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On the Continuity of the Trace THE PUBLISHING HOUSE PROCEEDINGS OF THE ROMANIAN ACADEMY, Series A, OF THE ROMANIAN ACADEMY Volume 5, Number 2/2004, pp. 000-000 ON THE CONTINUITY OF THE TRACE Victor ALEXANDRU*, Elena-Liliana POPESCU*, Nicolae POPESCU** *University of Bucharest, Academiei 14, Bucharest, Romania **Institute of Mathematics, Romanian Academy, PO Box 1-764, 70700, Bucharest, Romania Corresponding author: Nicolae POPESCU, Tel: +40213135403. E-mail: [email protected] The notion of trace for an element in an algebraic extension of a field can be extended for (not necessary algebraic) elements of same extensions of a complete valued field (see [2]). However there are two different ex- tensions of such notion, namely, the trace as was defined in [7] and continuous trace defined in [2]. According to Example 4.3 in [2] there results that these two kinds of trace seem to be generally different. In this paper, we try to put in light same fundamental relations between there concepts of trace. 1. PRELIMINARIES ⋅ 1. Let p be a prime number. As usual (see [A]) denote Q p the field of p-adic numbers and by = the p -adic module, normalized such that p 1/ p . Denote Q p a fixed algebraic closure of Q p and ⋅ continue to denote by the same symbol the unique extension of p-adic module to Q p . Furthermore, ⋅ ⋅ denote C p the completion of Q p with respect to module and denote by the same symbol the canonical extension of p-adic module to C p . 2. Denote G = Gal(Q /Q ) and consider on G the Krull topology. It is well know (see for p p ( ) example [APZ1]) that G is canonically isomorphic to Galcant C p / Q p , the group of all continuous Q p = ( ) authomorphism of C p . We shall assume G Galcant C p / Q p . One know that G , endowed with Krull topology, is a compact and totaly disconected group, and so it is equiped with a Haar measure χ , normalized such that χ()G = 1 . ∈ ()=σ{} () σ∈ Let x C p . Denote Ox x| G the orbit of x with respect to G and by H ()x = {}σ ∈G | σ ()x = x . Then H ()x is a closed subgroup of G and the map Ψ→() σ→σ () x :,GOx x identifies G / H ()x (endowed with the quotient topology) with Ox() (endowed to the topology induced by () C p ) (see [APZ1]. In such a way Ox is equiped with quotient Haar measure. The description of this measure, denoted also by χ is given in [2]. Also in [2] is show how one can consider χ as a ” p -adic measure”. Although this “ p -adic measure” is not a measure in usual sense, some ()→ χ functions fOx: Cp can be integrated with respect of . Then in [APZ2] (see also [7]) is defined the p- adic integral: _______________________ ** Member of the Roamnian Academy Victor ALEXANDRU, Elena-Liliana POPESCU, Nicolae POPESCU 2 ()= χ () Tr x ∫td t (1) O (x) called the “trace of x ”. However this integral do not exists always. If (1) is defined as usual, one says that ∈ the element x has a trace. For example any x Q p has a trace, namely one has: ()= 1 () Tr x tr () x (2) deg()x Qp x / Qp ()= [ [] ] () where deg x Qp x : Qp and tr () x means the usual trace i.e. the sum of all conjugates of x over Qp x / Qp Qp . In [PVZ] are given the general conditions such that the integral (1) there exists, i.e. x has a trace. Also () in [APZ2] are indicate some classes of elements x of C p such that Tr x is defined. For an element ∈ ![] [] x Cp , denote Qxp the closure of the polynomial ring Qp x in C p . ![] In [APZ1] is proved that Qxp is the smallest closed subfield of C p which contains x , and ![]∩= ![] " = ![] Qxppx Q K is called the algebraic part of Qx.p One has KQxxp, and for any closed subfield L ∈ = ![] of C p , there exists at least an element x L such that LQxp (see [1]). ∈ = {} One say that element x Cp has a pseudo-trace, if there exists a sequence S xn n of ![]∩ = {}() Qxpp Qsuch that lim xn x and that the sequence Tr xn of p -adic numbers is convergent. Then n n ()= ( ) one denote Tr cs x limTr xn the pseudo-trace of x with respect to the sequence S . Generally it is n possible that