Abstract: | We are interested in cyclic covers of the projective line which are totally ramified at all of their branch points. We begin with curves given by an equation of the form , where is a polynomial of degree . Under a mild hypothesis, it is easy to see that all of the branch points must be Weierstrass points. Our main problem is to find the total Weierstrass weight of these points, . We obtain a lower bound for , which we show is exact if and are relatively prime. As a fraction of the total Weierstrass weight of all points on the curve, we get the following particularly nice asymptotic formula (as well as an interesting exact formula): where is the genus of the curve. In the case that (cyclic trigonal curves), we are able to show in most cases that for sufficiently large primes , the branch points and the non-branch Weierstrass points remain distinct modulo . |