Families of curves with Galois action and their L-functions

We generalise results of Chris Hall on the L-function of curves E over characteristic p function fields K, by using equivariant L-functions and cohomologically trivial modules. In fact, K will be the rational function field over a fixed finite field most of the time. The curves which we can treat are superelliptic curves which come as Galois covers of prime degree of the projective line over K. We are thus able to determine the degree of the L-function (which is a polynomial in our situation), and sometimes we get upper bounds on the analytic rank.