aa r X i v : . [ m a t h . N T ] F e b A Dynamical Analogue of Sen’s Theorem
Mark A. Sing-SweeneyFebruary 22, 2021
Abstract
We study the higher ramification structure of dynamical branch extensions, andpropose a connection between the natural dynamical filtration and the filtration aris-ing from the higher ramification groups: each member of the former should, after alinear change of index, coincide with a member of the latter. This is an analogue ofSen’s theorem on ramification in p -adic Lie extensions. By explicitly calculating theHasse-Herbrand functions of such branch extensions, we are able to show that thisdescription is accurate for some families of polynomials, in particular post-criticallybounded polynomials of p -power degree. We apply our results to give a partial answerto a question of Berger [8] and a partial answer to a question about wild ramificationin arboreal extensions of number fields [1, 9]. Many guiding questions in arithmetic dynamics arise from or are inspired by analogies to well-studied objects in arithmetic geometry. Here, we formulate a tentative dynamical analogueof Sen’s theorem, and prove it in certain cases. Sen noticed that, for Galois extensions whoseGalois groups are p -adic Lie groups, there is a remarkable connection between the p -adic Liefiltration, which depends only on the Lie group, and the filtration by upper ramificationsubgroups: the two mutually refine each other in a precise way after a linear change ofindex [19]. In our dynamical setting, we replace p -adic Lie groups and the Lie filtrationwith “branch extensions” and their “branch filtration” (see Section 1.1 for definitions andnotation). For those familiar with arboreal representations, we are taking a single branch ofthe tree, filtered by height up the branch.Our dynamical version of Sen’s theorem says that, after possibly extending the groundfield and making a linear change in index, each member of the branch filtration coincidesexactly with a member of the upper ramification filtration. A more literal, and weaker,restatement in the dynamical setting would be that the two filtrations refine each other,again, after a linear change of index. However, for one of our applications, to a questionof Berger [8], we need the stronger formulation. We are able to give a general sufficientcriterion for our result to hold: it applies to extensions associated to so-called “tamelyramification-stable” branches. In our situation, “tamely” simply means that p does notdivide a certain quantity d , which is the limiting valuation of the members of the branch.Such branches are particularly striking from a dynamical perspective, exhibiting a kind1f stability in the structure of their higher ramification structure: neglecting scaling theintermediate Hasse-Herbrand functions associated to K n /K n − are identical up to accordingto small and well-controlled errors. For these branches, we obtain our main result: Theorem 3.11.
Suppose our branch, associated to the polynomial P ( x ) and base point α ,is tamely ramification-stable over K . Then K ∞ /K is arithmetically profinite, and there is aconstant V such that for all n , K n = K (( V − n +1) ∞ . We are able to give a general sufficient criterion for a branch to be tamely ramification-stable, Proposition 3.4. This criterion consists of two pieces: that p does not divide d ,and that an inequality depending only on the valuations of the coefficients of P ( x ) and thevaluation of α is satisfied. Some branches which are not tamely ramification-stable maybecome so after extending the ground field and re-indexing the branch; we call such branches potentially tamely ramification-stable.Using these criteria, we are able to that if P ( x ) is either post-critically bounded or primedegree, and we take a branch such that p does not divide the associated constant d , thenit is potentially tamely ramification-stable, and use this information to characterize higherramification in the associated extension: Corollary 3.12.
Let P ( x ) be a polynomial which either has degree p , or is post-criticallybounded and has degree p r . Take any nontrivial branch for P ( x ) , and suppose p does notdivide the constant d associated to the branch.Then the dynamical branch extension K ∞ /K is arithmetically profinite, and there areconstants N and V such that after replacing K by K N , K n = K (( V − n − N )+1) ∞ , for all n . For any particular branch, it is not difficult to apply our criteria to check whether or notit is (potentially) tamely ramification-stable, so long as one knows that p does not divide d . In fact, our criterion is almost entirely effective: only the stipulation that p does notdivide d is not known to be effective. Each branch determines certain “limiting ramificationdata” from which one can completely recover the Hasse-Herbrand function of the associatedbranch extension in the tamely ramification-stable case when d is known. The calculationof the limiting ramification data depends only on P ( x ) and some of the initial entries of thebranch (the number of entries needed is itself effective). While we lack a general algorithmto determine d , it can be calculated in many particular instances.We apply our results to provide a partial to answer two questions. One is raised byBerger [8], who asks: is it possible to show by elementary methods that if K ∞ /K is Galoisand the base point is a uniformizer then its Galois group is abelian? This is known to betrue by Berger [7] using quite sophisticated methods from p -adic Hodge theory. Our maintheorem involves more elementary tools, and proves allows us to re-prove this fact in somesituations: Theorem 4.1.1.
Assume p is odd. Suppose α is a uniformizer for K , P ′ (0) is nonzero,and we are given a branch associated to P ( x ) and α which is tamely ramification-stable.If K ∞ /K is Galois, it is also abelian. but not wildly ramified . It turns out that this is notpossible: Theorem 4.2.1.
Let F be a number field and p a prime of F lying over a rational prime p .Let P ( x ) ∈ O F [ x ] be a monic polynomial of degree p r such that P ( x ) ≡ x p r mod p , and let α ∈ F .Then the arboreal representation associated to P ( x ) and α is infinitely wildly ramified.If, further, P ( x ) has prime degree and v ( α ) = 0 , or is post-critically bounded withno restriction on v p ( α ) , and there is a branch over α whose associated constant d is notdivisible by p , then every higher ramification subgroup over p of the arboreal representationis nontrivial. In fact, while we have stated our main result as an analogy to Sen’s theorem, our initial mo-tivation comes from arithmetic dynamics and the structure of arboreal representations asso-ciated to post-critically finite maps. Arboreal representations, first introduced by Odoni [18],have been a subject of significant focus in arithmetic dynamics. This recently culminatedin the resolution of (one version of) Odoni’s conjecture over number fields, in prime degreeby Looper [15], in all even degrees and certain odd degrees by Benedetto and Juul [6], andfinally for all degrees by Specter [21]. The branch extensions we tackle are exactly that: theextensions associated to a single branch of the full preimage tree. The extensions we studyappear within the full arboreal representation and the ramification along such branches isquite important to the aforementioned results on Odoni’s conjecture. Additionally, Andrewsand Petsche [4] as well as Ferraguti and Pagano [13] have also used ramification informa-tion to prove interesting results about abelian arboreal representations over number fields.Our results are finer than necessary for any of the papers mentioned, but the importantrole ramification plays in those results suggests the potential value of the more detailed anddelicate ramification information that we obtain. Though arboreal extensions over globalfields are still quite mysterious, even less is known over local fields. Recently Anderson,Hamblen, Poonen, and Walton [3] studied full arboreal extensions in the local setting forpolynomials of the form x n + c . In fact, they even produce an example which shows that aliteral dynamical analogue of Sen’s theorem cannot hold in full generality, even in the caseof prime degree.The case of post-critically bounded polynomials is of particular dynamical interest be-cause it includes the post-critically