Noether’s theorem for plane domains with hyperelliptic double

This paper is motivated by the observation that Noether’s theorem for quadratic differentials fails for hyperelliptic Riemann surfaces. In this paper we provide an appropriate substitute for Noether’s theorem which is valid for plane domains with hyperelliptic double. Our result is somewhat more explicit than Noether’s, and, in contrast with the case of nonhyperelliptic surfaces, it provides a basis for the (even) quadratic differentials which holds globally for all domains with hyperelliptic double. An important fact which plays a significant role in these considerations is that no two normal differentials of the first kind can have a common zero on a domain with hyperelliptic double. © 1977 Pacific Journal of Mathematics. All rights reserved.