C e n t r e  d e  r e c h e r c h e s  m a t h é m a t i q u e s

Dans le cadre de l'Année thématique 1998-1999

As part of the 1998-1999 Theme Year

Chaire André Aisenstadt 1998-1999

In his Montreal Notes, Grothendieck included a letter to Barsotti in which he announes a conjecture. In this course I will present a proof for this conjecture.

To every p-divisible group (also called a Barsotti-Tate group) in characterstic p one can attach a discrete invariant, its Newton Polygon. Grothendieck showed that under specialization Newton Polygons "go up"; his conjecture says that, conversely, for a given p-divisible group, and a given "lower" Newton Polygon such a specialization should be possible.

My main interest comes from the study of moduli spaces of abelian varieties in positive characteristic. These spaces have an incredibly rich structure. Several properties can be studied by "going to the boundary" (a useful method in algebraic geometry), by which we mean in this case that the abelian variety does not degenerate at all, but that the p-structure becomes more special. The Grothendieck conjecture (in the polarized case) tells us exactly which strata should be in the boundery of a given stratum.

In proving this conjecture one encounters the problem that the variation of the Newton Polygon under a deformation is very difficult to follow (this is why it took us so long to give a proof for this reasonable conjecture). I shall explain this in several ways in my course. This causes that a direct approach to the problem does not seem to lead anywhere. My proof consists of several very different stages, each of them developing very interesting new techniques.

In my course I shall define, study and prove:

• Finite group schemes, p-divisible groups and deformation theory.
• Moduli of abelian varieties.
• A conjecture by Manin ("every symmetric Newton Polygon is algebraizable").
• This conjecture by Grothendieck.
• Techniques involving "catalogues" (a new, useful notion).
• Study various strata in many different moduli spaces and catalogues.
• Determine the structure of Newton Polygon strata (dimension, irreducibility, etc.).

In my first course (May 14) I shall review definitions and results. In my second (May 18) I shall sketch ingredients used and the main lines of the proofs. The first two lectures give a complete survey of methods and results. I shall try to make these 6 lectures accessible for a wide audience (although, on some occasions I am using advanced methods).

In each of the last four courses (May 20-28) I shall concentrate on a particular, interesting aspect. Catalogues will be introduced and discussed. Some combinatorial problems have to be solved. A mixture of deep algebraic geometry and easy computations will be presented, eventually leading to a complete understanding of this complex of problems, giving a rather precise description of the spaces involved.

A. Grothendieck, Groupes de Barsotti-Tate et cristaux de Dieudonné. Sém. Math. Sup., Univ. Montréal, Presses Univ. Montréal, 1974.

XX mois 1998, webmaster@CRM.UMontreal.CA