5-torsion points on curves of genus 2

摘要

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 ≥ 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(i) ∩ J_4~* are empty (see [3]). The purpose of this paper is to study curves C with C(k) ∩ J_5~* non-empty.