| In this paper, we will investigate the L-functions defined by the symmetric powers of the F-crystal associated to the family of exponential sums of xd+lambda x where lambda is the parameter. Using techniques of Dwork, this L-function will be a rational function and a bound for its degree will be given. When d = 3, by analyzing the singular point at infinity of the Picard-Fuchs differential equation, we will see that this L-function is a polynomial and compute its trivial factor explicitly; further, when the symmetric power is odd and smaller than the characteristic of the base field, we are able to determine the precise degree of this polynomial and a uniform quadratic lower bound for the Newton polygon.; These L-functions are closely related to Dwork's unit root L-functions. They also have arithmetic applications like that of equidistribution of angles of exponential sums and power moments. |
