Higher-order Weierstrass weights of branch points on superelliptic curves

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In this paper we consider the problem of calculating the higher-order Weierstrass weight of the branch points of a superelliptic curve C. For any q>1, we give an exact formula for the q-weight of an affine branch point. We also find a formula for the q-weight of a point at infinity in the case where n and d are relatively prime. With these formulas, for any fixed n, we obtain an asymptotic formula for the ratio of the q-weight of the branch points, denoted BWq, to the total q-weight of points on the curve: →∞BWq/g(g-1)²(2q-1)²≥ n+1/3(n-1)²(2q-1)², with equality when the limit is taken such that (n,d)=1.