p-adic Fourier Theory

By P. Schneider, Jeremy Teitelbaum. 2001

75 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/3fec6ba5aa28c4af021d508137a2eb859ead0dc9

How researchers see this paper

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In Geometry and Topology

1. M. Emerton
2. David Loeffler
3. Vytautas Paškūnas
4. Jan Kohlhaase
5. M. Kisin
6. K. Lai
7. T. Gee
8. Kiran S. Kedlaya

In Mathematical Physics

1. M. Emerton
2. David Loeffler
3. M. Kisin
4. K. Lai
5. Vytautas Paškūnas
6. Jan Kohlhaase
7. T. Gee
8. Frank Calegari

Citations in the corpus

• Locally analytic vectors in representations of locally p-adic analytic groups (2004, background): “s it remains to show that theˆ Zˆ 0 is representable by a strictly σ-affinoid rigid analytic space. Since Z0 is compact and locally L-analytic, it contains a finite index subgroup isomorphic to Od L. In [23] it is shown that OˆL is representable on affinoid spaces by a twisted form of the open unit disk,”
• The cohomology of locally analytic representations (2010, methodology): “Indeed, by [34], 220 Kohlhaase, The cohomology of locally analytic representations”
• Overconvergent Hilbert modular forms (2005, background): “rigid analytic space W over L [ ST , §2], and (4.5.1) gives rise to a map”
• Overconvergent algebraic automorphic forms (2009, methodology): “This can be extended to general Fp using the explicit description of ÔF,p given in [26].”