Cyclic automorphisms beyond the canonical bound

Let X be a smooth projective curve of genus g >= 2 over an algebraically closed field K, and suppose that Aut(X) contains a cyclic subgroup G of order N > 2g - 2. The cases N >= 2g + 1 are known. We treat the two boundary values N = 2g and N = 2g - 1. In the tame case we obtain an explicit finite li