ON THE NUMBER OF POINTS OF GIVEN ORDER ON ODD-DEGREE HYPERELLIPTIC CURVES

Abstract

For integers N≥2 and g≥1, we study bounds on the cardinality of the set of points of order dividing N lying on a hyperelliptic curve of genus g embedded in its jacobian using a Weierstrass point as base point. This leads us to revisit division polynomials introduced by Cantor in 1995 and strengthen a divisibility result proved by him. Several examples are discussed.