Sytems of polynomials with integral coefficients are studied in number theory. It is often very difficult to find the integral solutions of such a system. Instead, we simultaneouly deal with the solutions in various commutative rings. The solutions in various rings forms a scheme, which provides geometric methods for studying the system of polynomials.
Hasse-Weil L-functions of an arithmetic scheme are defined using geometric cohomology. I am interested in the special values of these L-functions. The special values are believed to be related to motivic cohomologies, which are defined by using algebraic cycles or algebraic K-theory and are usually they hard to know explicitly. It is a very deep prediction to expect that such abstract objects should be related to more concrete L-functions.
It is expected that motives are related to automorphic representations. The expectation is important since we have various methods for studying automorhic L-functions. Some relations between motives and automorphic representations are realized by using Shimura varieties. In a joint work with Satishi Kondo, I have proved a equality relating motivic cohomologies and special values of Hasse-Weil L-functions for some function field analogues of Shimura varieties.
Hasse-Weil L-functions are defined via some Galois representations. We need to study such Galois representations. For some technical reasons it is important to study Galois representations of p-adic fields with p-adic coefficients, and p-adic Hodge theory provides some tools for studying such representations. For recent years there have been much development in p-adic Hodge theory, and a lot of beautiful theories have been constructed. However the theory is not fully established and many aspects of the theory remains mysterious. I am now trying to make the integral p-adic Hodge theory more convenient for practical study.