An equivariant isomorphism theorem for mod
reductions of arboreal Galois representations
Let be a quadratic, monic polynomial with coefficients in , where is a localization of a number ring . In this paper, we first prove that if is non-square and non-isotrivial, then there exists an absolute, effective constant with the following property: for all primes such that the reduced polynomial is non-square and non-isotrivial, the squarefree Zsigmondy set of is bounded by . Using this result, we prove that if is non-isotrivial and geometrically stable then outside a finite, effective set of primes of the geometric part of the arboreal representation of is isomorphic to that of . As an application of our results we prove R. Jones’ conjecture on the arboreal Galois representation attached to the polynomial .
Key words and phrases:Arithmetic dynamics, arboreal Galois representations, Zsigmondy set.
2010 Mathematics Subject Classification:Primary: 37P05, 37P15; Secondary: 11G05, 20E08.
Let be a field, let be transcendental over and let be a rational function of degree . Suppose that is separable for every , where is the -th iterate. We fix separable closures of and of , and we let . One can associate to an infinite, regular, -ary tree rooted in 0: the nodes at distance from the root are labeled by the roots of and a node at level is connected to a node at level if and only if . An automorphism of is a bijection of the set of nodes such that is connected to if and only if is connected to ; the full automorphism group coincides with , where is the set of nodes at distance from the root. It follows that carries a natural profinite topology.
The group acts on through tree automorphisms; the corresponding continuous group homomorphism is called the arboreal Galois representation attached to . The study of such maps over various ground fields is a central topic in modern arithmetic dynamics, as witnessed by the many papers on the topic such as [3, 4, 8, 9, 10, 11, 12, 13, 14, 16]. In particular, in the setting described above the coefficients of belong to the rational function field over , and it is therefore important to distinguish the image of from its geometric part, as explained below. Boston and Jones  and Hindes  studied in detail some of the features of this framework.
Let and be as above.
The geometric part of is the image of the subgroup via . We denote it by , and it is identified, via , with .
We say that has geometrically finite index if . In particular, we say that is geometrically surjective if .
In this paper, we will focus on monic, quadratic polynomials with coefficients in the polynomial ring , where is a field of characteristic different from 2. The arithmetic of the arboreal representations attached to such polynomials is deeply related to that of their adjusted post-critical orbit, which is the sequence defined by: , for . In particular, linear dependence relations among the ’s in the -vector space are of crucial importance, as shown for example in [10, 16, 22]. For this reason, one is interested in studying primitive prime divisors for , namely irreducible elements such that and for every such that . We call a primitive prime divisor squarefree if, in addition, . The Zsigmondy set of is the set , see for example [11, 20] for more on the topic.
The squarefree Zsigmondy set of is defined by:
Notice that if , then , where is the squarefree Zsigmondy set of . For this reason, there is no ambiguity when talking about the Zsigmondy set of a quadratic polynomial in , when seen as an element of the bigger ring .
From now on, we let be a number field with number ring . Let be a finite set of primes of containing all primes dividing 2, and denote by the localization of at (i.e. we allow denominators which belong to some ). Let , where . For every the reduction of modulo yields a monic quadratic polynomial in , where . Recall that a polynomial is called isotrivial if there exists such that . Equivalently, is isotrivial if .
In [14, Conjecture 6.7] Jones conjectured that the arboreal Galois representation of is surjective for any field of characteristic different from . When has characteristic 0 or 3 modulo 4, such conjecture can be proven by adapting an argument of Stoll [22, §2] that was used by the author to show that the arboreal Galois representation attached to certain polynomials of the form , with , is surjective. However, this is a highly “ad hoc” argument, and fails when the characteristic is 1 modulo 4.
In this paper we show that the aforementioned conjecture is an instance of a much more general fact, of arithmetic and geometric nature, concerning the squarefree Zsigmondy set attached to a quadratic polynomial satisfying suitable hypotheses. More specifically, we will show how it is possible to compare the geometric part of the arboreal Galois representation attached to and the geometric part of the arboreal Galois representation attached to the reduced polynomial . This can be viewed as an instance of “arithmetic specialization” of the representation. The key idea is to provide, given a non-isotrivial, non-square , a uniform and effective bound on the squarefree Zsigmondy set of , for all but finitely many ’s. This will allow to show that the geometric part of the arboreal representation attached to does not change (in a suitable sense, cf. Definition 1.4) after reducing modulo , for all primes outside an effective, finite set. In turn, an application of our results yields a complete proof of Jones’ conjecture (cf. Theorem 4.1).
The first main result of the paper is the following theorem.
