Local statistics for zeros of Artin-Schreier -functions

We study the local statistics of zeros of L L -functions attached to Artin-Scheier curves over finite fields. We consider three families of Artin-Schreier L L -functions: the ordinary, polynomial (the p p -rank 0 stratum) and odd-polynomial families. We compute the 1-level zero-density of the first and third families and the 2-level density of the second family for test functions with Fourier transform supported in a suitable interval. In each case we obtain agreement with a unitary or symplectic random matrix model.