Trace functions and Galois invariant p-adic measures

Publ. Mat. 50 (2006), 43–55 TRACE FUNCTIONS AND GALOIS INVARIANT p-ADIC MEASURES Marian Vˆ ajˆ aitu and Alexandru Zaharescu Abstract Let pbe a prime number, Qpthe field of p-adic numbers, Qpa fixed algebraic closure of Qp, and Cpthe completion of Qpwith respect to the p-adic valuation. We study trace functions associated to p-adic measures defined on compact subsets of Cpwhich are invariant under the action of the Galois group G=Galcont(Cp/Qp). 1. Introduction Let pbe a prime number, Qpthe field of p-adic numbers, Qpa fixed algebraic closure of Qp, and Cpthe completion of Qpwith respect to the p-adic valuation. The notion of a trace function associated to an element Tfrom Cpwas introduced and investigated in [APZ3]. If Tis algebraic over Qp, and Lis a finite field extension of Qpcontained in Qpsuch that Tlies in L, then the p-adic number (1) T r T := T rL/Qp(T) [L:Qp] depends on Tonly, and not on L. The significance of T r T is that of the average value of the conjugates of Tover Qp. 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 Tfrom Cpwhich are transcendental over Qp. Given an element Tof Cp, one takes its Galois orbit C(T) = {σ(T) : σ∈Galcont(Cp/Qp)}, which is a compact subset of Cp, and one considers the p-adic Haar distribution πTdefined on C(T). Then, by analogy with the case when Tis algebraic and T r T is given by the average value of the conjugates of Tover Qp, one defines T r T 2000 Mathematics Subject Classification. 11S99. Key words. Trace functions, rigid analytic functions, p-adic measures. This work was partially supported by the Program ceex 2005 of the Romanian Ministry of Education and Research. 44 M. Vˆ ajˆ aitu, A. Zaharescu for a general T∈Cpby the equality (2) T r T =ZC(T) xdπT(x), provided that the integral on the right side of (2) is well defined. This is the case, for example, when the distribution πTis bounded, that is, when πTis a measure. The integral is also well defined when Tis a Lipschitzian element, in the sense of [APZ3]. The trace function F(T, z) is defined by (3) F(T, z) = ZC(T) 1 1−zxdπT(x), for all those z∈Cpfor which the integral is well defined. In the present paper we introduce and study some natural generalizations of the above objects. Let Gdenote the group of continuous automorphisms of Cpover Qp. A compact subset Mof Cpis G-invariant provided that σ(x)∈M for any x∈Mand σ∈G. If Mis a G-invariant compact subset of Cpand µis a distribution on Mwith values in Qp, we say that µis G-invariant if µ(B) = µ(σ(B)) for any ball Band any σ∈G. By a probability measure (or distribution) we mean a measure (respectively distribution) µon Mfor which µ(M) = 1. If Mis a G-invariant compact subset of Cpand µis a G-invariant probability distribution on M, we define the trace of µby the formula (4) T r µ =ZM xdµ(x), provided that the integral on the right side of (4) is well defined. We further associate a trace function F(µ, z) to µby letting (5) F(µ, z) = ZM 1 1−zxdµ(x), for all zin Cpfor 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 Cp(see [T], [S], [A]), closed subgroups of Gare in one-to-one correspondence with the closed subfields of Cp. If Eis a closed subfield of Cpon which the trace map T r is defined and continuous, and if Tis a generating element of Eover Qp (see [IZ], [APZ1], [APZ2]), then the trace map on the entire field E is determined by the Taylor series expansion F(T, z) = P∞ n=0 T r T nzn. Naturally, if