Bounding the order of automorphisms of certain curves

We study upper bounds on the order of automorphisms of non-singular curves $X$ satisfying at least one of the following hypothesis: 1) $X$ is an $m$-sheeted covering of exactly one non-singular curve of genus $γ$, where $m$ is prime; 2) the center of the group of automorphisms of $X$ is non-trivial.

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