P-adic Banach Spaces and Families of Modular Forms, by Robert F. Coleman

This paper has appeared in Invent. Math. 127 (1997) no. 3, 417-479, and so the dvi version has been removed. This paper is a new version of preprint 0007. In this paper, we prove qualitative versions of the Gouv\^ea-Mazur conjectures on families of modular forms of non-zero slope. We do this by first extending Serre's $p$-adic Banach space theory from spaces over complete normed fields to modules over complete normed algebras.

Robert F. Coleman <coleman@math.berkeley,edu>