Study On P-adic Galois Representations And Zeta Functions Of Curves Over Finite Fields

In this paper, we study several problems on p-adic Hodge theory, and L-functions associate to exponential sums.In Chapter 1, we briefly recall p-adic Hodge theory and give a simple proof of Hyodo’s result that H1g=H1st for a potentially semi-stable p-adic Galois representation over a finite extension of Qp.In Chapter 2, for any f∈Fq[χ], we extend Davis-Wan-Xiao’s result about the Newton polygons of L(f,χ,t). This extends Lazard’s result that B[m1,;m2] is a PID, where B [m1,m2] is the ring of Laurent series over discrete valuation field converging on m1 ≤v(T)≤m2. We extend this result to the complete discrete valuation ring case.In Chapter 3, we strengthen Davis-Wan-Xiao’s result about the p-adic Newton polygon of L-function of exponential sums. We also show that if a polynomial f con-tains a global permutation polynomial of degree great than 1 then lim N Pp (f) does not pâ†'∞ exists, which is one half of Wan’s conjecture.