Abstract
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 $$\mathbb{C}^{2}$$ for special cases and some implications on the components of the singular locus of the moduli space of smooth curves of genus g.
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