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$$p$$-adic Fourier theory. (English) Zbl 1028.11069
Summary: In this paper we generalize work of Y. Amice [Bull. Soc. Math. Fr. 92, 117-180 (1964; Zbl 0158.30203), Proc. Conf. on $$p$$-adic analysis, Nijmegen 1978, 1–15 (1978)] and M. Lazard [Publ. Math., Inst. Hautes Étud. Sci. 14, 223–251 (1962; Zbl 0119.03701)] from the early sixties. Amice determined the dual of the space of locally $$\mathbb{Q}_p$$-analytic functions on $$\mathbb{Z}_p$$ and showed that it is isomorphic to the ring of rigid functions on the open unit disk over $$\mathbb{C}_p$$. 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 $$\mathbb{Q}_p$$. 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 $$\mathbb{C}_p$$, 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 $$L=\mathbb{Q}_p$$ 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.

##### MSC:
 11S31 Class field theory; $$p$$-adic formal groups 11G40 $$L$$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture 14G22 Rigid analytic geometry 46S10 Functional analysis over fields other than $$\mathbb{R}$$ or $$\mathbb{C}$$ or the quaternions; non-Archimedean functional analysis
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