Finite index theorems for iterated Galois groups of unicritical polynomials
Abstract.
Let be the function field of a smooth, irreducible curve defined over . Let be of the form where is a power of the prime number , and let . For all , the Galois groups embed into , the fold wreath product of the cyclic group . We show that if is not isotrivial, then unless is postcritical or periodic. We are also able to prove that if and are two such distinct polynomials, then the fields and are disjoint over a finite extension of .
Key words and phrases:
Arithmetic Dynamics, Arboreal Galois Representations, Iterated Galois Groups2010 Mathematics Subject Classification:
Primary 37P15, Secondary 11G50, 11R32, 14G25, 37P05, 37P301. Introduction and Statement of Results
Let be a field. Let with and let . For , let be the field obtained by adjoining the th preimages of under to . (We declare that .) Set . For , define . In most of the paper, we will write and , suppressing the dependence on if there is no ambiguity.
The group embeds into , the automorphism group of an infinite ary rooted tree . Recently there has been much work on the problem of determining when the index is finite. The group is the image of an arboreal Galois representation, so this finite index problem is a natural analog in arithmetic dynamics of the finite index problem for the adic Galois representations associated to elliptic curves, resolved by Serre’s celebrated Open Image Theorem [Ser72]. By work of Odoni [Odo85], one expects that a generically chosen rational function has a surjective arboreal representation, i.e., that .
In this paper we study the family of polynomials for , which up to change of variables represents all polynomials with precisely one (finite) critical point. If the field contains a primitive th root of unity, then it is easy to show that for in this family, sits in , the infinite iterated wreath product of the cyclic group (with elements). As , this means that if , then . Thus it is impossible for to have finite index within this family (except when ). However, this simply means that, given the constraint on the size of , we should ask a different finite index question. We turn to the problem of when has finite index in .
Before stating our main results, we set some notation. Throughout this paper, unless otherwise indicated, will refer to a function field of transcendence degree over its field of constants . In other words, is the function field of a smooth, projective, irreducible curve over . We say that is isotrivial if is defined over up to a change of variables, that is, if for some of degree . In the special case of a unicritical polynomial , we have that is isotrivial if and only if . We say is periodic for if for some , and we say is preperiodic for if is periodic for some . Finally, we say that is postcritical for if for some and some critical point of .
With this notation, our first main theorem is as follows.
Theorem 1.1.
Let () be a power of the prime number , let , let and let . Then the following are equivalent:

The point is neither periodic nor postcritical for .

