Applications of p-Adic Methods to Group Theory

Abstract In the late nineteenth century, Sophus Lie studied ‘transformation groups’ in an attempt to understand various geometries from more group-theoretic point of view. His pioneering work forged what has been known for nearly a century as the theory of Lie groups, i.e. topological groups with the underlying structure of a real manifold with respect to which the group operations are analytic. The importance of the theory of Lie groups can be judged by the diversity of its applications in topology, differential geometry and arithmetic. There have also been a host of parallel theories where the underlying real manifold is replaced by such structures as algebraic varieties, schemes and p-adic manifolds. It is the last of these which lies at the heart of this chapter. Such groups were encountered for the first time in 1907 in the writings of Hensel (17] on p-adic analytic functions. By 1936 they had appeared in work of Weil and Lutz on p-adic elliptic curves (see (50] and (29]).