$p$-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.