5-TORSION POINTS ON CURVES OF GENUS 2

Abstract

Let C be a smooth proper curve of genus 2 over an algebraically closed field k . Fix a Weierstrass point ∞in C ( k ) and identify C with its image in its Jacobian J under the Albanese embedding that uses ∞ as base point. For any integer N [ges ]1, we write J N for the group of points in J ( k ) of order dividing N and J * N for the subset of J N of points of order N . It follows from the Riemann–Roch theorem that C ( k )∩ J 2 consists of the Weierstrass points of C and that C ( k )∩ J * 3 and C ( k )∩ J * are empty (see [ 3 ]). The purpose of this paper is to study curves C with C ( k )∩ J * 5 non-empty.