Oberwolfach Seminar: Finite Group Schemes and p-divisible Groups

Date:

May 15th - May 21st, 2005

Organizers:

Fabrizio Andreatta, Padova

Brian Conrad, Michigan

Rene Schoof, Rome

Program:

Finite group schemes and p-divisible groups are key notions in number theory and arithmetic algebraic geometry. They play an important part in the theory of abelian varieties. The proofs of important number theoretic results such as the Mordell Conjecture (Faltings' Theorem) and the Shimura-Taniyama conjecture make extensive use of the theory of group schemes and p-divisible groups.

In this seminar we present the basic theory of finite group schmemes and p-divisible groups, as well as the theory of Dieudonne' modules and Tate's description of local p-divisible groups. Finite group schemes and p-divisible groups naturally arise in the context of abelian varieties, and we give several applications to the theory of abelian varieties. We deduce Tate's local proof of a formula of Shimura-Taniyama that is fundamental in the theory of complex multiplication, and we obtain a p-adic version of Tate's isogeny theorem on abelian varieties over finite fields. In addition we show how p-divisible groups are used to understand the p-part of the Honda-Tate classification of simple abelian varieties over finite fields (up to isogeny).

Prerequisites:

Basic commutative algebra and algebraic geometry; familiarity with the basics in the theory of schemes and with the properties of abelian varieties (as in Mumford's book "Abelian Varieties", Ch.2); some familiarity with the theory of central simple algebras.

Literature:

• Tate, J.T.: Finite flat group schemes, in Cornell, G., Silverman, J., Stevens, G.: "Modular forms and Fermat's Last Theorem", Springer-Verlag, New York 1997, pp. 121-154.
• Tate, J.T.: p-Divisible groups, in "Proceedings of a conference on local fields, Springer-Verlag, Berlin 1967, pp. 158-183
• Waterhouse, W.: "Introduction to affine group schemes", GTM 66, Springer-Verlag, New York 1979.
• Mumford, D.: "Abelian varieties", TIFR Bombay, Oxford University Press, Bombay 1970.
• Fontaine, J.-M.: "Groupes p-divisibles sur les corps locaux", Asterisque 47-48, Soc. Math. France, Paris 1977.
• Hartshorne, R.: "Algebraic Geometry", GTM 52, Springer-Verlag, New York 1977.
• Conrad, B. and Skinner, C.: 2004-2005 VIGRE Number Theory Working Group, seminar notes at the VIGRE web site of B. Conrad.
• Conrad, B. and Lieblich, M.: Galois representations arising from p-divisible groups, rough draft at the web site of B. Conrad.
• Milne, J.S.: Abelian varieties, in Cornell, G. and Silverman, J.: "Arithmetic Geometry", Springer-Verlag 1986.

Deadline for applications:

April 1st, 2005

The seminars take place at the Mathematisches Forschungsinstitut Oberwolfach. The number of participants is restricted to 24. The Institute covers accommodation and food. Travel expenses cannot be reimbursed. Applications including

• full name and address, including e-mail address
• present position, university
• name of supervisor of Ph.D. thesis
• a short summary of previous work and interest

should be sent as hard copy or by e-mail (.ps or .pdf file) to:

Prof. Dr. Gert-Martin Greuel
Universität Kaiserslautern
Fachbereich Mathematik
Erwin Schrödingerstr.
67663 Kaiserslautern, Germany
.