Automorphisms of double coverings of curves

We study automorphisms of curves that commute with each other. We prove that the order and the number of fixed points of one of them satisfy certain relations involving those of the other. Then, we specialize our results to the case of double coverings of curves. For instance, if the genus of the curve is at least 4γ + 2 and γ >= 1 (γ = the genus of the covered curve) we prove that the order of an automorphism is bounded above by 2γ + 1 (resp. 4γ + 2) provided it is prime (resp. it has at least five fixed points). We also improve Farkas' bound on the number of fixed points namely 4γ + 4 by showing that it involves the order of the automorphism except in the case of even order when such an improvement is obtained provided the automorphism and the γ-involution has at least one common fixed point. (author). 15 refs