Automorphisms with fixed points and Weierstrass points of compact Riemann surfaces
Let \(M\) be a compact Riemann surface of genus \(g \geq 2\) and \(T\) be a conformal automorphism of order (not necessarily prime!) with \(t\) fixed points \(P_ 1,\dots,P_ t\). For each \(m \in \{1,\dots,t\}\) and for every local coordinate \(z\) at \(P_ m\) the integer \(\nu_ m\) such that \(T^{-1}\) is given by \(z \to e^{2 \pi i \nu_ m/N}z\) near \(P_ m\) is uniquely defined and must be relatively prime to \(N\). Let \(g'\) be the genus of \(M':=M/ \langle T \rangle\). The first result of this article gives classification of the compact Riemann surfaces if \(T\) has a fixed point which is not a 1-Weierstrass point according to -- the number \(t\) of fixed points, -- the relation between \(g\), \(g'\) and \(N\), -- the rotation constants \(\{\nu_ 1,\dots,\nu_ t\}\). The next two theorems give the corresponding next classifications for the cases in which \(T\) has a fixed point \(P\) which is a normal Weierstrass point, i.e. the sequence of gaps in \(P\) is \(1,2,\dots,g-1\), \(g+1\), or a hyperelliptic Weierstrass point, i.e. the sequence of gaps in \(P\) is \(1,3,5,\dots,2g-1\). In theorem 4, 5 and 6 are presented conditions for fixed points of \(T\) to be \(q\)-Weierstrass points, \(q \geq 2\), for some surfaces which are classified in theorem 1, 2 and 3. These results are supplemented by some interesting examples. In the next paragraphs the dimension of the vector space \(H^ q_ k\) of all holomorphic \(q\)-differentials \(\theta\) with \(T(\theta)=e^{2 \pi ik/N} \theta\) is computed, and in this connection some examples are given. The main improvement in this very interesting paper -- in my opinion -- is to avoid the assumption on the order \(N\) to be prime.
scientific article; zbMATH DE number 515585