Resumen de Trace functions and Galois invariant p-adic measures

Marian Vâjâitu, Alexandru Zaharescu

• Let p be a prime number, Q~ the fleid of p-adic numbera, Q5 a fixed algebraic closure of Q~, and C15 the completion of Q~ with respect te the p-adic valuation. The notion of a trace fuoction asaeciated te an element T from C~ was introduced and investigated lo APZ3]. If T it algebraic over Q~, and L it a finite field extension of Q~ contained lo Q~ such that T lies in L, then the p-adic number (1) Tr T TrL¡Qc(T) [L:Q~] depends en T only, and lot en L. The significance of Tr T it that of the average value of the conjugates of T over Q~. This idea of taking the average value rather than the sum of conjugates, may alto be applied, as thown lo [APZ3], te a rich class of elements T from C~ which are transcendental over Q~. Given an element T of ~ ene takes ita Galois orbit C(T) {cr(T) ce ~ Gal~,0e(G~/Q~)}, which it a compact subset of C~, and ene considers the p-adic Haar distribution irí defined en 0(T). Then, by analogy with the case when T it algebraic and Tr T it given by the average value of the conjugates of T over Q~, ene defines Tr T