x has several pseudo-traces, or it has no any. If all the pseudo-traces of x are coincident, one say that x has a continuous trace and this will be denoted by Trc()x . If the function ! ∩→ Tr:[] Qppp x Q Q (3) ∈ [] () defined by (2) is continuous, then for any y Qp x there exists Trc y . By above considerations there results the fallowing result: ∈ Proposition 1. Assume that x C p has a pseudo-trace (respectively a continuous trace). Then any element of [] Qp x has a pseudo-trace (respectively a continuous trace). ∈ Up to now we do not know if any element x Cp has a pseudo-trace. By Example 4.3 of [2] there ∈ () () results the existence of elements x Cp such that Tr x do not exists, but Trc x is defined. In what follow we shall prove that the trace Tr()x is in fact a pseudo-trace (Theorem 2). Also in () () ![] Theorem 4 one give a result on the coincidence of Tr y and Tr c y for all elements of Qx.p 2. ANY TRACE IS A PSEUDO-TRACE ∈ () {} Theorem 2 Let x Cp be such that Tr x is defined. Then there exists a sequence xn n of elements of ![]∩ = Qxpp Q such that lim xn x and n ()= ( ) () Tr x limTr xn i.e. Trx is a pseudo-trace. n 3 On the continuity of the trace {}ε ≥ Proof. Let n n be a decreasing sequence of positive real numbers with zero limit. For any n 1 denote ()ε = {}σ ∈ −σ ()≤ ε H x, n G | | x x | . Then by [7], there results that ε lim n = 0 n []()ε G : H x, n Now let us fix a natural number n and denote = ()ε = {}∈ σ ()= σ ∈ (ε ) K n Fix H x, n z Q p such that z z for all H x, n ()⊆ (ε ) ⊆∩![] β ![]∩ Since H x H x, n , then KQxQnp p. Let n be a sequence in Qxpp Q such that β = ()(ε = β ε ) lim n x . For m large enough one has: H x, n H m , n . For such an m , denote n ∆()β =σ ()β − β σ∈ ()β ε mn,maxKH{} m m mn ,. According to Ax-Sen Theorem (see [5] or [6]), there exists xn ∈ Kn such that |x- xn| = |βm - xn| ≤ c∆(βm , Km)≤ cεn, where c is so-called Ax-Sen constant. Moreover one can choice xn in such way that Kn = Qp(xn).By above inequalities and (4) one obtain: x − x ε n ≤ c n ε ε |[G : H (x, n )]| |[G : H (x, n )]| Now let r = α χ ()()()ε An ∑ i H x, n / H x i=1 () χ()()()ε = be a Riemannean sum associated to Tr x (see [7]). Here H x, n / H x 1/ r , where = []()ε = () r G : H x, n deg xn . Then one has: () r r r cε 1 xn 1 n Tr(x ) − A = tr ()α − ∑α = ∑σ ()x − ∑α ≤ → 0 n n Qp n / Qp i i n i | r | i=1 | r | i=1 i=1 | r | σ ()σ () α α when 1 xn ,#, r xn are all conjugates of xn and 1,#, n belongs to correspondings balls ()σ ()ε ≤ ≤ B i x , n , 1 i r . This shows that = limTr(xn ) Tr(x) , i.e. x has a pseudo-trace. n ∈ () Corollary 3. Assume that x C p has a trace and the function (3) is continuous. Then Tr c x is defined and one has: Tr()x = Tr c ()x 3. MAIN RESULT ∈ {}() [] Let x Cp . According to [APZ1] there exists a sequence M n x n≥0 of polynomials of Qp x such that: ()= ()≤ ≥ i) deg M n x n , and M n x 1, n 0 . Victor ALEXANDRU, Elena-Liliana POPESCU, Nicolae POPESCU 4 ∈ ![] {} = ii) For any yQxp , there exists an unique sequence an ≥0 of p -adic numbers, such that liman 0 , n n and that: = () y ∑anM n x (4) n≥0 = () Moreover one has: y sup an M n x . n () ()() ≥ Now let us assume that Tr x is defined. Then Tr M n x is also defined for all n 0 , and so for ∈ [] y Qp x given by (4) one can consider the series: ()= () () S y ∑ anTr M n x (5) n≥0 Generaly one do not say anything on the convergence of such series. However one has the following results: ∈ () Theorem 4. Let x Cp be such that Tr x is defined. The following assertions on equivalent: ∈ ![] i) For any yQxp the series (5) is convergent. ii) The function ![]∩→ Tr:,() Qppp x Q Q y$ Tr y is continuous. ∈ ![] () () Then for any yQxp are defined both Tr y and Tr c y and one has: Tr()y = Tr c ()y = S ()y ⇒ ∈ [] Proof. i) ii) Since the series (5) is convergent for all y Qp x , there results that for a suitable real number M > 0 one has: ()()≤ Tr M n x M for all n ≥ 0 . Then the function ! → SQ:[]pp x Q , y$ Sy () ∈∩![] ()= () is continuous and lineary.
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