Zeta functions of totally ramified p-covers of the projective line

In this paper we prove that there exists a Zariski dense open subset U defined over the rationals ℚ in the space of all one-variable rational functions with prescribed l poles with fixed orders, such that for every geometric point f in u (ℚ), the L-function of the exponential sum of f at a prime p has Newton polygon approaching the Hodge polygon as p approaches infinity. As an application to algebraic geometry, we prove that the p-adic Newton polygon of the zeta function of a p-cover of the projective line totally ramified at arbitrary í points with prescribed orders has an asymptotic generic lower bound.