TRACE FUNCTIONS AND GALOIS INVARIANT p-ADIC MEASURES

Let p be a prime number, Qp the field of p-adic numbers, Qp a fixed algebraic closure of Qp, and Cp the completion of Qp with respect to the p-adic valuation. We study trace functions associated to p-adic measures defined on compact subsets of Cp which are invariant under the action of the Galois group G = Galcont(Cp/Qp).


Introduction
Let p be a prime number, Q p the field of p-adic numbers, Q p a fixed algebraic closure of Q p , and C p the completion of Q p with respect to the p-adic valuation.The notion of a trace function associated to an element T from C p was introduced and investigated in [APZ3].If T is algebraic over Q p , and L is a finite field extension of Q p contained in Q p such that T lies in L, then the p-adic number (1) T r T := T r L/Qp (T ) [L : Q p ] depends on T only, and not on L. The significance of T r T is that of the average value of the conjugates of T over Q p .This idea of taking the average value rather than the sum of conjugates, may also be applied, as shown in [APZ3], to a rich class of elements T from C p which are transcendental over Q p .Given an element T of C p , one takes its Galois orbit C(T ) = {σ(T ) : σ ∈ Gal cont (C p /Q p )}, which is a compact subset of C p , and one considers the p-adic Haar distribution π T defined on C (T ).Then, by analogy with the case when T is algebraic and T r T is given by the average value of the conjugates of T over Q p , one defines T r T for a general T ∈ C p by the equality (2) T r T = C(T ) xdπ T (x), provided that the integral on the right side of ( 2) is well defined.This is the case, for example, when the distribution π T is bounded, that is, when π T is a measure.The integral is also well defined when T is a Lipschitzian element, in the sense of [APZ3].The trace function F (T, z) is defined by for all those z ∈ C p for which the integral is well defined.In the present paper we introduce and study some natural generalizations of the above objects.Let G denote the group of continuous automorphisms of ) for any ball B and any σ ∈ G.By a probability measure (or distribution) we mean a measure (respectively distribution) µ on M for which µ(M ) = 1.If M is a G-invariant compact subset of C p and µ is a G-invariant probability distribution on M , we define the trace of µ by the formula provided that the integral on the right side of ( 4) is well defined.We further associate a trace function F (µ, z) to µ by letting ( 5) for all z in C p for which the integral is well defined.This is an analytic object that embodies a significant amount of algebraic data.For instance, recall that by Galois theory in C p (see [T], [S], [A]), closed subgroups of G are in one-to-one correspondence with the closed subfields of C p .If E is a closed subfield of C p on which the trace map T r is defined and continuous, and if T is a generating element of E over Q p (see [IZ], [APZ1], [APZ2]), then the trace map on the entire field E is determined by the Taylor series expansion F (T, z) =

Preliminaries
Let M be a compact subset of C p , and let Ω(M ) be the set of all open compact subsets of M .By a distribution on M we mean a map µ : Ω(M ) → C p that is finitely additive.If the set {µ(B) : B ∈ Ω(M )} is bounded in C p , µ is said to be a measure.Denote by G = Gal cont (C p /Q p ) the group of all continuous automorphisms of C p over Q p .Any T ∈ C p has a G-orbit C(T ) = {σ(T ) : σ ∈ G}, which is a compact subset of C p .Denote by N (T, ε) the number of open balls of C p of radius ε, any two disjoint, which cover C (T ) where | | stands for the p-adic absolute value of the integer N (T, ε).
for any D ∈ Ω(M ) and any σ ∈ G.If µ(M ) = 1 we call µ a probability distribution on M .For any T ∈ C p , the p-adic Haar distribution π T : Ω(C(T )) → Q p is the unique probability distribution on C(T ) with values in Q p which is G-invariant.For more details and more general types of p-adic spaces and measures see [Man], [Ka], [Ko], [Vi].Any continuous function f : M → C p is integrable with respect to any measure µ on M , where the notion of integrability is defined as usual in terms of Riemann sums (see [Ko] the case M = C(T ), with T ∈ C p Lipschitzian, any Lipschitzian function f : C(T ) → C p , is integrable (see [APZ3]).

