Explicit reciprocity laws for p-adic fields

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Local class field theory describes the abelian extensions of a local field K in terms of the structure of the unit group K*:- For every finite abelian extension L of K there exists a canonical isomomorphism r_{L/K}: Gal(L/K) -> K* / N_{L/K}L*, the local reciprocity map. It has been a long-standing problem to find an explicit description of r_{L/K} - the first results in this direction go back to Kummer and Hasse and Artin.Nowadays it is known that the local reciprocity map can be alternatively constructed as a cup product pairing in Galois cohomology. Using this description, I hope to extend the classical results to the case when K is a ramified extension of Q_p via the theory of (phi,Gamma)-modules. This theory associates to a representation of the absolute Galois group of K a module over a certain ring of power series which converge on some annulus on the open unit p-adic disk, and in the case when K is unramified one can use it to get a simple and conceptual proof of the explicit reciprocity formula of Hasse and Artin. The main difficulty in the ramified case is the more difficult structure of the module of power series as a Galois module (which can be described using Lubin-Tate theory). A description of the local reciprocity map in terms of (phi,Gamma)-modules would have far-reaching consequences. In the case when K is equal to Q_p, Colmez discovered that it is the link for proving the p-adic Langlands correspondence for 2-dimensional representations of the absolute Galois group of Q_p. One of my long-term goals is the proof of a p-adic Langlands correspondence for 2-dimensional representations of the absolute Galois group of K, when K is an arbitrary extension of Q_p. Using the theory of (phi,Gamma)-modules, it is possible to generalize the notion of an explicit reciprocity law to p-adic representations of the absolute Galois group of K. In this setting, an explicit reciprocity law gives an explicit description of the (phi,Gamma)-module of a representation V in terms of the dual Bloch-Kato exponential map. When V comes from a p-divisible formal group, then this exponential map agrees with the exponential map from the tangent space of the formal group to the first Galois cohomology group with coefficients in the Tate module. The proof of these general reciprocity laws explores the interplay between cyclotomic Iwasawa theory and p-adic Hodge theory. I am interested in generalizing the classical theory to a higher dimensional local fields of mixed characteristic (0,p). These fields can be described in terms of power series over ordinary local fields, and they arise naturally as domains for q-expansions of p-adic modular forms. In previous work I have extended the construction of the Bloch-Kato exponential to this case, and so far I have used it to prove explicit reciprocity laws for general p-adic representations. The next step is the construction of the Perrin-Riou logarithm map, which can be seen as an analogue for de Rham representations of Coleman's logarithmic derivatives describing norm-compatible systems in a certain Iwasawa tower, and its description in terms of the higher exponential map. This logarithm map should have interesting arithmetic applications. In particular, it should be possible to construct p-adic L-functions by applying it to Kato's Euler system.