MYRBERG’S NUMERICAL UNIFORMIZATION OF HYPERELLIPTIC CURVES

In this paper we derive an algorithm that computes, for a given algebraic hyperelliptic plane curve C of genus p, p> 1, defined by a polynomial y 2 = (x − λ1) · · · (x − λ2p+2), an approximation of a Fuchsian group G acting in the unit disk D such that C = D/G. The method allows us also to approximate the projection mapping π: D → D/G = C, thus giving a solution to the problem of numerical uniformization in the case of hyperelliptic curves. The method is based on work of P. J. Myrberg that appeared already in 1920 but did not get much attention at that time