Duality theory of $p$-adic Hopf algebras

Document Type : Research Paper

Author

University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki 305-8571 Japan

Abstract

We show the monoidal functoriality of Schikhof duality, and cultivate new duality theory of $p$-adic Hopf algebras. Through the duality, we introduce two sorts of $p$-adic Pontryagin dualities. One is a duality between discrete Abelian groups and affine formal group schemes of specific type, and the other one is a duality between profinite Abelian groups and analytic groups of specific type. We extend Amice transform to a $p$-adic Fourier transform compatible with the second $p$-adic Pontryagin duality. As applications, we give explicit presentations of a universal family of irreducible $p$-adic unitary Banach representations of the open unit disc of the general linear group and its $q$-deformation in the case of dimension $2$.

Keywords


[1] Berkovich, V.G., Spectral Theory and Analytic Geometry over non-Archimedean Fields", Mathematical Surveys and Monographs 33, Amer. Math. Soc. 1990.
[2] Bosch, S., Guntzer, U., and Remmert, R., Non-Archimedean Analysis: A Systematic Approach to Rigid Analytic Geometry", Springer, 1984.
[3] Mahler, K., An interpolation series for continuous functions of a p-adic variable, J. Reine Angew. Math. 199 (1958), 23-34.
[4] Mihara, T., Characterisation of the Berkovich spectrum of the Banach algebra of bounded continuous functions, Doc. Math. 19 (2014), 769-799.
[5] Monna, A.F., Analyse Non-Archimedienne", Springer, 1970.
[6] Schikhof, W.H., A perfect duality between p-adic Banach spaces and compactoids, Indag. Math. (N.S.) 6(3) (1995), 325-339.
[7] Schneider, P., Non-Archimedean Functional Analysis", Springer, 2002.
[8] Demazure, M. and Grothendieck, A., Seminaire de Geometrie Algebrique du Bois Marie - 1962-64 - Schemas en groupes - SGA3 - Tome 1", Springer, 1970.
[9] Schneider, P. and Teitelbaum, J., Banach space representations and Iwasawa theory, Israel J. Math. 127(1) (2002), 359-380.