Abstract
The best known example of a non-archimedean period domain is the Drinfeld upper half space \(\Omega _E^d\) of dimension d - 1 associated to a finite extension E of Q p (complement of all E-rational hyperplanes in the projective space Pd-1). Drinfeld [D2] interpreted this rigid-analytic space as the generic fibre of a formal scheme over O E parametrizing certain p-divisible groups. He used this to p-adically uniformize certain Shimura curves (Cherednik’s theorem) and to construct highly nontrivial étale coverings of \(\Omega _E^d\). This report gives an account of joint work of Zink and myself [RZ] that generalizes the construction of Drinfeld (Sections 1–3). In the last two sections these results are put in a more general framework (Fontaine conjecture) and the problem of the computation of ℓ-adic cohomology is addressed (Kottwitz conjecture). In this report we return to the subject of Grothendieck’s talk at the Nice congress [G, esp. Section 5] where he stressed the relation between the local moduli of p-divisible groups and filtered Dieudonné modules.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Atiyah, M. F., and Bott, R., The Yang-Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. Lond. Ser. A 308, 523–615 (1982).
Carayol, H., Non-abelian Lubin-Tate theory, in Clozel, L., and Milne, J. S. (eds.), Automorphic forms, Shimura varieties and L-functions, vol. II, Perspect. in Math. 11, 15–40, Academic Press, Boston, MA, 1990.
Chai, C. L., and Norman, P., Singularities of the I’o(p)-level structure, J. Alg. Geom. 1, 251–278 (1992).
Cherednik, I. V., Uniformization of algebraic curves by discrete subgroups of PGL 2 (k w ) ..., Math. USSR-Sb. 29, 55–78 (1976).
Drinfeld, V. G., Elliptic modules, Math. USSR-Sb. 23, 561–592 (1974).
Drinfeld, V. G., Coverings of p-adic symmetric regions, Functional Anal. Appl. 10, 29–40 (1976).
Faltings, G., Mumford-Stabilitat in der algebraischen Geometrie, these proceedings.
Fontaine, J.-M., Groupes p-divisibles sur les corps locaux, Astérisque 47–48 (1977).
Fontaine, J.-M., Modules galoisiens, modules filtrés at anneaux de Barsotti-Tate, Asterisque 65 3–80 (1979).
Hales, Th., Hyperelliptic curves and harmonic analysis, Contemp. Math. 177, 137–169 (1994).
Hopkins, M., and Gross, B., Equivariant vector bundles on the Lubin-Tate moduli space, Contemp. Math. 158, 23–88 (1994).
Grothendieck, A., Groupes de Barsotti-Tate et cristaux, Actes du Congr. Int. Math., Nice (Paris), 1, Gauthier-Villars, 431–436 (1971).
Jong de, A. J., The moduli spaces of principally polarized abelian varieties with I’o(p)-level structure, J. Alg. Geom. 2, 667–688 (1993).
Katsura, T., and Oort, F., Supersingular abelian varieties of dimension two or three and class numbers, Adv. Stud. Pure Math. 10, 253–281 (1987).
Katz, N. M., Travaux de Dwork, Sem. Bourbaki 409, Lecture Notes in Math. 317, 167–200, Springer, Berlin and New York, 1973.
Kottwitz, R. E., Isocrystals with additional structure, Comp. Math. 56, 201–220 (1985).
Kottwitz, R. E., Shimura varieties and A-adic representations, in Clozel, L., and Milne, J. (eds.), Automorphic forms, Shimura varieties and L-functions, vol. I, Perspect. Math. 10, 161–209, Academic Press, Boston, MA, 1990.
Kottwitz, R. E., Points on some Shimura varieties over finite fields, J. Amer. Math. Soc., 5, 373–444 (1992).
Messing, W., The crystals associated to Barsotti-Tate groups ..., Lecture Notes in Math. 264, Springer, Berlin and New York, 1972.
Mustafin, G. A., Nonarchimedean uniformization, Math. USSR-Sb. 34, 187–214 (1978).
van der Put, M., and Voskuil, H., Symmetric spaces associated to split algebraic groups over a local field, J. Reine Angew. Math. 433, 69–100 (1992).
Rapoport, M., and Zink, Th., Period spaces for p-divisible groups, preprint Wuppertal 1994.
Schneider, P., and Stuhler, U., The cohomology of p-adic symmetric spaces, Invent. Math. 105, 47–122 (1991).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1995 Birkhäser Verlag, Basel, Switzerland
About this paper
Cite this paper
Rapoport, M. (1995). Non-Archimedian Period Domains. In: Chatterji, S.D. (eds) Proceedings of the International Congress of Mathematicians. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-9078-6_35
Download citation
DOI: https://doi.org/10.1007/978-3-0348-9078-6_35
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9897-3
Online ISBN: 978-3-0348-9078-6
eBook Packages: Springer Book Archive