p-adic Fourier Theory, by Peter Schneider and Jeremy Teitelbaum

Abstract: In this paper we generalize work of Amice and Lazard from the early sixties. Amice determined the dual of the space of locally Qp-analytic functions on Zp and showed that it is isomorphic to the ring of rigid functions on the open unit disk over Cp. 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 Qp. 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 Cp, 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=Qp 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.

Peter Schneider
Jeremy Teitelbaum



Peter Schneider and Jeremy Teitelbaum <jeremy@uic.edu, pschnei@math.uni-muenster.de>