finite polynomials of prime-power degree. Currently, thearboreal representations of post-critically finite polynomials are not well-understood, but itis known that they have arboreal representations which are ramified at only finitely manyprimes [1], so one would expect their arboreal representations to largely be controlled bytheir local behavior at those primes. Our result reveals initially unexpected structure totheir wild ramification at the prime in question.Some other work has been done with extensions of the kind we consider. Both Berger [8]3nd Cais and Davis [10] study them (under the name “ φ -iterate extensions”) with the ma-chinery of p -adic Hodge theory, and show that if these extensions are Galois they must beabelian. Cais, Davis, and Lubin [11] study the ramification in a somewhat more generalsetting, using similar methods to ours to give a characterization of arithmetically profiniteextensions – it is an important corollary of Sen’s theorem that p -adic Lie extensions arearithmetically profinite. The dynamical case of their result applies to a broader class ofpolynomials than ours, with the restriction that the base point is a uniformizer. For thepolynomials considered in this paper, we are able to relax this restriction on the base pointand obtain more precise information about the ramification of our extensions. The structure of our paper is as follows. Section 2 establishes some preliminary facts, includ-ing weaker descriptions of ramification in branch extensions. Section 3 uses the preliminariesof Section 2 to furnish more detailed ramification information, then introduces the notionof (potentially) tamely ramification-stable extensions then proves our main result. Section 4discusses the two aforementioned applications and the effectivity of our results, using thosealgorithms to give an example of a tamely ramification-stable branch which is not primedegree or post-critically bounded. – p is a prime,– K is a valued field of characteristic zero complete for a discrete valuation and withperfect residue field of characteristic p . For example, a finite extension of Q p or of d Q urp ,– ¯ K a fixed algebraic closure of K ,– O K is the ring of integers of K and π K a uniformizer of O K ,– P ( x ) ∈ O K [ x ] is a monic polynomial of degree q = p r such that P (0) = 0 and P ( x ) ≡ x q mod π K ,– α ∈ K , is called the base point, and we assume v ( α ) = 0,– v is a valuation for which ¯ K is complete, such that v ( p ) and the valuations of thecoefficients of P are integers, and there is a subfield E of K such that [ K : E ] is finiteand v ( E ) = Z ,– e K/E is the ramification index of the extension
K/E ,– ( α n ) n ∈ N is a sequence in ¯ K such that P ( α n ) = α n − and not all entries are zero (suchsequences may also be called branches , in keeping with the arboreal nomenclature),– K n = K ( α n ) and K ∞ = S n K n , 4 π n are uniformizers of O K n , along with units u n ∈ O K n , and integers d n such that α n = u n π d n n ,– d = lim n →∞ d n , if this limit exists,– N n is the Newton polygon of P ( x + α n ) − α n − , and co N n is the copolygon,– φ n is the Hasse-Herbrand function for K n /K n − ,– Φ n is the Hasse-Herbrand function for K n /K ,– Γ K is the absolute Galois group of ¯ K over K ,– Γ νK the subgroup associated to a nonnegative real number ν by the upper-numberingof the higher ramification subgroups.One may take E = K with v an appropriately normalized valuation, but we separate E from K here in anticipation of changing the base field K .At times we treat the cases v ( α ) > v ( α ) < integral and non-integral cases, respectively. It is easy to see from the Newton polygon that v ( α n ) has the same sign as v ( α ) for all n .By conjugation we mean conjugation by a nonconstant linear map, rather than the moretypical (for dynamics) conjugation by a linear fractional transformation, because we workwith polynomials.For a polynomial f ( x ), we denote by f i the coefficient of x i .We assume that the reader is familiar with local fields and higher ramification. Serre [20]covers much of this (Chapter IV) but we do not assume that our extensions are Galois,which at times requires slightly different tools; fortunately, Lubin has collected these in anexcellent expository article [17]. One should take caution in passing between these sources:their ramification numberings differ, and in the present paper we adopt Lubin’s numbering.The reader should have a least a passing familiarity with the notion of arithmetically profiniteextensions, such as in Wintenberger [23].When L/K is arithmetically profinite, we denote its m th ramification break by b m andthe m th elementary subfield of L by L ( m ) . This is the subfield of L which is fixed by Γ b m K .By convention, we set K ( m ) ∞ = K when m is nonpositive. The results here are used primarily as tools for our main theorems. However, some may beof independent interest, which we have tried to mark as propositions.Any post-critically bounded polynomial of p -power degree can, after possibly extendingthe ground field, be put in the same form as the polynomials we consider: monic, withintegral coefficients, and fixing zero. In fact, after conjugation, it satisfies even strongerconstraints on its coefficients. A few other versions and proofs of this proposition appear inthe literature [2, 5, 12]. 5 roposition 2.1. If a polynomial has degree q = p r and is post-critically bounded, then ithas a conjugate P ( x ) which is monic, integral, fixes and further satisfies v ( P i ) + v ( i ) ≥ v ( q ) = rv ( p ) for all ≤ i ≤ q. Proof.
Let R ( x ) be the given polynomial.After conjugating, we may assume that R ( x ) is monic and fixes zero; conjugates alsoremain PCB. This conjugation may require taking a ( p − R ( x ) and adjoining a fixed point of R to the ground field. Call this conjugate P ( x ). Itdoes not necessarily have integral coefficients at this point, but we will show that P ′ ( x ) q is in O K [ x ], from which the last claim follows, and hence integrality as well.Suppose otherwise: that guarantees a positive slope in the Newton polygon for P ′ ( x ) q , thesteepest slope of which ends at the vertex associated to the leading term. This slope mustbe strictly steeper than the steepest slope of the Newton polygon of P ( x ) because everynon-leading vertex moves down in passing from P ( x ) to P ( x ) q . However, this means if we takea critical point associated to this steepest slope, v ( P ( c )) = qv ( c ) < v ( c ), hence v ( P ( c )) Fix a positive integers i, j, k with j ≥ i and j ≥ p k .(i) If p k ≤ i < p k +1 , then v (cid:18) ji (cid:19) ≥ v (cid:18) jp k (cid:19) . (ii) Additionally, v (cid:18) jp k +1 (cid:19) ≥ v (cid:18) jp k (cid:19) − v ( p ) , with equality if and only if v (cid:0) jp k (cid:1) = 0 .Proof. Both claims follow from Kummer’s theorem [14], which states that the p -adic valu-ation of a binomial coefficient (cid:0) ji (cid:1) is cv ( p ), where c is the number of carries when adding i and j − i in base p .Applying that theorem, we see that a lower bound for the valuation of (cid:0) ji (cid:1) when theleading base p digit of i is in the ℓ th place is the number of consecutive zeros in the base- p expansion of j starting at the ℓ th digit. Notice that if i = p ℓ then this is exact, but it canbe larger in general, from carries that occur before the ℓ th digit.The condition p k ≤ i < p k +1 says exactly that i ’s leading base p coefficient is in the k thplace. 