2001/02/01 by Peter Schneider, Schneider, Peter, Jeremy Teitelbaum +1 · 2 citations
Mathematics · #11S31 #11S80 #14G22 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT) #advanced mathematical theories #math.NT #msc:11S31 #msc:11S80 #msc:14G22
paper · pdf · doi:10.48550/arxiv.math/0102012
arxiv created 2001/02/01 · openalex publication_date 2001/02/01 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we generalize work of Amice and Lazard from the early (nineteen) 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.