The group has finite index in .
All the methods used in the proof of Theorem 1.1 work for unicritical polynomials of any degree , except that we need the degree to be a prime power for proving the eventual stability of (see Theorem 1.3 below and Section 6). In the case where , this means that has finite index in . For larger this index is infinite, as mentioned previously. The case of isotrivial polynomials (i.e., when in Theorem 1.1) is very different and will be dealt with in Section 10.
It is fairly easy to see that the conditions on in Theorem 1.1 are necessary. If is periodic or postcritical, then by a straightforward argument (see Proposition 3.2). Most of the paper is devoted to the showing that these conditions are sufficient.
Remark 1.2.
In general one needs to rule out postcritically finite (PCF) maps in order to obtain a finite index result, as in the main result of [BT18b]. The reason we do not need to do this in Theorem 1.1 is that a PCF polynomial of the form is automatically isotrivial. This is because satisfies an equation of the form for some , and so . For isotrivial polynomials the PCF distinction regains its importance; see Section 10.
One of the key steps in the proof of Theorem 1.1 is an eventual stability result. As is usual in arithmetic dynamics, we say that the pair is eventually stable over the field if the number of irreducible factors of is uniformly bounded for all .
Theorem 1.3.
Let () be a power of the prime number . Let be a polynomial of the form where . Then for any nonperiodic , the pair is eventually stable over .
We also prove the following disjointness theorem for fields generated by inverse images of different points under different maps.
Theorem 1.4.
For let , where , and let . Suppose that there are no distinct with the property that lies on a curve in that is periodic under the action of . For each , let denote . Then for each , we have that
Theorem 1.4 also has a natural interpretation as a finite index result across preimage trees of several points (see Section 9).
Remark 1.5.
In light of Odoni’s work, unicritical polynomials with degree cannot be considered generic from the point of view of arboreal Galois theory (indeed, they are not a generic family in the moduli space of degree polynomials in any reasonable sense). There are other families of polynomials and rational functions (such as postcritically finite maps) that arise as obstructions to any potential classification of finite index arboreal representations – see [Jon13, Section 3] and [BT18b, Prop 3.3] for examples. One might hope that in these “exceptional” families, something similar to Theorem 1.1 could hold, in that a broad finite index result could be established for a natural overgroup other than . The authors will explore this in future work.
Acknowledgments. D.G. was partially supported by an NSERC Discovery grant. L.C. H. was partially supported by MOST Grant 1062115M003014MY2.
2. Wreath products
In this section we give a brief introduction to wreath products, which arise naturally from the Galois theory of the preimage fields .
Let be a permutation group acting on a set , and let be any group. Let be the group of functions from to with multiplication defined pointwise, or equivalently the direct product of copies of . The wreath product of by is the semidirect product , where acts on by permuting coordinates: for and we have
for each . We will use the notation for the wreath product, suppressing the set in the notation. (Another common convention is or if we wish to call attention to .)
Fix an integer . For , let be the complete rooted ary tree of level . It is easy to see that , and standard to show that satisfies the recursive formula
Therefore we may think of as the “th iterated wreath product” of , which we will denote . In general, for of degree and , the Galois group embeds into via the faithful action of on the th level of the tree of preimages of (see for example [Odo85] or [BT18b, Section 2]).
Assume now that , where is a field of characteristic that contains the th roots of unity. For such that is not a th power in , we have and . For any , the extension is a Kummer extension attained by adjoining to the th roots of where ranges over the roots of . Thus we have
This is clear if has distinct roots in . If has repeated roots, then sits inside a direct product of a smaller number of copies of , so the stated containments still hold.
Considering the Galois tower
we see that
where the implied permutation action of is on the set of roots of . By induction, embeds into , the th iterated wreath product of . Observe that sits as a subgroup of via the obvious action on the tree. Taking inverse limits, embeds into , which sits as a subgroup of .
We summarize our basic strategy for proving that has finite or infinite index in as Proposition 2.1.
Proposition 2.1.
Let . Then if and only if for all sufficiently large .
Proof.
Consider the projection map . The restriction of maps to . By basic group theory,
Therefore if is a proper subgroup of for infinitely many , then is unbounded as , and .
Conversely, by appealing to the profinite structure of we see that distinct cosets of in must project to distinct cosets in under for some . If there exists such that for all , then by induction,
for all . Thus as well. ∎
3. Necessary conditions
We prove that the conditions in Theorem 1.1 are necessary for finite index. For this part of the theorem, we do not need to assume that has prime power degree, or that is not isotrivial. The argument relies on a basic fact of algebra known as Capelli’s Lemma, which we will use many times throughout the paper. We state it below without proof.
Lemma 3.1 (Capelli’s Lemma).
Let be any field and let . Suppose is any root of . Then is irreducible over if and only if both is irreducible over and is irreducible over .
Proposition 3.2.
Suppose with , and let . If is either periodic or postcritical for , then
Proof.
First assume that is postcritical for , i.e., that there is some critical point of with for some . This means that the tree of preimages of is degenerate at the th level: as has at least one repeated root, we have
for every . As in Section 2, the Galois group embeds into the direct product of copies of . In particular,
for all sufficiently large . By Proposition 2.1, we conclude that has infinite index in .
Now assume that is periodic for and not postcritical, so that the tree of preimages of can be identified with the complete ary tree . The pair cannot be eventually stable by a straightforward argument using Capelli’s Lemma [BT18b, Prop 4.2]. This implies that the number of Galois orbits in is unbounded as , and thus that there are an infinite number of orbits in the action of on , where is the boundary (or the “ends”) of the tree , which can be identified with the set of infinite paths starting from the root of the tree [JL17, Prop 2.2]. But as acts transitively on the th level of the tree for every , we see that acts transitively on . A simple argument in group theory then implies that [JL17, Prop 3.3]. ∎
4. Height Estimates
In this section we present two lemmas that give key height inequalities, which will be used in the proofs of the main theorems. For background on heights, see [HS00, GNT13, BT18a].