6rom these observations, ( i ) and the inequality of ( ii ) are immediate by taking ℓ = k and ℓ = k + 1. As to the last claim: the quantities in question are integers, so equality isimpossible if v (cid:0) jp k (cid:1) is zero, and conversely if v (cid:0) jp k (cid:1) is nonzero then the change from p k to p k +1 loses exactly one of the aforementioned zeros. (cid:3) The next proposition is our first dynamical result, a kind of ramification semi-stability,although much less refined than our main result. Proposition 2.3. For all sufficiently large n :(a) v ( α n + k ) = v ( α n ) q k ,(b) the sequence ( d n + k ) k ∈ N is constant,(c) K n /K n − is totally ramified of degree q .Proof. Consider the polynomial P ( x ) − α n − , of which α n is a root. We may, by taking n large enough, assume α n − is nonzero. Inspecting the Newton polygon of P ( x ) − α n − , wesee that, in the integral case where v ( α n ) > v ( α n ) ≤ max { v ( α n − ) − , v ( α n − ) / } . Thus, for n large enough, we obtain v ( α n ) < 1, in which case the Newton polygon of P ( x ) − α n has just one segment, whose slope is v ( α n ) q , which is also less than 1. Hence v ( α n +1 ) = v ( α n ) /q , and inductively this yields v ( α n + k ) = v ( α n ) /q k .In the non-integral case, the Newton polygon immediately has a single slope, which givesus v ( α n +1 ) = v ( α n ) q < , and again the claim follows inductively. Together, these two cases verify ( a ).Next, let e n be the ramification index of K n /K n − . Take n − a )holds, and so we have both v ( α n ) = v ( u n π d n n ) = d n v ( π n ) = d n v ( π n − ) e n , and v ( α n ) = v ( α n − ) q = v ( u n − π d n − n − ) q = d n − v ( π n − ) q . Comparing the two yields the following relation: d n = e n q d n − . (1)From (1), we see that if e n = q , then d n = d n − , and so to prove both ( b ) and ( c ) itsuffices to verify e n = q for n large enough. Evidently e n ≤ q , so we wish to show that thisinequality is strict at most finitely often. Indeed, each time the inequality is strict, the p -adicvaluation of d n is strictly smaller than the p -adic valuation of d n − . Since the sequence ofvaluations ( v p ( d n )) is a sequence of nonnegative integers, these strict drops can happen onlyfinitely many times. (cid:3) N n . Lemma 2.4. For n sufficiently large, the Newton polygon N n of P ( x + α n ) − α n − has atmost r + 1 vertices, whose x -coordinates can only be powers of p .Thus N n is the lower convex hull of the points ( p k , y p k ) , where the height y p k is given by y p k = min p k ≤ j ≤ q (cid:26) v (cid:18) jp k (cid:19) + v ( P j ) + ( j − p k ) v ( α n ) (cid:27) . Proof. Let Q ( x ) = P ( x + α n ) − α n − . Expanding and collecting terms, we see that Q i = q X j = i (cid:18) ji (cid:19) P j α j − in . Hence v ( Q i ) ≥ min i ≤ j ≤ q (cid:26) v (cid:18) ji (cid:19) + v ( P j ) + ( j − i ) v ( α n ) (cid:27) (2)The fractional parts of the terms in the minimum, which come from ( j − p k ) v ( α n ), areall distinct so long as 0 < | v ( α n ) | ≤ q , and from Proposition 2.3 we know this is the casefor all sufficiently large n . As such, the terms themselves are distinct and so the inequality(2) is actually an equality. Additionally, v ( Q ) = ∞ since the minimum above is evidentlyfinite.Since Q = 0, but g = 0, the Newton polygon has a vertical line through (1 , v ( Q )).The leading coefficient is 1, so there is another vertex at ( q, N n only has vertices at prime powers, we will prove something slightlystronger: that v ( Q i ) for i between p k and p k +1 has valuation at least v ( Q p k ) + ( p k − i ) v ( α n ),or, in other words, such points ( i, Q i ) are above the line through ( p k , v ( Q p k )) with slope − v ( α n ). Because | v ( α n ) | ≤ q , the slope of that line through ( p k , v ( Q p k )) is so shallow, thatthis line always passes above ( q, 0) and so no point above this line can be a vertex.Thus vertices can only occur at prime powers, where v ( Q p k +1 ) might jump below thatline.And so we compute: v ( Q i ) = min i ≤ j ≤ q (cid:26) v (cid:18) ji (cid:19) + v ( P j ) + ( j − i ) v ( α n ) (cid:27) = min i ≤ j ≤ q (cid:26) v (cid:18) ji (cid:19) + v ( P j ) + ( j − p k ) v ( α n ) (cid:27) + ( p k − i ) v ( α n ) ≥ min p k ≤ j ≤ q (cid:26) v (cid:18) ji (cid:19) + v ( P j ) + ( j − p k ) v ( α n ) (cid:27) + ( p k − i ) v ( α n ) (3)8his is nearly the desired inequality, but with v (cid:0) ji (cid:1) rather than v (cid:0) jp k (cid:1) . To resolve this issue,we apply Lemma 2.2, which tells us that if p k ≤ i < p k +1 , then v (cid:18) ji (cid:19) ≥ v (cid:18) jp k (cid:19) . Continuing where we left off at (3): v ( Q i ) ≥ min p k ≤ j ≤ q (cid:26) v (cid:18) jp k (cid:19) + v ( P j ) + ( j − p k ) v ( α n ) (cid:27) + ( p k − i ) v ( α n )= v ( Q p k ) + ( p k − i ) v ( α n )as was to be shown.Lastly, y p k is simply v ( Q p k ), which is given by (2). (cid:3) In the preceding description of the heights of the points defining N n , one might noticethat for sufficiently large n , the “error terms” ( j − p k ) v ( α n ) appearing in the minimum arevery small. So we should expect the polygons N n to be quite similar when n is large. Thisis the case, as we will prove shortly, although tracking these error terms make the proof lessclear than we might like.The main idea is that the height of each point defining N n has a main term and an errorterm. Sometimes, one can identify a vertex or non-vertex simply by the position of its mainterm relative to the other main terms, because the error is small. When vertices are notdistinguished by the main term, it must be the error term distinguishing the vertex, andthere is sufficient regularity in these error terms that when that happens for N n , it continuesto do so for N n +1 and so on.This important, but technical, geometric fact is made precise by the following lemma. Lemma 2.5. Let m, m ′ , m ′′ and ≤ e, e ′ , e ′′ ≤ q − be nonnegative integers, ≤ s < t
1, so if such a linedoesn’t pass through some lattice point, the closest it can approach that lattice point is ata vertical distance of q − .With that in mind, P ′ n lies below the line connecting P n and P ′′ n if and only if m ′ + e ′ Cq n < p t − p s p u − p s (cid:18) m + e Cq n (cid:19) + p u − p t p u − p s (cid:18) m ′′ + e ′′ Cq n (cid:19) . (4)Our goal is to show that (4) holds with n + 1 in place of n : m ′ + e ′ Cq n +1 < p t − p s p u − p s (cid:18) m + e Cq n +1 (cid:19) + p u − p t p u − p s (cid:18) m ′′ + e ′′ Cq n +1 (cid:19) . (5)We can see that inequality (4) roughly decomposes into two pieces: one involving onlythe main terms m, m ′ , m ′′ , and one involving just the error terms e, e ′ , e ′′ . This leads us toconsider two cases: m ′ ≤ p t − p s p u − p s m + p u − p t p u − p s m ′′ (6)and m ′ > p t − p s p u − p s m + p u − p t p u − p s m ′′ . (7) Case 1. If (6) holds, then subtracting it from (4) and dividing by q yields e ′ Cq n +1 < p t − p s p u − p s e Cq n +1 + p u − p t p u − p s e ′′ Cq n +1 . (8)Adding(8) back to our assumption (6) yields the desired inequality (5). These manipulationscan be reversed, so (6) is equivalent to (5) in this case. Case 2. If (7) holds instead, we will have a contradiction. By our key observation, thefact that (7) is a strict inequality means that m ′ − p t − p s p u − p s m − p u − p t p u − p s m ′′ ≥ q − m ′ − p t − p s p u − p s m − p u − p t p u − p s m ′′ < − e ′ Cq n + p t − p s p u − p s e Cq n + p u − p t p u − p s e ′′ Cq n . (10)The left hand side is at least q − by (9), but the right hand side is too small to allow10his: (cid:12)(cid:12)(cid:12)(cid:12) − e ′ Cq n + p t − p s p u − p s e Cq n + p u − p t p u − p s e ′′ Cq n (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) − e ′ + p t − p s p u − p s e + p u − p t p u − p s e ′′ (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) Cq n (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) p t − p s p u − p s ( q − 1) + p u − p t p u − p s ( q − (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) Cq n (cid:12)(cid:12)(cid:12)(cid:12) = | q − | (cid:12)(cid:12)(cid:12)(cid:12) Cq n (cid:12)(cid:12)(cid:12)(cid:12) ≤ ( q − 1) 1 q < q . (11)Together, (9), (10), and (11) give q − < q , clearly a contradiction. (cid:3) With Lemma 2.5 in hand, we are ready to prove the final result of this section, a crucialinput to our main results. Proposition 2.6. There is a positive integer V depending only on the polynomial P ( x ) andthe sign of v ( α ) such that for all n sufficiently large the Newton polygon N n of P ( x + α n ) − α n − has exactly V vertices.In fact, there are nonnegative integers r i , a p ri , e p ri , for ≤ i ≤ V , depending only on P and v ( α ) , and a constant C which depends only on the degree q and sequence of valuations ( v ( α n )) n ∈ N , such that, for all sufficiently large n , the vertices of N n are all of the form (cid:18) p r i , m i + e i q n C (cid:19) . Proof. Apply Proposition 2.3 and Lemma 2.4, which together tell us that there is some N such that | v ( α N ) | ≤ q and all the conclusions of both Proposition 2.3 and Lemma 2.4 holdfor n ≥ N . For the remainder of the proof, we only discuss n ≥ N . Set C = q N v ( α N ); thisis independent of our choice of N , which we can see by again applying Proposition 2.3: q n v ( α n ) = q n v ( α N ) q n − N = q N v ( α N ) = C, (12)from which it also follows that, for all n ≥ vN , v ( α n ) = Cq n . Now, recall the description of N n given by Lemma 2.4: it is the lower convex hull of thepoints ( p k , y p k ), where y p k = min p k ≤ j ≤ q (cid:26) v (cid:18) jp k (cid:19) + v ( P j ) + ( j − p k ) v ( α n ) (cid:27) . | v ( α n ) | ≤ q and | j − p k | ≤ q − | ( j − p k ) v ( α n ) | < , while v (cid:0) ji (cid:1) + v ( P j ) is an integer. Moreover, all the terms ( j − p k ) v ( α n ) have the same sign,and so the index j which achieves the minimum is determined entirely by the “main term” v (cid:0) ji (cid:1) + v ( P j ) except when ties must be broken. The ties always break the same way, anddepend only on the sign: in the integral case, one takes the smallest index j achieving thetie value, while in the non-integral case one takes the largest such index.So for each k , let m p k be the value of v (cid:0) jp k (cid:1) + v ( P j ) and e p k the value of j − p k associatedto the point over p k ; what was just argued is that these quantities are independent of n .Clearly all are positive. Moreover, because v ( α n ) = Cq n , we see that y p k = m p k + e p k q n C. (13)From (13), what remains to be shown is that the number of vertices and the x -coordinatesof the vertices do not depend on n . This will follow by an induction, which has two parts:showing that if the Newton polygon N n has a vertex over p t then the Newton polygon N n +1 does too, and conversely that if N n has no vertex over p t , then neither does N n +1 .We know that N n has a vertex over p t if and only if for all s and u such that s < t < u the point over p t lies below the line segment connecting the vertices over p s and p u . If we let m = m p s , m ′ = m p t , m ′′ = m p t , e = e p s , e ′ = e p t , e ′′ = e p u , then we are exactly in the situation to which Lemma 2.5 applies: by (13) the points P n , P ′ n , P ′′ n are the points over p s , p t , and p u defining N n , while P n +1 , P ′ n +1 , P ′′ n +1 are the points over p s , p t , and p u that are used to define N n +1 . So the lemma tells us that N n has a vertex over p t ifand only if N n +1 also has a vertex over p t .Thus, by induction, all of the vertices lie over the same x -coordinates for all n ≥ N ,and hence their number, which we call V , is constant. We let r i be the exponents of theprime powers which appear as x -coordinates; m i be the associated main term m p vi ; e i theassociated error coefficient e p vi . The arguments above show that these do not depend on thechoice of branch, only the valuations of the coefficients of P ( x ) and the sign of v ( α ). Wenote that the subscripts indexing m i and e i are incompatible with the subscripts indexing m p k and e p k ; this should not cause confusion.We also let C be the constant q N v ( α N ). As was shown in (12), if we made a differentchoice of N , the constant C would be unchanged. Thus we can define C = lim n →∞ q n v ( α n ) , which makes it clear that C depends only on q and the sequence of valuations ( v ( α n )) n ∈ N . (cid:3) efinition 2.7. In the notation of the preceding proposition, we define the limiting ram-ification data associated to P and the branch: V ( P, ( α n ) n ∈ N ) = the number of vertices V,R ( P, ( α n ) n ∈ N ) = ( r , ..., r V ) ,M ( P, ( α n ) n ∈ N ) = ( m , ..., m r B ) ,E ( P, ( α n ) n ∈ N ) = ( e , ..., e B ) ,C ( P, ( α n ) n ∈ N ) = the constant C. (“number of vertices”, “vertex exponents”, “main terms”, “error factors”, “error coefficient”)When n is large, N n is the polygon with vertices (cid:18) p r i , m i + e i q n C (cid:19) . Remark. Since the first vertex is over 1 and the last vertex is ( q, r = 0 and r V = r and m V = e V = 0. Observation (effectivity of our constants) . As was pointed out in Proposition 2.6, V , R , M , and E , only depend on the (ordered) valuations of the coefficients of P and the sign of v ( α ), while C depends only on the degree q of P ( x ) and the sequence ( v ( α n )) n ∈ N ) of themembers of the branch. The calculation of these parameters is effective.Since the quantity Cq n tends to 0, we see that the shape of the Newton polygon N n isnearly independent of the branch. In fact, one can even bound C in terms of v ( α ) and thevaluations of the coefficients of P ( x ), so the discrepancy between branches is even “uniformlybounded” across branches. This also makes the computation of C relatively straightforward:there is an effective constant N depending only on P and v ( α ) such that for any branchassociated to P and α , the value of C is q N v ( α N ).In fact, the only ineffective step in our results occurs in Proposition 2.3 – the proof of(b) and (c) does not give an effective determination of “sufficiently large”. There are somecases where this can be circumvented; for instance, if v ( α ) = 1 then it is straightforward tosee that, for all n , P ( x ) − α n is Eisenstein, which implies (b) and (c) hold for all n . Moregenerally, it follows from our proof of Proposition 2.3 that if there is some N such that v ( α N )is not divisible by p and has smaller valuation than any coefficient of P ( x ), then (b) and (c)hold for all n ≥ N .One can see quite readily from Proposition 2.6 that the polygons N n have a pointwiselimit (viewing them as functions on R ≥ N n (this occurs when main termsof vertices, ( p r i , m i ), are collinear). The number of vertices V is extremely important for ourmain result and applications, because V − N n .13 Main Results As mentioned in the introduction, our main results may not be true without first extendingthe ground field and adjusting the index. For present technical simplicity, we prefer to avoidthis trouble, and will instead assume that the conclusions of all facts proven above to holdimmediately for P and α . At the end we will explicitly work out these adjustments. Thisleads us to introduce the following property: Definition 3.1. A pair ( P, α ) satisfies (H) if they satisfy the conclusions of Propositions 2.3and 2.6, or, in other words, that those propositions are true for P and α without thequalification “for sufficiently large n ”.And so Propositions 2.3 and 2.6 tell us that even if P and α do not satisfy (H), thereis some N such that P and α N do. In the Galois case, this is equivalent to replacing an(infinite) group with a finite-index subgroup which, hopefully, retains a lot of informationabout the original group.Besides this, it is also necessary to introduce a “tameness” assumption, that d (whichwe know exists from Proposition 2.3) is not divisible by p . In what follows, we will want totake a d th root of the unit u n inside K n . Recall that the unit u n was defined by α n = u n π d n n ,and so the presence of this d th root allows us to take a different choice of uniformizer π n ,such that α n = π dn . This d th root is not necessarily in K n , but if p does not divide d , thenwe can obtain a d th root of u n after an unramified extension of K n , which does not changethe ramification along the branch. However, if p divides d then the d th root of u n may onlyappear in a ramified extension of K n , and this extra ramification interferes with our abilityto extract information about ramification prior to including the d th root. We hope that thisrestriction can be relaxed in some or all cases – the study of some special cases suggests thatif d = d p m where p ∤ d , then our results still hold with d in place of d . An unfortunatedownside of this restriction is that it means our results are not base-change invariant – if wereplace K by an extension with ramification index divisible by p and linearly disjoint from K ∞ , then p is guaranteed divide d . Luckily, we at least have invariance under tame basechange.Given these assumptions, our next goal is verify that the extension K ∞ /K is arith-metically profinite and compute its Hasse-Herbrand function, under (H) and the tamenessassumption. We will break up the computation of the Hasse-Herbrand function of K ∞ /K into calculating the Hasse-Herbrand functions for the intermediate extensions K n /K n − ,composing those functions to obtain the Hasse-Herbrand function of K n /K , and then passto the limit. As mentioned in the introduction, we avoid assuming any of our extensions areGalois (indeed, one would expect this to be rare in general) so to study higher