Some Work On P-adic Galois Representations

In this thesis, we mainly discuss some work on p-adic Galois representation theory.In the first chapter, we review several classical results of p-adic Langlands program (the Qp case), which reveals the connection between the p-adic repre-sentations of GQp and admissible representations of some Qp-analytic groups. On the Galois side, we review Fontaine's theory of (φ,Γ)-modules. On the analytic side, we review Schneider and Teitelbaum's notion of admissible p-adic Banach representations. At last, we review Colmez's result on (φ,Γ)-modules and repre-sentations of the mirabolic subgroup of GL2(Qp) as an example to see how to build p-adic Langlands correspondence.In the second chapter, we use the theory of Lubin-Tate formal groups to make a generalization of Fontaine's theory of (φ,Γ)-modules, i.e., we get the notion of (φ,O*L)-modules, where L is a finite extension of Qp and thisφis the corresponding Frobenius.In the third chapter, we work on p-adic analysis. Combined with the theory of Lubin-Tate formal groups, we get characterization of the space of continuous functions on OL and its dual space (the space of measures on OL). And we get some interesting properties of them.In the fourth chapter, we review Dieudonne-Manin's classification theorem ofφ-isocrystals and give a new and simple proof of it. From the aspect of p-adic Hodge theory, one can relate some important representations (e.g crystalline representations) of GK toφ-isocrystals and thus this theorem can help us study p-adic Galois representations.