Riemann surfaces of genus g with an automorphism of order p prime and p > g

The present work completes the classification of the compact Riemann surfaces of genus g with an analytic automorphism of order p (prime number) and p > g. More precisely, we construct a parameterization space for them, we compute their groups of uniformization and we compute their full automorphism groups. Also, we give affine equations in C-2 for special cases and some implications on the components of the singular locus of the moduli space of smooth curves of genus g.