ramification,we employ the techniques explained by Lubin [17]. The reader is advised to take some carein passing between this and other sources (such as Serre [20]) since the ramification groupsmay be numbered differently; we adopt Lubin’s convention.For convenience, we remind the reader of two important polygons: Definition 3.2. The level n Newton polygon of pair ( P, α ) which satisfies (H), denoted N n , is the Newton polygon of P ( x + α n ) − α n − . Explicitly, it is the lower convex hull of the14ollowing points determined by the limiting ramification data: (cid:18) p r i , m i + e i Cq n (cid:19) ≤ i ≤ V. The level n Newton copolygon , denoted co N n , is the dual of the level n Newtonpolygon. Explicitly, for each slope of N n , it has a vertex whose x -coordinate is the negativeof that slope, and its slopes are the values p r i (in order of decreasing size).Note that the copolygon co N n has one fewer vertex than the polygon N n . Proposition 3.3. Suppose the pair ( P, α ) satisfies (H) and that p does not divide d . Thenthe Hasse-Herbrand transition function φ n for K n /K n − relative to K can be obtained byapplying the following three transformations to the copolygon co N n :(1) Increase the x -coordinates of each vertex by sgn( v ( α ))( d − v ( π n ) , keeping the sameslopes.(2) Stretch horizontally by a factor of e K/E q n .(3) Stretch vertically by a factor of e K/E q n − .The first slope of φ n is and the last slope of φ n is /q . The x -coordinates of the firstand last vertices of φ n , are respectively, − e K/E q n (shallowest slope of N n ) + sgn( v ( α ))( d − v ( α ) and − e K/E q n (steepest slope of N n ) + sgn( v ( α ))( d − v ( α ) . Proof. We will prove this in full for the integral case, where v ( α ) > d ≥ 1, andat the end indicate the minor modifications necessary for the non-integral case.Let f ( x ) be the minimal polynomial for π n over K n − . The Hasse-Herbrand functionfor K n /K n − can be obtained by applying stretches (2) and (3) to the Newton copolygonof f ( x + π n ) [17]. So we only need to show that the copolygon of f ( x + π n ) can itself beobtained by applying (1) to co N n .In terms of Newton polygons, (1) is equivalent to decreasing all of the slopes of N n by( d − v ( π n ) (there is a sign change in the duality between polygon and copolygon!). TheNewton polygons of f ( x + π n ) and P ( x + α n ) − α n − encode the valuations of the roots of thecorresponding polynomials. These roots are of the form π σn − π n and α σn − α n , respectively,for σ ∈ Γ K , and our task is to relate their valuations.In the integral case, we want to show that, for all σ ∈ Γ K , v ( π σn − π n ) = v ( α σn − α n ) − ( d − v ( π n ) . Recall that we selected uniformizers π n and units u n such that α n = u n π d n n . By (H), d n = d does not vary with n , and we also assumed it is not divisible by p . As such, u n admitsa d th root after at most an unramified extension; the transition function is insensitive tounramified extensions, so we may assume that u n has a d th root in the ground field. So after15ossibly altering our choice of π n , we may write α n = π dn . This allows us to compare thevaluations more directly: α σn − α n = ( π σn ) d − π dn = Y ζ d =1 ( π σn − ζ π n ) (14)Of the terms in the product (14), we are only interested in v ( π σn − π n ). To manage theothers, notice that v ( π σn − ζ π n ) = v ( π n ) + v (cid:18) π σn π n − ζ (cid:19) . (15)If v (cid:16) π σn π n − ζ (cid:17) is positive, then π σn π n is necessarily a d th root of unity modulo π n . On theother hand, the norm from K n to K of π σn π n is just 1; but viewed in the residue field, the normis just the q th power. Therefore, in the residue field, π σn π n is both a d th root of unity and a q throot of unity. Because p ∤ d , this is only possible if ζ = 1. In all other cases, v (cid:16) π σn π n − ζ (cid:17) = 0.Thus, (15) simplifies to just v ( π n ) whenever ζ = 1, and so the valuation of (14) becomes v ( α σn − α n ) = v ( π σn − π n ) + ( d − v ( π n )or equivalently v ( π σn − π n ) = v ( α σn − α n ) − ( d − v ( π n ) , which is exactly the statement to which we reduced the main part of this proposition for theintegral case.For the non-integral case, when d is negative, we must instead work with1 α σn − α n = ( π σn ) | d | − π | d | n . The left hand side can be written as α n − α σn α n α σn which has valuation v ( α n − α σn ) − v ( α n ) . Recall too that v ( α n ) = dv ( π n ). Inserting these into the arguments of the integral caseleads to the claimed conclusion.Finally, by inspecting the transformation of co N n into φ n , one can see that the first andlast slopes of φ n are e K/E q n − e K/E q n = q multiplied by the first and last slopes of co N n . The firstand last slopes of co N n are the first and last x -coordinates of vertices of N n , which are 1 and q , so together we see that the first and last slopes of φ n are 1 and q , as claimed. Likewise, the x -coordinates can be obtained from the duality of co N n , which turns negative slopes of N n into x -coordinates of vertices, then modified according to the first two transformations. (cid:3) Remark. We use the assumption p ∤ d in two places: to take a d th root of u n , and thatthe d th roots of unity are distinct modulo p to control v ( π n − ζ π n ). The former seems to bemore difficult to handle than the latter. 16he essence of the preceding proposition is that the ramification-theoretic properties ofthese extensions are somewhat stable. Neglecting scaling, all the Hasse-Herbrand functionslook like a small shift of co N n , and the copolygon itself changes little as a function of n , ina way which is described very precisely by Proposition 2.6.The most difficult step would appear to be composing the intermediate Hasse-Herbrandfunctions φ , φ , ..., φ n to obtain the Hasse-Herbrand function Φ n for K n /K . However, thisis straightforward if we can ensure that the φ n ’s behave sufficiently well. Since φ n is theidentity along its first segment, one might hope that the domain on which it is the identityincludes all of the vertices of Φ n − .Unfortunately this is too much to expect in general, but we can give a characteriza-tion in terms of the limiting copolygon and show it is satisfied for post-critically boundedpolynomials (of the appropriate form) and polynomials of prime degree. Proposition 3.4. Suppose ( P, α ) satisfy (H) and that p does not divide d .For n ≥ , the leftmost vertex of φ n has strictly larger x -coordinate than that of therightmost vertex of φ n − if the limiting Newton polygon has just one slope, or if q m V − m V − p r V − p r V − > m − m p r − p r + 2 p − | v ( α ) | . Proof. By the final statement of Proposition 3.3, we can rewrite the claim about the x -coordinates of those vertices in terms of the slopes of N n and N n − . We want − e K/E q n (shallowest slope of N n ) + sgn( v ( α ))( d − v ( α )to be strictly larger than − e K/E q n − (steepest slope of N n − ) + sgn( v ( α ))( d − v ( α ) . For convenience, let’s name the negatives of these slopes s and s ′ , respectively, in whichcase we can simplify and rewrite the preceeding as qs > s ′ . When there is just one slope, s = s ′ and the inequality obviously holds. Otherwise, thereare two slopes.Now, observe that the quantities m V − m V − p r V − p r V − and m − m p r − p r in the statement of the proposition are the (negative) contributions of the “main terms” ofthe vertices on the segments corresponding to the slopes s and s ′ . For simplicity, let thembe t and t ′ , respectively. In light of this interpretation, we can write s − t = e V − − e V p v B − p v V − v ( α ) q n ,s ′ − t ′ = e − e p v − p v v ( α ) q n − . 