On the p-adic Geometry of Modular Curves

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ABSTRACT Robert Coleman University of California, Berkeley 98 01389 Professor Robert Coleman plans to continue his study of classical modular forms with particular attention to the Mazur-Gouvea conjecture. The project revolves around a recent discovery of the investigator's of a new geometric structure on certain modular forms. Coleman's "eigencurve" provides a new view of some of the most studied mathematical objects in modern number theory. If only a part of the planned project is completed, it could change the face of 150 year old branch of Number Theory. The experts are eagerly awaiting the first reports from this work. When mathematicians first used the techniques of calculus to study modular forms, they were interested in their interesting analytical properties. Modular forms were first seen as mathematical functions with incredibly strong symmetries. While these aspects of modular forms are still of intense interest, modern number theorists are mainly interested in the arithmetical information that modular forms contain. Modular forms encode information about the size of other mathematical constructions that at first (and even second) glance are totally unrelated. Numbers you can obtain from modular forms have a meaning quite separate from the forms themselves. Basically, modular forms count things. A large part of Number Theory in the second half of the twentieth century has been devoted to understanding why and how this strange counting works. Professor Coleman has some very new ideas about modular forms that should increase our understanding of a major mathematical mystery.