Mand µare G-invariant, one would like to be able to use the action of the Galois group Gto express and compute the above integrals. We achieve this goal in Theorems 1 and 2 from Section 4 below. Trace Functions and Galois Invariant p-Adic Measures 45 2. Preliminaries Let Mbe a compact subset of Cp, and let Ω(M) be the set of all open compact subsets of M. By a distribution on Mwe mean a map µ: Ω(M)→Cpthat is finitely additive. If the set {µ(B) : B∈Ω(M)}is bounded in Cp,µis said to be a measure. Denote by G=Galcont(Cp/Qp) the group of all continuous automorphisms of Cpover Qp. Any T∈Cp has a G-orbit C(T) = {σ(T) : σ∈G},which is a compact subset of Cp. Denote by N(T, ε) the number of open balls of Cpof radius ε, any two disjoint, which cover C(T). If ε < ε′then N(T, ε′) divides N(T, ε). An element T∈Cpis called Lipschitzian if limε→0ε |N(T,ε)|= 0, where | | stands for the p-adic absolute value of the integer N(T, ε). A compact subset Mof Cpis G-invariant provided that σ(x)∈Mfor any x∈Mand σ∈G. If Mis a G-invariant compact subset of Cp, given a distribution µ: Ω(M)→Cp, we say that µis G-invariant if µ(D) = µ(σ(D)) for any D∈Ω(M) and any σ∈G. If µ(M) = 1 we call µa probability distribution on M. For any T∈Cp, the p-adic Haar distribution πT: Ω(C(T)) →Qpis the unique probability distribution on C(T) with values in Qpwhich 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→Cpis integrable with respect to any measure µon M, where the notion of integrability is defined as usual in terms of Riemann sums (see [Ko] the case M=C(T), with T∈CpLipschitzian, any Lipschitzian function f:C(T)→Cp, is integrable (see [APZ3]). 3. The case M=∪r j=1C(Tj) In this section we briefly discuss the case when Mis a finite union of distinct Galois orbits, M=C(T1)∪ · · · ∪ C(Tr) say. Our first objective is to obtain a characterization of all the G-invariant probability distributions on Mwith values in Qpin terms of the Haar distributions πT1,...,πTrdefined on the Galois orbits C(T1),...,C(Tr). Let µ: Ω(M)→Qpbe finitely additive, satisfying µ(M) = 1, and G-invariant. Write any D∈Ω(M) as D=D1∪ · · ·∪ Dr, where Dj=D∩C(Tj). Clearly each Djbelongs to Ω(M). Thus µ(D) = Pr j=1 µ(Dj).Let α1,...,αr∈Qpbe given by (6) µ(C(Tj)) = αj, j = 1,...,r. Note that (7) α1+···+αr= 1. 46 M. Vˆ ajˆ aitu, A. Zaharescu Also, (8) µ(D) = r X j=1 αjπTj(Dj) = r X j=1 αjπTj(D∩C(Tj)) for any D∈Ω(M). This expresses µin terms of πT1,...,πTr. Conversely, for any α1,...,αrof Qpsatisfying (7), formula (8) defines a G-invariant probability distribution on M. If πT1,...,πTrare bounded, so is µ. Then any continuous function f:M→Cpis integrable with respect to µ, and (9) ZM fdµ = r X j=1 αjZC(Tj) fdπTj. The equality (9) also holds if T1,...,Trare Lipschitzian, provided fis Lipschitzian. Applying (9) with f(x) = xgives 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=1C(Tj)with T1,...,TrLipschitzian elements of Cpfor which the Galois orbits C(T1),...,C(Tr)are distinct. Let µbe a G-invariant probability on Mwith values in Qp, and let α1,...,αrbe given by (6). Then (i) T r µ is well defined, and satisfies the equality T r µ = r X j=1 αjT r Tj. (ii) The trace series F(µ, Z)is well defined, and is given by F(µ, Z) = r X j=1 αjF(Tj, Z). As a consequence, it follows by [APZ3] that under the hypothesis from Proposition 1, the function z7→ F(µ, z) is well defined and rigid analytic on P1(Cp)\{u∈P1(Cp) : 1 u∈M}(see also [E], [Ko], [Kr], [B]). This is its maximal domain of rigid