  We study the variation of the trace of the Frobenius endomorphism associated
to a cyclic trigonal curve of genus g over a field of q elements as the curve
varies in an irreducible component of the moduli space. We show that for q
fixed and g increasing, the limiting distribution of the trace of the Frobenius
equals the sum of q+1 independent random variables taking the value 0 with
probability 2/(q+2) and 1, e^{(2pi i)/3}, e^{(4pi i)/3} each with probability
q/(3(q+2)). This extends the work of Kurlberg and Rudnick who considered the
same limit for hyperelliptic curves. We also show that when both g and q go to
infinity, the normalized trace has a standard complex Gaussian distribution and
how to generalize these results to p-fold covers of the projective line.