17s was remarked previously, r = 0, r V = r and e V = 0, because the first vertex lies over 1,while the last vertex is ( q, qt > t ′ + 2 p − v ( α ) , and we have some s, s ′ such that s − t = e V − q − p v V − v ( α ) q n ,s ′ − t ′ = e − e p v − v ( α ) q n − , and our goal is qs > s ′ . But then it is enough for our two errors q ( s − t ) and s ′ − t ′ to be small enough thattheir sum is less than p − | v ( α ) | in absolute value, as then adding these error terms to theinequality we initially assumed will preserve the inequality up to the loss of margin of error, p − | v ( α ) | , that we allowed ourselves. To prove that the sum of q ( s − t ) and s ′ − t ′ is smallenough, it suffices to show that each is at most | v ( α ) | p − . And indeed: | q ( s − t ) | = q e V − q − p r V − | v ( α ) | q n ≤ q q − q − p r − | v ( α ) | q n < q − p r − | v ( α ) | q n − ≤ | v ( α ) | p − , and | s ′ − t ′ | = | e − e | p v − | v ( α ) | q n − ≤ q − p v − | v ( α ) | q n − ≤ q − p − | v ( α ) | q n − < p − | v ( α ) | q n − < | v ( α ) | p − , where, on the second line, we use | e − e | ≤ q − ≤ q − 1) because we knowthat e and e are both nonnegative. (cid:3) orollary 3.5. Assume (H) and that p ∤ d . If P has degree q = p , then it satisfies Proposi-tion 3.4.Proof. Immediate, as in this case the limiting Newton polygon can only have vertices over 1and p , hence it has just a single slope. (cid:3) Corollary 3.6. Assume (H) and that p ∤ d . If P is post-critically bounded, and | v ( α ) | < p − v ( p ) , then the pair satisfies Proposition 3.4.Proof. Adopt the notation of Proposition 3.4. Recall Proposition 2.1, which says that P ′ ( x ) q has integral coefficients. The first vertex of N n is (1 , v ( P ′ ( α n )), and so Proposition 2.1 tellsus that its height is at least v ( q ) = rv ( p ).From Lemma 2.2, we know that the height drop between vertices over p s and p u is atmost ( u − s ) v ( p ). The steepest slope of N n is the first one, so s = 0, and so the bound onthe height drop also bounds that slope: t ′ ≤ uv ( p ) p u − ≤ p − v ( p ) . As for the shallowest slope, it is the last slope of N n and hence it ends at the vertex( q, 0) = ( p r , t ≥ rv ( p ) − sv ( p ) p r − p s ≥ v ( p ) p r − p r − = pq ( p − v ( p ) . (16)To apply Proposition 3.4, we would like qt > t ′ + 2 p − v ( α ) . And indeed, by (16): qt ≥ pp − v ( p )= 1 p − v ( p ) + v ( p ) ≥ t ′ + v ( p ) . Certainly v ( p ) > p − | v ( α ) | , which is simply a rearrangement of our assumption about | v ( α ) | . (cid:3) Remark. Notably, when p is at least 5, the inequality in the proposition is always satisfiedfor v ( α ) = 1. 19t still remains to compose our Hasse-Herbrand functions. The conclusion of Proposi-tion 3.4 describes the “good behavior” that we want in order for the Hasse-Herbrand func-tions to compose well: the first vertex of φ n should have larger x -coordinate than the lastvertex of φ n − . When this happens, the higher ramification behavior of the branch is quitewell-controlled, and highly regular. From working with explicit examples, it is clear thatthis happens in many situations besides those described by Proposition 3.4 or Corollaries 3.5and 3.6. This leads us to introduce the following definition: Definition 3.7. A branch associated to P and α over K is said to be tamely ramification-stable if p ∤ d , and the pair satisfies (H) and the conclusions of Propositions 3.3 and 3.4.A branch is said to be potentially tamely ramification-stable if there is some N suchthat upon replacing K by K N and re-indexing the branch to be based at α N it is tamelyramification-stable. Remark. In our definition, “tamely” refers to the restriction that p ∤ d . We expect that evenif p | d , such branch extensions would still exhibit this kind of ramification stability. Howeverprecise expressions given in Proposition 3.3, particularly the ( d − v ( π n ) term, may notcorrectly describe these cases. Proposition 3.8. Suppose that p ∤ d . If P ( x ) has prime degree or is post-critically bounded,then any branch associated to P ( x ) is potentially tamely ramification-stable.Proof. Propositions 2.3 and 2.6 ensure that for all sufficiently large N , (H) is satisfied when-ever K is replaced by K N and the branch is modified to start at α N .For polynomials of prime degree and post-critically bounded polynomials, Corollaries 3.5and 3.6, respectively, prove that this branch also satisfies the conclusion of Proposition 3.4after possibly increasing N . (cid:3) From the proof of Proposition 3.4, we know that if p ∤ d , a branch is potentially tamelyramification stable when, roughly, the first and last slopes of N n do not differ by more thana factor of q . This property depends only on P ( x ), not the branch. For this to fail, the firstvertex of N n must be relatively high compared to the others, which seems unlikely based onthe structure of the minima that describe the heights of these vertices.Before proceeding, recall the following definition: Definition 3.9 ( [17]) . The altitude of an extension E/K with transition function Ψ( x ) isthe height of the rightmost vertex of Ψ( x ); at times we may abbreviate this as the altitudeof Ψ( x ). Proposition 3.10. Suppose our branch, associated to ( P, α ) , is tamely ramification-stableover K . Let V be the number of vertices from the limiting ramification data.Then the Hasse-Herbrand transition function Φ n ( x ) for K n /K is a piecewise linear func-tion which satisfies the following properties:1. it has ( V − n vertices,2. its last vertex has the same x -coordinate of the last vertex of φ n ,3. its final slope is /q n , . Φ n ( x ) coincides with Φ n − ( x ) for x smaller than the last coordinate of Φ n − ,5. the altitude of Φ n ( x ) is strictly greater than the altitude of Φ n − and is unbounded asa function of n .Proof. By transitivity, Φ n ( x ) = Φ n − ◦ φ n ( x ), so it is natural to proceed by induction. Thebase case is Φ = φ , where there is nothing to prove.The first vertex of φ n ( x ) is after the last vertex of Φ n − ( x ), and φ n ( x ) is the identityup to its first vertex, so properties (2) and (4) follow. Moreover, after that point, we add V − φ n , yielding (1). By inspection, the final slope of Φ n isthe product of the final slope of Φ n − , which is q n − and the final slope of φ n , which is q , sotogether the final slope is q n , which is (3).Finally, the altitude is the height of the last vertex of Φ n ( x ), which lies over the last vertexof φ n . By Proposition 3.3 combined with the limiting ramification data, we can express the x -coordinates of the last vertices of Φ n and Φ n − as Aq n + B and Aq n − + B, respectively, where A and B are positive constants which do not depend on n . The constant A comes from the part of the slope associated to the main terms, while B comes from theerror terms plus the shift by sgn( v ( α ))( d − v ( π n ), and both incorporate the scaling by e K/E .Between these two vertices, the slopes of Φ n ( x ) must be at least pq n because the outgoingslope is q n and the slopes are all powers of p . Then we can estimate the difference in altitudesas follows altitude(Φ n ) − altitude(Φ n − ) ≥ pq n ( Aq n + B − ( Aq n − + B )) ≥ A (cid:18) p − pq (cid:19) . Thus the gap between consecutive altitudes is bounded below by a positive constant whichdoes not depend on n , and so the altitudes are unbounded as n grows. (cid:3) With this setup, our main theorem falls readily into place: Theorem 3.11. Suppose our branch, associated to the polynomial P ( x ) and base point α ,is tamely ramification-stable over K . Then K ∞ /K is arithmetically profinite, and there is aconstant V such that for all n , K n = K (( V − n +1) ∞ . Proof. We first show that K ∞ /K is arithmetically profinite. As explained in Winten-berger [23], we simply need a filtration of elementary extensions whose altitudes tend toinfinity. Because Φ n restricts to Φ n − , the elementary subextensions of K n inside K n − areall of the elementary subextensions of K n − , which gives us our tower. The altitude of K n tends to infinity by Proposition 3.10, hence the heights of these elementary subextensionsdo as well. From this we see that the extension is arithmetically profinite, and that its21asse-Herbrand function Φ( x ) is given by the pointwise limit of the intermediate Hasse-Herbrand functions Φ n ( x ). Further, by Proposition 3.10, Φ n ( x ) coincides with Φ n − ( x ) upto the last vertex ofΦ n − ( x ), and so the same holds for Φ( x ): whenever x is smaller than the x -coordinate of the last vertex of Φ n ( x ), we have Φ( x ) = Φ n ( x ).The altitude of K n over K is the same as the height of the ( V − n th vertex of Φ, againby our assumption that the branch is tamely ramification-stable. That altitude is strictlyless than the height of the (( V − n + 1)th vertex of Φ, and so K n ⊆ K (( V − n +1) ∞ . On theother hand, the slope going into the (( V − n +1)th vertex is q n , so the degree of K (( V − n +1) ∞ over K is q n , which is the same as the degree of K n over K . Thus the two fields are