Suppose that is not isotrivial and . Then there exists an effective constant with the following property: let be a prime of such that is not isotrivial and , and suppose that . Then .
One of the key ingredients of the proof is the notion of dynamical inseparability degree of a quadratic polynomial in positive characteristic (which we introduce with Definition 2.4) that allows to transfer methods for height bounds in characteristic zero to positive characteristic via a version of the ABC theorem for function fields. The dynamical inseparability degree of a non-isotrivial quadratic map is a way of measuring the degeneracy of the problem in positive characteristic by a non-negative integer, which is then used in the height bounds.
The constant mentioned in Theorem 1.3 can be easily made completely explicit: it just depends on the reduction of modulo a finite, effectively computable set of primes of . Notice that if is not isotrivial and , then is not isotrivial and for all but finitely many primes . Finally, observe that the hypotheses of Theorem 1.3 are sharp: if is isotrivial then it is post-critically finite modulo for every , and therefore is infinite. If , then is a square for every , and .
Next, we will use Theorem 1.3 in order to compare the geometric part of the arboreal representation attached to and the geometric part of the representation attached to the reduced polynomial . In doing so, we must take care of the following subtlety: the tree on which acts and the tree on which acts are isomorphic as trees, but they are not the same object. Therefore, in order to compare the two representations one needs to choose an identification between the trees. This motivates the following definition.
Let and let be infinite, regular, rooted -ary trees. Let and let be the trees truncated at level . Let be topological groups acting continuously on , respectively. An equivariant isomorphism is a pair , where is an isomorphism of topological groups and is a tree isomorphism such that for every and every one has . If there exists an equivariant isomorphism between the two pairs, we write .
Definition 1.4 aims to encapture the structure of the tree as a -set. For example, the group can act on the binary tree truncated at level 2 in two different ways: transitively or non-transitively. Of course in our analysis we want to consider these as two distinct objects.
We will make use of Theorem 1.3 to prove the following result. Recall that is called geometrically stable if is irreducible in for every .
Let be geometrically stable and non-isotrivial. Then there exists a finite, effective set of primes of such that if and only if .
In particular, the arboreal representation attached to a polynomial satisfying the hypotheses of Theorem 1.5 has geometrically finite index, as shown in . We will re-obtain this result in the course of our proof.
Notice that non-isotriviality is a necessary condition for the statement of Theorem 1.5 to hold: if for example , it is immediate to check that is geometrically surjective (cf. Theorem 4.2); however, is post-critically finite for every prime , and therefore the geometric part of has infinite index in (see [16, Theorem 3.1]). Moreover, it is crucial to consider the geometric part, and not the whole image of the representation. In fact, for instance one can show that has surjective arboreal representation, but is certainly not surjective for every .
Finally, we will show how to use Theorem 1.5 to prove the following conjecture.
Conjecture 1.6 ([14, Conjecture 6.7]).
Let be a field of characteristic , and let be transcendental over . Then the polynomial has surjective arboreal representation.
Here is a brief outline of the paper: in Section 2 we will prove Theorem 1.3, by providing (cf. Theorems 2.6 and 2.8) a suitable completely effective height bound for certain integral points on elliptic curves over function fields in positive characteristic associated to quadratic polynomials. In Section 3 we will prove Theorem 1.5, using Theorem 1.3 and the relation bewteen the squarefree Zsigmondy set and the image of an arboreal Galois representation (cf. Corollary 3.4). Finally, in Section 4 we will prove Conjecture 1.6.
Notation and conventions
When is a finite extension, we will denote by a complete set of valuations of . All valuations in are normalized, and all residue fields have degree 1, since the base field is algebraically closed. We will denote by the set of finite valuations, i.e. the set of valuations of not extending the infinite valuation of , and by the set of infinite valuations, so that .
For an element , the logarithmic height of , with respect to , is defined by:
When , we will omit from the notation for height, i.e. we will write for .
If , the degree of coincides with . We denote by the radical of , namely the product of its distinct irreducible factors.
If is a finite, separable field extension, we will denote by the norm map.
If is a group acting on a set , the action of on will be denote in the upper left corner, as in .
In the rest of the paper whenever we take a square root of an element we are implicitly chosing one of the roots and of . Whenever we do that, such choice is fixed for the rest of the paper.
The goal of this section is to prove Theorem 1.3. We will start with some auxiliary lemmas. Throughout the whole section, will be an odd prime and will be a fixed algebraic closure of .
Lemma 2.1 (ABC in positive characteristic).