The case
In this section we briefly discuss the case when M is a finite union of distinct Galois orbits, M = C(T 1 ) ∪ • • • ∪ C(T r ) say.Our first objective is to obtain a characterization of all the G-invariant probability distributions on M with values in Q p in terms of the Haar distributions π T1 , . . ., π Tr defined on the Galois orbits C(T 1 ), . . ., C(T r ).Let µ : Ω(M ) → Q p be finitely additive, satisfying µ(M ) = 1, and G-invariant.Write any D ∈ Ω(M ) as Clearly each D j belongs to Ω(M ).Thus µ(D) = r j=1 µ(D j ).Let α 1 , . . ., α r ∈ Q p be given by ( 6) µ(C(T j )) = α j , j = 1, . . ., r.
The equality (9) also holds if T 1 , . . ., T r are Lipschitzian, provided f is Lipschitzian.Applying (9) with f (x) = x gives a formula for the trace of µ.Similarly, ( 9) and ( 5) give an expression for the trace function associated to µ.We collect the results in the following proposition.
Proposition 1.Let M = ∪ r j=1 C(T j ) with T 1 , . . ., T r Lipschitzian elements of C p for which the Galois orbits C(T 1 ), . . ., C(T r ) are distinct.Let µ be a G-invariant probability on M with values in Q p , and let α 1 , . . ., α r be given by (6).Then (i) T r µ is well defined, and satisfies the equality (ii) The trace series F (µ, Z) is well defined, and is given by As a consequence, it follows by [APZ3] that under the hypothesis from Proposition 1, the function z → F (µ, z) is well defined and rigid analytic on P 1 (C p )\{u ∈ P 1 (C p ) : 1 u ∈ M } (see also [E], [Ko], [Kr], [B]).This is its maximal domain of rigid analyticity if α 1 , . . ., α r are nonzero.Returning to (9), we may view its right side as an iterated integral, if we introduce an appropriate measure on the finite set T = {T 1 , . . ., T r }.Specifically, for any T ∈ C p , denote by ν T the normalized Dirac measure on C p concentrated at T .Next, with α 1 , . . ., α r given by ( 6), consider the measure ν on T defined as the linear combination Thus T gdν = r j=1 α j g(T j ) for any g : T → C p .If now f and g are related by g(T ) = C(T ) f (y)dπ T (y), then T g(T )dν(T ) = r j=1 α j g(T j ), provided the two sides are well defined.We deduce that (11) which, to simplify the notation, we also write as Applying (12) to the function f from (9) gives M f dµ as an iterated integral.We state the result in the following proposition.and for any z ∈ P 1 (C p ) for which both sides of (14) are well defined.

The general case
Proposition 2 above expresses the integral M f dµ as an iterated integral.We achieved this result by an ad-hoc construction, which made use of a set T that is not intrinsically needed in our problem.We took advantage of the initial appearance of the elements T 1 , . . ., T r in the definition of the set M , but they do not have any special significance to the problem, and the set T may be replaced by any set of the form {σ 1 (T 1 ), . . ., σ r (T r )}, with σ 1 , . . ., σ r ∈ G. Thus, rather than to insist that T be a subset of C p , it is more natural to define this set to be a set of Galois orbits.On C p we have a natural equivalence relation: two elements T 1 and T 2 of C p are equivalent if and only if there exists σ ∈ G such that T 2 = σ(T 1 ).Denote by C p the set of equivalence classes.Thus an element t of C p is a Galois orbit.Let Ψ : C p → C p the canonical map, which sends each element of C p to its Galois orbit.On C p we introduce a distance function d, by We state some of the basic properties of C p and its compact opens in the following lemma.
