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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 √Čtud. 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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