analyticity if α1,...,αrare 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={T1,...,Tr}. Specifically, for any T∈Cp, denote by νTthe normalized Dirac measure on Cpconcentrated at T. Next, with α1,...,αrgiven by (6), consider the measure νon Tdefined as the linear combination (10) ν= r X j=1 αjνTj. Trace Functions and Galois Invariant p-Adic Measures 47 Thus RTgdν =Pr j=1 αjg(Tj) for any g:T → Cp. If now fand gare related by g(T) = RC(T)f(y)dπT(y),then RTg(T)dν(T) = Pr j=1 αjg(Tj), provided the two sides are well defined. We deduce that (11) ZTZC(T) f(y)dπT(y)dν(T) = r X j=1 αjZC(Tj) f(y)dπTj(y), which, to simplify the notation, we also write as (12) ZTZC(T) fdπTdν = r X j=1 αjZC(Tj) fdπTj. Applying (12) to the function ffrom (9) gives RMfdµ as an iterated integral. We state the result in the following proposition. Proposition 2. Let M=∪r j=1C(Tj)with T1,...,TrLipschitzian elements of Cpfor which the Galois orbits C(T1),...,C(Tr)are distinct. Let µbe a G-invariant probability distribution on Mwith values in Qp, and define the measure νon T={T1,...,Tr}by (10) and (6). Then, for any Lipschitzian function f:M→Cp, (13) ZM fdµ =ZTZC(T) fdπTdν. Corollary 1. Let M,µ,Tand νbe as in Proposition 2. Then (i)T r µ =ZT T r T dν(T), and (ii)F(µ, z) = ZT F(T, z)dν(T),(14) for any z∈P1(Cp)for which both sides of (14) are well defined. 4. The general case Proposition 2 above expresses the integral RMfdµ as an iterated integral. We achieved this result by an ad-hoc construction, which made use of a set Tthat is not intrinsically needed in our problem. We took advantage of the initial appearance of the elements T1,...,Trin the definition of the set M, but they do not have any special significance to the problem, and the set Tmay be replaced by any set of the form {σ1(T1),...,σr(Tr)}, with σ1, . . ., σr∈G. Thus, rather than to insist that Tbe a subset of Cp, it is more natural to define this set to be a set 48 M. Vˆ ajˆ aitu, A. Zaharescu of Galois orbits. On Cpwe have a natural equivalence relation: two elements T1and T2of Cpare equivalent if and only if there exists σ∈Gsuch that T2=σ(T1). Denote by b Cpthe set of equivalence classes. Thus an element tof b Cpis a Galois orbit. Let Ψ: Cp→b Cpthe canonical map, which sends each element of Cpto its Galois orbit. On b Cpwe introduce a distance function d, by d(t1, t2) = inf{|x−y|:x∈Ψ−1(t1), y ∈Ψ−1(t2)}. We state some of the basic properties of b Cpand its compact opens in the following lemma. Lemma 1. With the above notations, b Cpis a complete ultrametric space, and the map Ψ: Cp→b Cpis 1-Lipschitzian. Moreover, for any D∈Ω(M) and E∈Ω(c M), we have Ψ(D)∈Ω(c M)and Ψ−1(E)∈Ω(M). To any G-invariant probability distribution µon Mwith values in Qp we now associate a probability distribution bµon c Mby bµ(E)=µ(Ψ−1(E)), for any E∈Ω(c M). If E∈Ω(c M) is written as a disjoint union E= E1∪E2∪· · ·∪Em, with E1, . . . , Em∈Ω(c M), then bµ(E) = µ(Ψ−1(E)) = Pm i=1 µ(Ψ−1(Ei)) = Pm i=1 bµ(Ei),so bµis finitely additive. Also, bµ(c M) = µ(M) = 1, so bµis a probability distribution on c M. We claim that the map from the set of G-invariant probability distributions on M with values in Qpto the set of probability distributions on c Mwith values in Qp, given by µ7→ bµ, is bijective. We first show that this map is injective. Assume that µ16=µ2and bµ1=bµ2. Choose Din Ω(M) for which µ1(D)6=µ2(D). Write Das a disjoint union of closed balls in M,D=B1∪B2∪ · · · ∪ Bn.Then Pn i=1 µ1(Bi) = µ1(D)6= µ2(D) = Pn