Lemma 1.With the above notations, C p is a complete ultrametric space, and the map Ψ : To any G-invariant probability distribution µ on M with values in Q p we now associate a probability distribution µ on M by µ so µ is a probability distribution on M .We claim that the map from the set of G-invariant probability distributions on M with values in Q p to the set of probability distributions on M with values in Q p , given by µ → µ, is bijective.We first show that this map is injective.Assume that µ 1 = µ 2 and µ and we obtain a contradiction.This shows that the map µ → µ is injective.The surjectivity is proved by similar reasonings.We collect the results in the following theorem.
Theorem 1.Let M be a G-invariant compact subset of C p .Then the G-invariant probability distributions on M with values in Q p are in oneto-one correspondence with the probability distributions on M with values in Q p , via the map µ → µ.
We note that µ may be bounded even if µ is not bounded.For instance, if M is a finite union of Galois orbits, then µ is bounded, regardless of whether µ is bounded or not.Note also that even if µ and µ are both bounded, we might not be able to obtain (13).For instance, if M = ∪ r j=1 C(T j ), with π Tj bounded for all j with the exception of j = 1, then any G-invariant probability distribution µ on M with values in Q p for which the corresponding α 1 defined as in ( 6) vanishes, will be bounded.In that case the left side of (13) will be well defined, for any continuous function f : M → C p .On the other hand, depending on f and π T1 , the right side of ( 13) might be undefined, as the inner integral in ( 13) may be undefined at the point T = T 1 .In what follows we will avoid such situations by restricting to the case when µ and the Haar distributions π Tj are bounded.
We return to the more general case of a G-invariant compact subset M of C p and make the following assumption: there exists a positive real number A, depending on M , such that for any T ∈ M and any D ∈ Ω(C(T )), one has Here the absolute value on the left side of ( 15) is the p-adic absolute value on Q p .The condition, in other words, says that the Haar distributions π T , with T ∈ M , are uniformly bounded.We remark that, although M is compact, it is not enough to assume that each π T with T in M is bounded in order to conclude that these distributions are uniformly bounded.
As an example, choose a sequence (α n ) n∈N of algebraic elements over Q p , which is convergent to an algebraic element α, and such that the exponent of p in the degree deg α n of α n over Q p tends to infinity as Then M is compact and Ginvariant.Also, π α and π αn are bounded.On the other hand, for any point U in C(α n ), π αn ({U }) = 1 deg αn .Since the exponent of p in deg α n tends to infinity as n → ∞, the Haar distributions are not uniformly bounded.
We now return to the case of a general G-invariant compact set M for which the Haar distributions π T , with T in M , are uniformly bounded.Let µ be a G-invariant probability distribution on M for which µ is bounded.Then µ will also be bounded.We are interested to see whether an analogue of formula (13) still holds in this generality.For any t ∈ M we denote by C t the Galois orbit in C p which defines t, and by π t the Haar distribution on C t .Let f : M → C p , f continuous.The analogue of ( 13) reads ( 16) Since f is continuous and µ is bounded, the left side of ( 16) is well defined.On the right side of ( 16), the outer integral, over M , is with respect to the variable t.For each fixed t ∈ M , the inner integral is well defined, as one integrates on C t the restriction of f , which is continuous with respect to π t , which is bounded.In order to show that the right side of ( 16) is well defined, it will be enough to prove that the function defined on M by t → Ct f dπ t is continuous.Then, since µ is bounded, the outer integral, and then also the entire right side of ( 16) will be well defined.
