THE AUTOMORPHISM GROUP OF A p-ADIC CONVOLUTION ALGEBRA

Abstract

Throughout ℤ_p and ℚ_p will, respectively, denote the ring of p-adic integers and the field of p-adic numbers (for p prime). We denote by [Copf ]_p the completion of the algebraic closure of ℚ_p with respect to the p-adic metric. Let v_p denote the p-adic valuation of [Copf ]_p normalised so that v_p(p)=1. Put [ ]_p ={ω∈[Copf ]_p[mid ] ω^pn=1 for some n[ges ]0} so that [ ]_p is the union of cyclic (multiplicative) groups C_pⁿ of order pⁿ (for n[ges ]0).

Let UD(ℤ_p) denote the [Copf ]_p-algebra of all uniformly differentiable functions f[ratio ]ℤ_p→[Copf ]_p under pointwise addition and convolution multiplication *, where for f, g∈UD(ℤ_p) and z∈ℤ_p we have

formula here

the summation being restricted to i, j with v_p(i+j−z)[ges ]n.

This situation is a starting point for p-adic Fourier analysis on ℤ_p, the analogy with the classical (complex) theory being substantially complicated by the absence of a p-adic valued Haar measure on ℤ_p (see [5, 6] for further details).