1 p - adic Fourier theory

In the early sixties, Amice ([Am1], [Am2]) studied the space of K-valued, locally analytic functions on Z Z p and formulated a complete description of its dual, the ring of K-valued, locally Q p-analytic distributions on Z Z p , when K is a complete subfield of C p. She found an isomorphism between the ring of distributions and the space of global functions on a rigid variety over K parameterizing K-valued, locally analytic characters of Z Z p. This rigid variety is in fact the open unit disk, a point z of C p with |z| < 1 corresponding to the locally Q p-analytic character κ z (a) = (1 + z) a for a ∈ Z Z p. The rigid function F λ corresponding to a distribution λ is determined by the formula F λ (z) = λ(κ z). Amice's description of the ring of Q p-analytic distributions was complemented by results of Lazard ([Laz]). He described a divisor theory for the ring of functions on the open disk and proved that, when K is spherically complete, the classes of closed, finitely generated, and principal ideals in this ring coincide. In this paper we generalize the work of Amice and Lazard by studying the space C an (o, K) of K-valued, locally L-analytic functions on o, and the corresponding ring of distributions D(o, K), when Q p ⊆ L ⊆ K ⊆ C p with L finite over Q p and K complete and o = o L the additive group of the ring of integers in L. To clarify this, observe that, as a Q p-analytic manifold, the ring o is a product of [L : Q p ] copies of Z Z p. The K-valued, Q p-analytic functions on o are thus given locally by power series in [L : Q p ] variables, with coefficients in K. The L-analytic functions in C an (o, K) are given locally by power series in one variable; they form a subspace of the Q p-analytic functions cut out by a set of " Cauchy-Riemann " differential equations. These facts are treated in Section 1. Like Amice, we develop a Fourier theory for the locally L-analytic functions on o. We construct (Section 2) a rigid group variety o, defined over L, whose closed points z in a field K parameterize K-valued locally L-analytic characters κ z of o. We …