Comparison isomorphisms and arithmetic applications

点击翻译按钮获取中文摘要

It is universally accepted that geometric objects are difficult to understand and classify. In order to describe them one usues linear-algebraic approximations of them, known as cohomology groups. They appear as families of groups (or vector spaces) with certain linear (or semi-linear) structure given by filtrations, gradings and operators. There are a number of cohomology theories each one attaching, by a specific and usually rather complicated recipe, to a geometric object a family of cohomology groups with their respective structures. A. Grothendieck predicted that there should be certain ''mysterious functors", or comparison isomorphisms relating these cohomology theories. The importance of comparing cohomology groups coming from different cohomology theories, beyond the fact that it is a natural question, is that certain questions about the geometric object are best formulated in terms of one cohomology theory and easier answered in terms of another. Our project is to use comparison isomorphisms in order to, on the one hand, construct and describe overconvergent modular forms and their Galois representations and on the other hand describe the geometry of curves over local fields.