equal,as claimed. (cid:3) Corollary 3.12. Let P ( x ) be a polynomial which either has degree p , or is post-criticallybounded and has degree p r . Take any nontrivial branch for P ( x ) , and suppose p does notdivide the constant d associated to the branch.Then the dynamical branch extension K ∞ /K is arithmetically profinite, and there areconstants N and V such that after replacing K by K N , K n = K (( V − n − N )+1) ∞ , for all n .Proof. If P ( x ) has prime degree or is post-critically bounded and p ∤ d , then any nontrivialbranch associated to it is potentially tamely ramification-stable by Proposition 3.8. Recallthat this means that there is an N such that after restricting our branch to start at α N it istamely ramification-stable over K N .To keep our indexing clear, set β n = α N + n , L = K ( α N ), L n = K ( β n ), and L ∞ = S L n .Clearly L ∞ = K ∞ . Then our main result, Theorem 3.11, applies to this branch, and so K ∞ = L ∞ /L = K N is arithmetically profinite and L n = L (( V − n +1) ∞ . Translating from L to K , we see that K n = L when n ≤ N , while K n = L n − N for n ≥ N .So making this change of index, ( V − n + 1 becomes ( V − n − N ) + 1. Our convention fornegative-indexed elementary subfields handles the split between n ≤ N and n ≥ N , becausereplacing K by K N means that K = K n for n ≤ N . (cid:3) As our first application, we can offer a partial answer to a question raised by Berger [8].That paper considers extensions of the same type studied here, though with two restrictions:the degree is the size of the residue field, and the base point is a uniformizer. An importantinput to that paper is that K ∞ /K Galois ⇒ K ∞ /K abelian . K ∞ /K is abelian, then K n /K is also abelian,and in particular normal. When K n /K is normal and the degree of K n /K n − is q , onecan define, for each σ ∈ Γ K , a power series Col σ ∈ K [[ x ]] such that Col σ (0) = 0 andCol σ ( α n ) = α σn (generalized Coleman power series). This power series commutes with P ,and so by a result of Lubin [16], that power series is determined by the coefficient of itslinear term, which gives a character from Γ K to O ∗ K . This character is injective, because theaction on the branch determines the action everywhere in the extension, since the branchgenerates the extension. Berger then goes on to study this character in detail.But the logic flows the other way too: if we know that K n /K is normal for some otherreason, then we can construct these power series and the associated injective character, whichwould prove that K ∞ /K is abelian. And indeed, the elementary subfields of K ∞ over K areall normal over K if K ∞ /K is normal. Thus if one were to know that for all n there exists an m such that K n = K ( m ) ∞ for some m , as in our main theorem, then K ∞ /K must be abelian. Theorem 4.1.1. Assume p is odd. Suppose α is a uniformizer for K , P ′ (0) is nonzero,and we are given a branch associated to P ( x ) and α which is tamely ramification-stable.If K ∞ /K is Galois, it is also abelian.Proof. Because α is a uniformizer, all of the polynomials P n ( x ) = α are Eisenstein, so theyare irreducible and give rise to a totally ramified extension of degree q n . This means that d = 1 and that [ K n : K n − ] = q for all n .The branch is tamely ramification stable, so we may apply Theorem 3.11, to concludethat for all n , the extension K n /K is elementary, and therefore also Galois because K ∞ /K is Galois.Now let σ ∈ Gal( K ∞ /K ). Because K n /K is normal, α σn is in K n = K ( α n ). Thesequence ( α σn ) n ∈ N is itself a branch, and by our assumption that p is odd and the irreducibilityof P ( x ) − α n − , we see that N K n K n − ( α n ) = α n − . This means that we can use Berger’sconstruction (Theorem 3.1 [8]) to produce a uniquely determined series Col σ ∈ O K [[ T ]]which acts by Col σ ( α n ) = α σn and commutes with P ( x ). This gives rise to a character χ fromGal( K ∞ /K ) to O ∗ K given by χ ( σ ) = Col ′ σ (0) . Since Col σ commutes with P ( x ) and P ′ (0) is neither zero nor a root of unity, the seriesCol σ is determined by Col ′ σ (0) by Proposition 1.1 of Lubin [16]. Since Col σ also determinesthe action of σ on α n , and hence on the whole extension K ∞ , the character χ is injective.Since Gal( K ∞ /K ) embeds into an abelian group, it is itself abelian. (cid:3) Corollary 4.1. Assume p is odd. Suppose α is a uniformizer for K , P ′ (0) is nonzero, andwe are given a branch associated to P ( x ) and α which is potentially tamely ramification-stable.If K ∞ /K is Galois, it has a finite-index abelian subgroup.Proof. Select N such that the branch is tamely ramification-stable over K N . Since it is stillthe case that the polynomials P n ( x ) − α are Eisenstein, the new base point α N remains auniformizer. Therefore, Theorem 4.1.1 applies over this larger field, and hence Gal( K ∞ /K N )is abelian. Its index in Gal( K ∞ /K ) is exactly q N . (cid:3) 23e cannot relax the assumption that α is a uniformizer, as this is crucial to Berger’sconstruction of the Coleman power series. Moreover, the fact that α is a uniformizer meansthat every α n will also be a uniformizer of the field it generates over K , and so d = 1for any branch based at α . As a result, whether or not the branch is potentially tamelyramification-stable depends entirely on P ( x ).Theorem 4.1.1 is not vacuous; there are tamely ramification-stable branches associatedto Galois extensions. For example, it is straightforward to check that Berger’s example(Theorem 6.5 [8]) P ( x ) = x + 6 x + 9 x α = − K = Q satisfies Theorem 4.1.1 by combining our observation that d = 1 with the effective results ofSection 4.3.In fact, because d = 1 and the polynomial in question has prime degree, the branchis guaranteed to be potentially tamely ramification-stable, so we could have applied Corol-lary 4.1, without making any calculations, to determine that the Galois group has a largeabelian subgroup (applying our effective results, one can see that this would have proven K ∞ /K is abelian). This can be done for any other examples involving a post-criticallybounded or prime degree polynomial. Both Aitken, Hajir, and Maire [1] (Question 7.1) and Bridy, Ingram, Jones, Juul, Levy,Manes, Rubinstein-Salzedo, and Silverman [9] (Conjecture 6) raise questions about wildramification in arboreal extensions. Namely: are there arboreal extensions over numberfields which are ramified at only finitely many primes but not wildly ramified?We answer this negatively for all arboreal extensions associated to polynomials of prime-power degree. Under some restrictions on the base point, we can also show that such arborealextensions are not only infinitely wildly ramified, but that all of their higher ramificationsubgroups are nontrivial. For the latter, we do not need the full strength of our results, onlythat K ∞ /K is arithmetically profinite (which, for certain base points, already follows fromCais, Davis, and Lubin [11]). Theorem 4.2.1. Let F be a number field and p a prime of F lying over a rational prime p .Let P ( x ) ∈ O F [ x ] be a monic polynomial of degree p r such that P ( x ) ≡ x p r mod p , and let α ∈ F .Then the arboreal representation associated to P ( x ) and α is infinitely wildly ramified.If, further, P ( x ) has prime degree and v ( α ) = 0 , or is post-critically bounded withno restriction on v p ( α ) , and there is a branch over α whose associated constant d is notdivisible by p , then every higher ramification subgroup over p of the arboreal representationis nontrivial.Proof. It suffices to work over the completion K of F at a prime lying over p , and we mayalso take finite extensions of the ground field as necessary. Thus by Proposition 2.1 we mayassume that P ( x ) is monic, has integral coefficients, and fixes 0. Replacing P by P s for asufficiently large integer s , we may assume that the the size of the residue field of K dividesthe degree of P . 