First we set some notation. We continue with the assumption that is a function field of transcendence degree over . Choose a place of and set
Let be a nonarchimedean prime of , which gives a prime of . Let be the residue field ; note that is naturally isomorphic to . Then for each point , we have its Weil height
For with , let be the CallSilverman canonical height of relative to [CS93], defined by
We will often write sums indexed by primes of that satisfy some condition. As an example of our indexing convention, observe that
for all by the product formula for . Also define the forward orbit of under to be
With this notation we have the following two lemmas.
Lemma 4.1.
Let with . Let with . Let such that . For , let denote the set of primes of such that
for some . Then for any , we have
for all . (Note that, with the notation as in [BT18b], we have that for each place since is isomorphic to the field of constants.)
Proof.
See [BT18b, Section 5]. Note that and need not be distinct. ∎
Lemma 4.2.
Let with , and assume that is not isotrivial. Let be such that and that is also not postcritical. For every , there exists a constant such that
for all .
Proof.
This follows immediately from [BT18a, Lemma 4.2]. ∎
5. Finiteness of GCD
Proposition 5.1.
Let be the function field of a smooth, projective, irreducible curve defined over . For , let be nonisotrivial with for any such that . Let be such that for and for all nonnegative integers . Then there are at most finitely many places of such that there are positive integers with the property that
(5.1.1) 
In order to prove Proposition 5.1, we need a Bogomolovtype version of the main theorem of [GKNY17].
Theorem 5.2.
Let , let be a number field, let be the function field of a smooth, projective, geometrically irreducible curve defined over , and let such that is not a st root of unity. We let for , and for each point which is not a pole for either or , we consider the specialization of the polynomials at , denoted as ; for each such , we denote by the corresponding canonical heights. Then there exists such that there are finitely many points for which .
Proof.
We argue by contradiction and therefore assume there exists an infinite sequence of points such that
(5.2.1) 
We proceed as in [GKNY17] and for each , we construct adelic metrized line bundles on the curve corresponding to the families of dynamical systems , respectively (parametrized by the points ). In general, given a rational function (defined over ), there exists an adelic metrized line bundle associated to the family of dynamical systems (as we vary ), where is the line bundle on obtained by pullingback through the morphism ; for more details, see [GKNY17, Sections 3.2 and 4.1]. In particular, this gives rise to height functions (associated to the metrized line bundles , for ) for which we have:
(5.2.2) 
for each , where is the degree of the rational function ; see [GKNY17, Proposition 3.5]. Our hypothesis (see (5.2.1)), coupled with (5.2.2), yields that for the infinite sequence , we have
(5.2.3) 
Using (5.2.3) coupled with [CL11, Proposition 3.4.2] (which uses crucially the inequalities established by Zhang [Zha95] regarding the successive minima associated to a metrized line bundle), we derive that there exist positive integers (for ) such that the two line bundles and are linearly equivalent and, moreover, the two heights are proportional. In particular, this means that for each , we have that if and only if . Using this last equivalence along with equation (5.2.2), we obtain that for each point ,
(5.2.4) 
Using (5.2.4) and the fact that only preperiodic points have canonical height equal to (for a rational function defined over ), we get that is preperiodic for the dynamical system if and only if is preperiodic for the dynamical system . In other words, is a PCF (postcriticallyfinite) parameter for the family of unicritical polynomials (parametrized by ) if and only if is a PCF parameter for the same dynamical system . Because there exist infinitely many PCF parameters for the family of polynomials , we conclude that there exist infinitely many such that
(5.2.5) 
Now, we let be the Zariski closure in the plane of the image of under the rational map given by ; then (5.2.5) yields that there exist infinitely many points with both coordinates PCF parameters for the unicritical dynamical system . But then [GKNY17, Theorem 1.1] yields that is given by an equation of the form for some st root of unity , where are the coordinates of (note that is neither a horizontal line, nor a vertical line because both and are nonconstant rational functions, and so possibilities (1)(2) in [GKNY17, Theorem 1.1] cannot occur). However, our hypothesis regarding not being a st root of unity prevents from satisfying such an equation and this contradiction proves that there exists no infinite sequence as in (5.2.1). This concludes our proof of Theorem 5.2. ∎
We now state a simple lemma that follows from work of Call and Silverman [CS93]. We recall the following lemma from [BT18b, Lemma 8.3].
Lemma 5.3.
Let be the function field of a smooth, irreducible curve defined over . For each and each which is not a pole of , we denote by the specialization of at . Let have degree . Let with (where is the canonical height associated to the rational function ). Let be a sequence of points of satisfying for a sequence of positive integers with . Then
Proof of Proposition 5.1..
Let be a number field such that is a geometrically irreducible curve defined over . If there were infinitely many places of the function field such that (5.1.1) holds, then this means there exists an infinite sequence of points such that there exist some nonnegative integers and , for which we have
(5.3.1) 
Also, since for all integers and for all integers , we derive that the integers and appearing in (5.3.1) must tend to infinity. But then Lemma 5.3 yields that
contradicting thus Theorem 5.2. ∎
6. Eventual Stability
The results of this section are valid (with only a few changes) in the more general setting of any function field of a curve defined over a finitely generated field of characteristic . However, we will restrict to the case relevant for our results. So, let be a number field, let be the function field of a smooth, projective, geometrically irreducible curve defined over , let be a power of a prime number and let for some . (Note that our hypothesis yields that is algebraically closed in .) Let be a point which is not periodic under ; then we will prove that the pair is eventually stable over .
We note that the places of correspond to points of . For any element and any point of such that does not have a pole at , we let denote the specialization of to at (see [CS93] for more details); in other words, seeing as a rational function defined over , then . Likewise for a rational function , we let denote the specialization of to at for any such that the coefficients of do not have poles at ; so, for our polynomial , we simply have that for any point which is not a pole of . Finally, for any , we let be the field of definition for the point ; in particular, this means that and furthermore, for each , we have that .
With notation as above, the following lemma follows from the work of [CS93].
Lemma 6.1.
Let be a nonisotrivial rational function of degree greater than one and let . Then we have the following:

