Elliptic subcovers of a curve of genus 2. I. The isogeny defect

Abstract

Let C / K be a curve of genus 2 over an arbitrary field K. The purpose of this paper is to relate the set of equivalence classes of elliptic subcovers of C / K to the set of primitive representations of an intrinsic quadratic form $$q_C$$ called the refined Humbert invariant. Furthermore, the discriminant of this quadratic form is determined here by relating it to a quantity called the isogeny defect.