Selmer Group Estimates Arising from the Existence of Canonical Subgroups

Compositio Mathematica 71 (1989) 121-137.

Andrew Klapper, 779A Anderson Hall, Dept. of Computer Science, University of Kentucky, Lexington, KY, 40506-0046, klapper at

Abstract Generalizing the work of Lubin in the one dimensional case, conditions are found for the existence of canonical subgroups of finite height commutative formal groups of arbitrary dimension over local rings of mixed characteristic (0,p). These are p-torsion subgroups which are optimally close to being kernels of Frobenius homomorphisms. The F_p ranks of the first flat cohomology groups of these canonical subgroups are found. These results are applied to the estimation of the F_p rank of the Selmer group of an Abelian variety over a global number field of characteristic zero, and the lim sup of these ranks as the Abelian variety varies in an isogeny class.