i=1 µ2(Bi).Choose a Bifor which µ1(Bi)6=µ2(Bi). One has µ1(Ψ−1(Ψ(Bi))) = bµ1(Ψ(Bi)) = bµ2(Ψ(Bi)) = µ2(Ψ−1(Ψ(Bi))).Let σ1,...,σN∈Gbe such that Ψ−1(Ψ(Bi)) = ∪N j=1σj(Bi),with σj(Bi) disjoint. Then PN j=1 µ1(σj(Bi)) = µ1(Ψ−1(Ψ(Bi))) = µ2(Ψ−1(Ψ(Bi))) = PN j=1 µ2(σj(Bi)).Since µ1and µ2are G-invariant, PN j=1 µ1(σj(Bi)) = Nµ1(Bi),and PN j=1 µ2(σj(Bi)) = Nµ2(Bi).Hence µ1(Bi) = µ2(Bi), and we obtain a contradiction. This shows that the map µ7→ bµis injective. The surjectivity is proved by similar reasonings. We collect the results in the following theorem. Theorem 1. Let Mbe a G-invariant compact subset of Cp. Then the G-invariant probability distributions on Mwith values in Qpare in oneto-one correspondence with the probability distributions on c Mwith values in Qp, via the map µ7→ bµ. Trace Functions and Galois Invariant p-Adic Measures 49 We note that bµmay be bounded even if µis not bounded. For instance, if Mis a finite union of Galois orbits, then bµis bounded, regardless of whether µis bounded or not. Note also that even if µand bµare both bounded, we might not be able to obtain (13). For instance, if M=∪r j=1C(Tj), with πTjbounded for all jwith the exception of j= 1, then any G-invariant probability distribution µon Mwith values in Qpfor which the corresponding α1defined 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→Cp. On the other hand, depending on f and πT1, the right side of (13) might be undefined, as the inner integral in (13) may be undefined at the point T=T1. In what follows we will avoid such situations by restricting to the case when bµand the Haar distributions πTjare bounded. We return to the more general case of a G-invariant compact subset Mof Cpand make the following assumption: there exists a positive real number A, depending on M, such that for any T∈Mand any D∈Ω(C(T)), one has (15) |πT(D)| ≤ A. Here the absolute value on the left side of (15) is the p-adic absolute value on Qp. The condition, in other words, says that the Haar distributions πT, with T∈M, are uniformly bounded. We remark that, although Mis compact, it is not enough to assume that each πTwith Tin Mis bounded in order to conclude that these distributions are uniformly bounded. As an example, choose a sequence (αn)n∈Nof algebraic elements over Qp, which is convergent to an algebraic element α, and such that the exponent of pin the degree deg αnof αnover Qptends to infinity as n→ ∞. Set M=C(α)∪ ∪n∈NC(αn).Then Mis compact and Ginvariant. Also, παand παnare bounded. On the other hand, for any point Uin C(αn), παn({U}) = 1 deg αn.Since the exponent of pin deg αn 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 Mfor which the Haar distributions πT, with Tin M, are uniformly bounded. Let µbe a G-invariant probability distribution on Mfor which bµis bounded. Then µwill also be bounded. We are interested to see whether an analogue of formula (13) still holds in this generality. For any t∈c M we denote by Ctthe Galois orbit in Cpwhich defines t, and by πtthe Haar distribution on Ct. Let f:M→Cp,fcontinuous. The analogue 50 M. Vˆ ajˆ aitu, A. Zaharescu of (13) reads (16) ZM fdµ =Zc MZCt fdπtdbµ. Since fis continuous and µis bounded, the left side of (16) is well defined. On the right side of (16), the outer integral, over c M, is with respect to the variable t. For each fixed t∈c M, the inner integral is well defined, as one integrates on Ctthe restriction of f, which is continuous with respect to πt, 