0 , there are only finitely many α ∈ Q satisfyingbothh(α) ≤ A

andRafe Jones[Q(α) : Q] ≤ B.
Post-critically finite rational functions over number fields

Post-critically finite maps
Galois representations and finite ramification
A finiteness theorem for PCF maps

Multipliers and McMullen’s theorem
Heights
A non-archimedean version of a theorem of Fatou

A non-archimedean version of a theorem of Fatou
Theorem (Fatou, 1920)
Let φ ∈ C(z). A cycle of φ whose multiplier satisfies |λ| < 1strictly attracts a critical point of φ.
Key observation: a PCF rational function φ cannot have any
critical points strictly attracted to a cycle. So every multiplier of φsatisfies |λ| ≥ 1.
Hope: prove that if φ is defined over a number field K , in fact
|λ| ≥ 1 for every absolute value on K .
Example: φ(z) = z p . Every cycle is attracting with respect to thep-adic absolute value.
Rafe Jones

Post-critically finite rational functions over number fields

Post-critically finite maps
Galois representations and finite ramification
A finiteness theorem for PCF maps

Multipliers and McMullen’s theorem
Heights
A non-archimedean version of a theorem of Fatou

Example: φ(z) = z 2 − 4z. The fixed point 0 has multiplier 4, andso is 2-adically attracting. But 2 7→ −4 7→ 0, so 0 does not strictly
attract a critical point.

Theorem (Benedetto-Ingram-J-Levy)
Let φ ∈ L(z), where L has characteristic zero, residue characteristicp, and is complete with respect to a non-archimedean absolutevalue | · |p . There exists p ≤ 1 such that any cycle whose
multiplier satisfies |λ|p < p strictly attracts a critical point.
Moreover, if p > d, then p = 1.
More precisely, we can take
p = min{|m|dp : 1 ≤ m ≤ d}.

Rafe Jones

Post-critically finite rational functions over number fields

Post-critically finite maps
Galois representations and finite ramification
A finiteness theorem for PCF maps

Multipliers and McMullen’s theorem
Heights
A non-archimedean version of a theorem of Fatou

Example: When d = 2, 2 = 1/4 and p = 1 for p ≥ 3. Note thebound of 1/4 cannot be improved.
This shows that if λ is the multiplier of a fixed point of a quadratic
rational map φ ∈ C(z), then h(λ) ≤ log 4.

Rafe Jones

Post-critically finite rational functions over number fields