Abstract
We investigate the temperate p-adic Riemann–Hilbert functor defined by André on abelian varieties that are analytic tori. We show that this functor induces an equivalence between the category of discrete and integral representation of the temperate fundamental group of the torus on finite dimensional \({{\mathbb C}_{p}}\) -vector spaces as well as the category of homogeneous p-adic vector bundles on the torus.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Y. André, Period mappings and differential equations. From \({{\mathbb C}}\) to \({{\mathbb C} _{p}}\) , Tôhoku-Hokkaidô lectures in arithmetic geometry (2003), 246 p.
V. Berkovich, Spectral theory and analytic geometry over non-Archimedean fields, Mathematical Surveys and Monographs 33, A.M.S., (1990).
V. Berkovich, Étale cohomology for non-Archimedean analytic spaces, Publ. Math. IHES, tome 33, (1993).
Deninger C., Werner A.: Vector bundles on p-adic curves and parallel transport. Ann. Scient. Éc. Norm. Sup., 4e série 38, 553–597 (2005)
Faltings G.: Semi-stable vector bundles on Mumford curves. Invent. Math. 74, 199–212 (1983)
Herz G.: Representations and Vector Bundles on Mumford Curves. Abhandlungen Math. Sem. Universität Hamburg 77, 81–96 (2007)
T. Ludsteck, p-adic vector bundles on curves and abelian varieties and representations of the fundamental group, dissertation, Universität Stuttgart (2008).
Ludsteck T.: Homogeneous p-adic vector bundles on abelian varieties that are analytic tori. Arch. Math. 92, 585–592 (2009)
Mukai S.: Semi-homogeneous vector bundles on an abelian variety. J. Math. Kyoto Univ. 18, 239–272 (1978)
van der Put M.: Reversat M. Fibres vectoriels semi-stables sur un courbe de Mumford. Math. Ann. 273, 573–600 (1986)
van der Put M., Reversat M.: Fibres vectoriels homogenes sur les variétés abéliennes qui sont des tores analytiques (rigides). Manuscripta Math. 60, 71–88 (1988)
Author information
Authors and Affiliations
Corresponding author
Additional information
The author wants to thank C. Deninger, C. Florentino, G. Herz, and A. Werner for useful discussions on vector bundles and the anonymous referee for useful comments. The paper was partly prepared in the Math Department of the University of Stuttgart and during a visit at IST Lisbon. The author wants to thank both institutions for their support and hospitality.
Rights and permissions
About this article
Cite this article
Ludsteck, T. Homogeneous p-adic vector bundles on abelian varieties that are analytic tori II. Arch. Math. 93, 323–330 (2009). https://doi.org/10.1007/s00013-009-0048-x
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00013-009-0048-x