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 c Mby t7→ RCtfdπtis continuous. Then, since bµis 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 t0∈c M. We need to show that RCtfdπt→RCt0 fdπt0,as t→t0.Fix ε > 0. Since fis continuous and Mcompact, from uniform continuity, there is a δ > 0 such that, for any x, y ∈Mwith |y−x| ≤ δ, one has |f(y)−f(x)| ≤ ε. Let tin c Mwith d(t, t0)< δ. Write Mas a disjoint union of closed balls of radius δ. Some of them intersect Ct0. Denote these balls by B1, B2,...,BN. Since d(t, t0)< δ, each ball Bj, with 1 ≤j≤N, has a nonempty intersection with Ct. Let xj∈Ct0∩Bjand yj∈Ct∩Bj, for j= 1,2,...,N. Let S1=PN j=1 f(xj)πt0(Ct0∩Bj) and S2=PN j=1 f(yj)πt(Ct∩Bj). The sums S1and S2are Riemann sums for the integrals RCt0 fdπt0and respectively RCtfdπt. We need to show that these two integrals are close to each other, provided that εis small enough, and for this it is enough to show that S1and S2are close to these integrals, and that S1and S2 are close to each other. The balls Bjbeing conjugate, πt0(Ct0∩Bj) = πt(Ct∩Bj) = 1 N,hence S1=1 NPN j=1 f(xj),and S2=1 NPN j=1 f(yj). Therefore, |S1−S2|=1 NPN j=1(f(xj)−f(yj))≤1 |N|max1≤j≤N|f(xj)− f(yj)|.Here xj, yj∈Bj,|xj−yj| ≤ δ, so |f(xj)−f(yj)| ≤ ε, and consequently |S1−S2| ≤ ε |N|.By (15), the p-power in Nis bounded in terms of Monly. It follows that (17) |S1−S2| → 0,as ε→0. Next, with εand δfixed, take a small δ′>0 and write each closed ball Bjof radius δas a disjoint union of closed balls of radius δ′,Bj= ∪r i=1Bji. The balls Bjbeing conjugate, each of them is a disjoint union of the same number rof closed balls of radius δ′. For any j∈ {1,...,N}, Trace Functions and Galois Invariant p-Adic Measures 51 out of the rballs Bji, 1 ≤i≤r, the same number of them, r′say, have a nonempty intersection with Ct0. After redenoting the balls if necessary, assume that the balls which intersect Ct0are Bji, 1 ≤i≤r′, 1≤j≤N. Choose xji in Bji ∩Ct0, and consider the Riemann sum S′ 1=PN j=1 Pr′ i=1 f(xji)πt0(Bji ∩Ct0). Let δ′be small enough so that for any choice of xji,S′ 1−RCt0 fdπt0< ε. The sets Bji ∩Ct0are conjugate, and S′ 1=1 Nr′PN j=1 Pr′ i=1 f(xji).By our assumptions, πt0(Bji ∩Ct0)=1 Nr′is bounded by a number which depends on Monly. Using the fact that |f(xj)−f(xji)| ≤ ε, since xj, xji ∈Bjfor all iand j, we have |S1−S′ 1| ≤ 1 Nr′max 1≤N 1≤i≤r′f(xj)− f(xji)≤1 Nr′ε, so |S1−S′ 1| → 0,as ε→0.We deduce that (18) S1→ZCt0 fdπt0,as ε→0, regardless of the choice of the parameters appearing in the definition of S1. Similarly, (19) S2−ZCt fdπt→0,as ε→0. By (17), (18), and (19), we conclude that RCtfdπt→RCt0 fdπt0,as t→t0,which proves the continuity of the map t7→ RCtfdπt. It remains to show that the two sides of formula (16) are equal. We first reduce to the case of step functions. Fix a continuous function f:M→Cp. By the uniform continuity of f, for any ε > 0, there is a δ > 0 such that |f(x)−f(y)| ≤ εfor any x, y ∈Mwith |x−y| ≤ δ. Write Mas a disjoint union of closed balls of radius δ,M=∪m j=1Bj.Choose zj in Bj, and let g=Pm j=1 f(zj)χBj,where χBjdenotes the characteristic function of the ball Bj. Denote h=f−g. Then |h(z)|=|f(z)−g(z)| ≤ ε, for any z∈M. It follows that for any Riemann sum Sassociated with the integral RMhdµ,S=PL i=1 h(xi)µ(Di),where D1,...,DL∈Ω(M) form a partition of M, and xi∈Difor 1 ≤i≤L, one has (20) |S| ≤ max 1≤i≤L|h(xi)µ(Di)| ≤ εA(µ), where A(µ) = supD∈Ω(M)|µ(D)|<∞.By applying (20) to a sequence of Riemann sums converging to RMhdµ, we obtain (21) ZM hdµ≤εA(µ).