24ecall that our results require v ( α ) = 0. If v ( α ) = 0, then after possibly extending F ,we will conjugate by a translation to make its valuation positive. In particular, P ( x ) has afixed point congruent to α modulo π K , because P ( x ) − x ≡ x p r − x mod π K , and the size of the residue field divides p r , so that every element of the residue field is a zeroof P ( x ) − x modulo π K . Let α be such a fixed point, then replace P ( x ) by its conjugate by x x − α .This leaves us with a final pair P ( x ) , α where v ( α ) = 0. It follows from Proposition 2.3that (every) branch extension K ∞ /K is infinitely wildly ramified, hence the full arborealextension K arb /K is also infinitely wildly ramified.Because being post-critically bounded is conjugation and composition invariant, we may always assume when P ( x ) is post-critically bounded that v p ( α ) = 0.We can say more if P ( x ) has prime degree with v p ( α ) = 0 or P ( x ) is post-criticallybounded and v p ( α ) = 0, and there is a branch such that p ∤ d , as then Corollary 3.12applies: there is an N such that after replacing K by K N , K n = K (( V − n − N )+1) ∞ . Those are the subfields of K ∞ fixed by Γ b ( V − n − N )+1 K . The branch extension K ∞ /K iscontained in the full arboreal extension K arb /K , which, combined with the functoriality ofthe upper numbering, means K n is the subfield of K ∞ /K that is fixed by the subgroupsΓ b ( V − n − N )+1 arb . But the fields K n are all distinct, and hence the subgroups which fix themmust all be distinct too. Finally, it was shown that the ramification breaks b ( V − n − N )+1 areunbounded as a function of n , and so every upper-numbered higher ramification subgroupof Γ arb is nontrivial. (cid:3) Observation. Bridy, Ingram, Jones, Juul, Levy, Manes, Rubinstein-Salzedo, and Silver-man [9] showed that an finitely ramified arboreal extension over a number field necessarilycomes from a post-critically finite, and hence post-critically bounded map. This means thatthe preceding theorem applies in full strength as soon as one checks that p does not divide d . The theorem tells us that, at least in some cases, the higher ramification subgroups of Γ arb are all nontrivial, so we are led to wonder how large or small these subgroups might be. Inparticular, is K arb /K arithmetically profinite? We suspect not, and conjecture that if thereis no branch such that K ∞ /K is Galois, then wild ramification subgroup has infinite indexinside Γ arb (in other words, the tame part of K ∞ /K has infinite degree over K ). However,it seems plausible that this could be the only obstacle to the extension being arithmeticallyprofinite: is it the case that for any 1 < µ < ν , the index [Γ µarb : Γ νarb ] is finite? Almost every step of the proof is effective, and in practice straightforward to compute.Here we sketch the computation of the limiting ramification data. An implementation in25ageMath [22] is available upon request. The only ineffective step made to obtain our resultsoccurs in Propositon 2.3. The determination of “sufficiently large” to ensure that (b) and (c)of this proposition are satisfied is not effective. This also means that the value d = lim n →∞ d n is not effective. Knowing that p does not divide d is an important input to our main results,so from a computational perspective, this is a particularly unfortunate limitation.However, if d is known, then all of our constants are effective. For example: if α is auniformizer, such as in the previous section, then P n ( x ) − α is Eisenstein, so α n is also auniformizer, and so d = 1 and (H) is immediately satisfied at the first level. V , R , M , and E . We begin with the computation of V , R , M , and E . Interestingly, these depend only onthe valuations of the coefficients of P and on the sign of the valuation of α . They do notdepend on the choice of branch.All of the following steps can be extracted readily from the proof of Proposition 2.6.Roughly, the proposition tells us that when v ( α n ) is small, we can drop the small errorterms that show up in the minimum defining the Newton polygon N n as long as we carefullytrack which terms achieve that minimum. Step 1. For each 0 ≤ k ≤ r , compute the minimum m p k = min p k ≤ j ≤ q (cid:26) v (cid:18) jp k (cid:19) + v ( P j ) (cid:27) . (17) Step 2. For each 0 ≤ k ≤ r : if v ( α ) is positive (resp. negative), let j be the first (resp.last) index achieving the minimum (17) which defines m p k . Then set e p k = j − p k . Step 3. Let N be the lower convex hull of the following vertices: (cid:26)(cid:18) p k , m p k + e p k q (cid:19) : 0 ≤ k ≤ r (cid:27) . Step 4. Let V be the number of vertices of the polygon N , and write the x -coordinates ofthe vertices of N as p r , ..., p r V . Then the limiting ramification data is: V ( P, α ) = VR ( P, α ) = ( r , ..., r V ) M ( P, α ) = ( m p r , ..., m p rV ) E ( P, α ) = ( e p r , ..., e p rV )26 .3.2 Calculating C . The constant C requires slightly more information to calculate. Unlike V , R , M , and E , thisconstant depends on the branch. However, the dependence is weaker than one might expect:if α = 0, there is a constant N which is uniform among all branches when the valuation ofthe base point, v ( α ), is fixed, such that C = q N v ( α N ). In fact, this constant N does noteven depend on P ( x ), only its degree. When α = 0, there is still such a constant, but itdepends on the valuations of the coefficients of P ( x ) and the number of leading zeros of thebranch.Inspecting the proof of Proposition 2.6, we see that if we have an N such that ( P, α N )satisfy Proposition 2.3(a), then the constant C is given by q N v ( α N ). So we simply need togive a bound on this N in terms of P and v ( α ).We can extract this from the proof of Proposition 2.3(a). If v ( α ) < v ( α ) > 0, more work is required.If α = 0, then the decrease in valuation is partly controlled by the following estimate: v ( α n ) ≤ max { v ( α n − ) − , v ( α n − ) / } . In the maximum, it is easy to see that v ( α n − ) − ≥ v ( α n − ) / v ( α n − ) ≤ , and when that occurs, it must be that v ( α n ) ≤ 1. So after N = v ( α ) steps, we areguaranteed to be in a situation where Proposition 2.3(a) applies, and hence C = q N v ( α N ).Otherwise, α = 0. Let k be the number of leading 0s in the branch, which means α k = 0and α k − = 0. and by inspecting the Newton polygon of P ( x ) − α k − = P ( x ), a generousbound for v ( α k ) is ℓ = max { v ( P j ) } , as long as α = 0. Then we may apply our reasoningfor the case α = 0, but with α k in place of α to see that C = q k + ℓ v ( α k + ℓ )This gives us a remarkably simple process for computing the index N such that C = q N v ( α N ), and of course C itself: Step 1. If v ( α ) < 0, then let N = 0. Step 2. If v ( α ) > α = 0, then let N = v ( α ). Step 3. If α = 0, let k be the number of leading zeros in the branch and let ℓ =max { v ( P j ) } . Then let N = k + ℓ . Step 4. Set C = q N v ( α N ).Evidently, the value N is independent of the branch except when α = 0, and in that casethe dependence is only on the number of leading zeros. Usually this index is much largerthan necessary. 27 .3.3 Sample calculation. In any particular case, it is almost straightforward to check that a pair is tamely ramification-stable, except for the tameness component, since we do not have an effective way to compute d . However, it is still possible to do so in some cases.The following example is small enough that one can carry out the calculation by hand.Let K = E = Q ( √ 3) with valuation v normalized so that v ( √ 3) = 1. Consider thepolynomial P ( x ) = x + 12 √ x + 18 x + 3 √ x + 35 x + 9 x, with any branch whose initial sequence of valuations looks like (4 , / , / , ... ).We readily obtain our effective constants: V = 3 ,R = (0 , , ,M = (3 , , ,E = (3 , , . as well as C = 9 v ( α ) = 9 ∗ 23 19 = 6 . Inspecting the first few levels of such a branch in Sage, we see that d = 2, which is notdivisible by p = 3. Combined with this limiting ramification data, one can see directly that( P, α ) is tamely ramification-stable. Therefore, when we replace K by K , we may applyTheorem 3.11 to obtain K n = K (( V − n − ∞ . So, even though P ( x ) is not prime-degree or post-critically bounded, it is an example ofa polynomial whose branch extensions are amenable to study by our methods. Acknowledgments I would like to thank my advisor, Joseph H. Silverman, for many helpful discussions on thisproject and his careful comments on early versions of the paper. References [1] Wayne Aitken, Farshid Hajir, and Christian Maire. Finitely ramified iterated extensions. International Mathematics Research Notices , 2005(14):855–880, 01 2005.[2] Jacqueline Anderson. Bounds on the radius of the p-adic mandelbrot set. 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