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 p n =1 for some n [ges ]0} so that [ ] p is the union of cyclic (multiplicative) groups C p n of order p n (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).