Schneider, P.; Teitelbaum, J. \(p\)-adic Fourier theory. (English) Zbl 1028.11069 Doc. Math. 6, 447-481 (2001). 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; Zbl 1031.11505)] 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. Cited in 3 ReviewsCited in 27 Documents MSC: 11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.) 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 Keywords:Fourier transform; character group; locally analytic distribution; Mahler expansion; \(p\)-adic \(L\)-function Citations:Zbl 0158.30203; Zbl 1031.11505; Zbl 0119.03701 PDFBibTeX XMLCite \textit{P. Schneider} and \textit{J. Teitelbaum}, Doc. Math. 6, 447--481 (2001; Zbl 1028.11069) Full Text: arXiv EuDML