$p$-adic Fourier theory

Abstract

In this paper we generalize work of Amice and Lazard from the early sixties. Amice determined the dual of the space of locally Q_p -analytic functions on Z_p and showed that it is isomorphic to the ring of rigid functions on the open unit disk over C_p . Lazard showed that this ring has a divisor theory and that the classes of closed, finitely generated, and principal ideals in this ring coincide. We study the space of locally L -analytic functions on the ring of integers in L , where L is a finite extension of Q_p . We show that the dual of this space is a ring isomorphic to the ring of rigid functions on a certain rigid variety X . We show that the variety X is isomorphic to the open unit disk over C_p , but not over any discretely valued extension field of L ; it is a “twisted form” of the open unit disk. In the ring of functions on X , the classes of closed, finitely generated, and invertible ideals coincide, but unless L=Q_p not all finitely generated ideals are principal. The paper uses Lubin–Tate theory and results on p-adic Hodge theory. We give several applications, including one to the construction of p-adic L -functions for supersingular elliptic curves.