Cycles of covers

摘要

We initially consider an example of Flynn and Redmond, which gives an infinite family of curves to which Chabauty’s Theorem is not applicable, and which even resist solution by one application of a certain bielliptic covering technique. In this article, we shall consider a general context, of which this family is a special case, and in this general situation we shall prove that repeated application of bielliptic covers always results in a sequence of genus 2 curves which cycle after a finite number of repetitions. We shall also give an example which is resistant to repeated applications of the technique. Keywords Coverings of curves - Descent - Jacobians - Method of Chabauty Mathematics Subject Classification (2000) Primary: 11G30 - Secondary: 11G10 - 14H40 Communicated by U. Zannier.