To show that the above function is continuous, fix t 0 ∈ M .We need to show that Ct f dπ t → Ct 0 f dπ t0 , as t → t 0 .Fix ε > 0. Since f is continuous and M compact, from uniform continuity, there is a δ > 0 such that, for any x, y ∈ M with |y − x| ≤ δ, one has |f (y) − f (x)| ≤ ε.Let t in M with d(t, t 0 ) < δ.Write M as a disjoint union of closed balls of radius δ.Some of them intersect C t0 .Denote these balls by B 1 , B 2 , . . ., B N .Since d(t, t 0 ) < δ, each ball B j , with 1 ≤ j ≤ N , has a nonempty intersection with C t .Let x j ∈ C t0 ∩ B j and y j ∈ C t ∩ B j , for j = 1, 2, . . ., N . Let ).The sums S 1 and S 2 are Riemann sums for the integrals Ct 0 f dπ t0 and respectively Ct f dπ t .We need to show that these two integrals are close to each other, provided that ε is small enough, and for this it is enough to show that S 1 and S 2 are close to these integrals, and that S 1 and S 2 are close to each other.The balls B j being conjugate, Next, with ε and δ fixed, take a small δ ′ > 0 and write each closed ball B j of radius δ as a disjoint union of closed balls of radius δ ′ , B j = ∪ r i=1 B ji .The balls B j being conjugate, each of them is a disjoint union of the same number r of closed balls of radius δ ′ .For any j ∈ {1, . . ., N }, out of the r balls B ji , 1 ≤ i ≤ r , the same number of them, r ′ say, have a nonempty intersection with C t0 .After redenoting the balls if necessary, assume that the balls which intersect C t0 are B ji , 1 ≤ i ≤ r ′ , 1 ≤ j ≤ N .Choose x ji in B ji ∩ C t0 , and consider the Riemann sum ).Let δ ′ be small enough so that for any choice of x ji , S ′ 1 − Ct 0 f dπ t0 < ε.
The sets B ji ∩ C t0 are conjugate, and N r ′ is bounded by a number which depends on M only.Using the fact that |f (x j ) − f (x ji )| ≤ ε, since x j , x ji ∈ B j for all i and j, we have  On the other hand, it is easy to see that π t (B ∩ C t ) = 1 N for any t ∈ E, while for t ∈ M \ E one has π t (B ∩ C t ) = 0. Combining this with (25) we find that (29) Finally, by ( 28) and ( 29) we see that formula (16) holds true for χ B .We collect the results in the following theorem.
Theorem 2. Let M be a G-invariant compact subset of C p and let µ be a G-invariant probability distribution on M with values in Q p .Assume that µ is bounded, and that the Haar distributions π t , with t ∈ M , are uniformly bounded.Then for any continuous function f : As a corollary we obtain an expression for the trace of µ and the trace function associated to µ in terms of the trace T r t and respectively the trace function F (t, z) associated to the Galois orbits C t , with t in M .
Corollary 2. Let M and µ be as in the statement of Theorem 2. Then for any z ∈ P 1 (C p ) for which both sides of (30) are well defined.
Fix a continuous function f : M → C p .By the uniform continuity of f , for any ε > 0, there is a δ > 0 such that |f (x)−f (y)| ≤ ε for any x, y ∈ M with |x−y| ≤ δ.Write M as a disjoint union of closed balls of radius δ, M = ∪ m j=1 B j .Choose z j in B j , and let g = m j=1 f (z j )χ Bj , where χ Bj denotes the characteristic function of the ball B j .Denote h = f −g.Then |h(z)| = |f (z)−g(z)| ≤ ε, for any z ∈ M .It follows that for any Riemann sum S associated with the integral M hdµ, S = L i=1 h(x i )µ(D i ), where D 1 , . . ., D L ∈ Ω(M ) form a partition of M , and x i ∈ D i for 1 ≤ i ≤ L, one has (20) |S| ≤ max 1≤i≤L |h(x i )µ(D i )| ≤ εA(µ), where A(µ) = sup D∈Ω(M) |µ(D)| < ∞.By applying (20) to a sequence of Riemann sums converging to M hdµ, we obtain (21) M hdµ ≤ εA(µ).
Ct f dπ t → Ct 0 f dπ t0 , as t → t 0 , which proves the continuity of the map t → Ct f dπ t .It remains to show that the two sides of formula (16) are equal.We first reduce to the case of step functions.