Let be a finite, separable extension, and let be non-zero elements such that . Let be a finite set such that for all and let be the genus of . Suppose for some . Then:
Let , for some . Then . Now apply [19, Lemma 10] to the triple : this yields . It is immediate to conclude by noticing that . ∎
From now on, we let and . We assume throughout the section that and , so that is non-square and non-isotrivial. Let be the adjusted post-critical orbit of . Following , we let .
Lemma 2.2 ([10, Lemma 2]).
The following height bounds hold.
Suppose that . Then:
for all , and ;
for all .
Suppose that and let . Then:
for all ;
for all ;
for all .
Let be such that and let be such that . Then is not a square in .
Suppose by contradiction that is a square. Then for some we have
If we can show that we are done, as the leading coefficient of cannot be deleted by the leading coefficient of both with and (and also none of the two factors can obviously be zero, as ). Observe now that can be seen as the evaluation of the polynomial
at . So . But this is greater than by assumption, so we are done. ∎
In order to state and prove the key result, we need to introduce the following definition.
The dynamical inseparability degree of is the non-negative integer defined in the following way:
Notice that the dynamical inseparability degree is well-defined: if is not a square, then cannot be constant. If is a square and is constant, then cannot be constant because otherwise would be isotrivial. Notice also that the definition does not depend on the choice of a square root of .
Definition 2.4 and Lemmas 2.5 and 2.7 constitute the technical heart of the argument in this paper. In fact, they allow to transfer a technique to obtain height bounds for integral points on hyperelliptic curves in characteristic zero due to Baker  and Mason  to heights bounds for elements in the adjusted post-critical orbit of in positive characteristic. This will be described in Theorems 2.6 and 2.8.
Now we need to distinguish two cases, according to whether or not. Although the arguments in the two cases are essentially the same, certain crucial elements constructed starting from are different. Hence, in order not to confuse the reader, we will split the two cases in two different subsections.
2.1. The case .
Let be non-isotrivial, with and such that . Let us fix and define the following quantities, which we will use in the rest of the subsection:
Notice that whenever and that consequently for every . Notice also that by construction, . Finally, let and .
Let be such that . Let , let be the dynamical inseparability degree of and assume that for some . Then .
First of all, notice that:
This immediately shows that, since for some , then:
Next, we claim that:
We will prove (5) by examining separately the cases (i.e. ) and (i.e. ).
Case 1): . We prove first that . In fact, suppose by contradiction that , where . Then and for some , where is the generator of . It follows that , which implies in particular that . By Lemma 2.3, this cannot happen. We now have to distinguish two subcases.
Subcase 1a): . In this case, is a Galois extension with Galois group . It follows immediately that there exists in such that and . Therefore, and , proving (5).
Subcase 1b): . Suppose for some . Writing for some , one sees immediately, again by Lemma 2.3, that must hold, so that . It is then a well-known Galois theoretic fact that is a Galois extension with cyclic Galois group of order 4. Since generates over , then there exists a generator of such group with the property that while . It follows immediately that , and hence and , proving (5) again.
Case 2): . Notice that if for some , then , which is impossible by Lemma 2.3. Moreover, if then , which is impossible by Lemma 2.3 again. Hence, is Galois with Galois group . It is immediate to see that are all Galois conjugates up to sign, and (5) follows.
It only remains to prove the lemma in the case where and is a non-zero constant. This forces for some . Since, as we showed in Case 2), are all Galois conjugate up to sign, it is enough to assume . By hypothesis, for some . Write for some . Then:
Simple algebraic manipulations, together with the fact that is a -basis of , show that one must have , implying that . An easy induction shows that for , for some . Now observe that is a -th power if and only if is, and that is coprime with . This forces to lie in , and therefore . ∎
From now on, for any we write , where and is squarefree. Let us define the following elliptic curve over :
The point lies on .
Let be non-isotrivial, non-square and having dynamical inseparability degree . Suppose that and let be such that . Then we have:
The idea of the proof is to start by observing the following identity:
Now the fact that if then immediately shows that:
Since we have, by construction, that:
where is the genus of . Now the rest of the proof will consist of obtaining suitable bounds on the terms of inequality (6).
We start by finding an upper bound on the right hand side of (6). In order to do so, first let . We claim that:
In fact, since the ’s and the ’s lie in the integral closure of inside , then for every we have for all . Thus if is finite, we have or for some . Now observe that ; it follows that .
relation (7) immediately shows that:
Next, we are going to deal with the term of (6). Since is a Galois extension, Hurwitz formula shows that:
where is the genus of , is the ramification index at , and is the set of valuations of that ramify in . We now claim that . To show that, let be such that . We will show that . Notice that if then there cannot be with such that , as otherwise we would have , yielding a contradiction (notice that