Character sums determined by low degree isogenies of elliptic curves

Abstract

We generalize the character sum formulas of McLeman-Rasmussen attached to isogenies of elliptic curves in positive characteristic. Two improvements are given: sums are evaluated for isogenies of degree greater than two, and over arbitrary finite fields. We prove a transfer formula to evaluate such sums quickly over the domain curve and use this to evaluate the character sums attached to several standard families of isogenies of low degree.