Covariant Classification of Two-Dimensional Smooth Commutative Formal Groups over an Algebraically Closed Field of Positive Characteristic

摘要

Some generalities in the covariant formal group theory are introduced. In this theory, the central place is taken by the group of p typical curves in a formal group. The covariant technique gives rise to some new isomorphism invariants of finite dimensional smooth commutative formal groups over an algebraically closed field of positive characteristic. For dimension two, an explicit classification up to isomorphism is obtained. The methods which have led to this result may in principle be extended to higher dimensions. The covariant classification of the two dimensional formal groups is compared with the existing contravariant classification of the same class of formal groups, due to Manin. The covariant classification turns out to be finer and more explicit than its contravariant counterpart. Nevertheless, the covariant techniques are relatively elementary, the preformation of the classification is less laborious and the arrangement of the results is more clear. The concept of a special module, which plays an important role in the contravariant theory, is generalized. This generalization leads to a refinement of the contravariant classification. It seems hard, however, to make the contravariant classification more explicit. Finally, an application of the covariant classification theory in algebraic geometry is given: The computation of the isomorphism invariants of the formal group arising from the Jacobian of a generic curve of genus two over an algebraically closed field of characteristic 2. These isomorphism invariants are, of course, birational invariants.