if , then the set of specializations from to such that has bounded height (with respect to some degree divisor on ); and

if is not periodic under , then the set of specializations from to such that is periodic under has bounded height.
Proof.
The statement of (a) follows directly from [CS93, Theorem 4.1]. We now prove (b). If is not preperiodic under , then , by [Bak09]. Then, from (a), it follows that the set of such that has bounded height. Now, if is strictly preperiodic, then there are at most finitely many such that is periodic under , so the set of such clearly has bounded height. ∎
The following lemma is a simple consequence of the main theorem of [JL17].
Lemma 6.2.
Let where is a element of a number field with the property that for some nonarchimedean place of such that . Then for any that is not periodic under , the pair is eventually stable over .
Proof.
Let be the residue field at . Then reducing at induces a map such that every point has exactly one inverse image under . Let denote the reduction of at . Then there is an such that . Theorem 1.7 of [JL17] states that must therefore be eventually stable over . ∎
Now we can state the main result of this section, which is instrumental in proving Theorem 1.3.
Proposition 6.3.
Let be the function field of a smooth, projective, geometrically irreducible curve defined over a number field . Let , where . Then for any that is not periodic under , the pair is eventually stable over .
Proof.
We may choose a specialization of to such that:

is an algebraic integer; and

is not periodic under .
Indeed, for all but finitely many , there is a such that ; furthermore, condition (ii) is achieved for all points of sufficiently large height (by Lemma 5.3), while on the other hand, if the algebraic integer has large height, then the point must have large height (on ) as well. Then, by Lemma 6.2, the pair is eventually stable over , which implies that is eventually stable over . ∎
7. Ramification and Galois theory
Let with . In this section we define Condition R and Condition U in terms of primes dividing certain elements of related to the forward orbits of 0. In Proposition 7.4 and 7.5 we show that these conditions control ramification in the extensions , with consequences for the Galois theory of these extensions. We begin with the following standard lemma from Galois theory.
Lemma 7.1.
Let and be fields all contained in some larger field. Assume that are finite extensions of .

If are Galois over with , then is Galois over and .

If are Galois over with for each , then .
Conditions R and U make use of the notion of good reduction of a map at a prime . A polynomial
has good reduction at if and for . See [MS94] or [Sil07, Theorem 2.15] for a more careful definition that also applies to rational functions. Clearly any has good reduction at all but finitely many . The idea behind the definition is that if has good reduction at , then commutes with the reduction mod map . This is clear for polynomials (see [Sil07, Theorem 2.18] for a proof for rational functions). We say that has good separable reduction at if the reduced map is separable.
Definition 7.2.
Let . We say that a prime of satisfies Condition R at for and if the following hold:

has good separable reduction at ;

for all ;

;

.
Definition 7.3.
Let . We say that a prime of satisfies Condition U at for and if the following hold:

has good separable reduction at ;

for all ;

.
Proposition 7.4.
Let . Let be a prime of that satisfies Condition U at for and . Then is unramified in .
Proof.
Proposition 7.5.
Let . Suppose that is a prime of that satisfies Condition R at for and that is irreducible over . Then
Furthermore, does not ramify in and does ramify in any field such that .
Proof.
Observe that Condition R at for implies Condition U at for . By Proposition 7.4, does not ramify in .
Let denote the image of under the reduction mod map, which is well defined as long as . Consider the map that comes from reducing at , and recall that Condition R assumes that has good reduction at . The unique critical point of is . By (b) of Condition R, we see that has no repeated roots. By (c) of Condition R, we see that , and is totally ramified over