N-COVERS OF HYPERELLIPTIC CURVES

Abstract

Abstract. For a hyperelliptic curve C of genus g with a divisor class of order n = g + 1, we shall consider an associated covering collection of curves Dδ, each of genus g 2. We describe, up to isogeny, the Jacobian of each Dδ via a map from Dδ to C, and two independent maps from Dδ to a curve of genus g(g − 1)/2. For some curves, this allows covering techniques that depend on arithmetic data of number fields of smaller degree than standard 2-coverings; we illustrate this by using 3-coverings to find all Q-rational points on a curve of genus 2 for which 2-covering techniques would be impractical. 1. Description of the Jacobian of the Covering Curves We shall consider a hyperelliptic curve of genus g = n − 1 ≥ 1, of the form (1) C: Y 2 = F (X) = G(X) 2 + kH(X) n, where G(X) is of degree n = g + 1 and H(X) is of degree 2, and where G(X), H(X), k are defined over the ring of integers O of a number field K. Here, and elsewhere, we shall adopt the usual convention that C is used to denote the non-singular curve, even though the equation given in (1) is singular; for the practical purpose of points on C, we can take these to be the affine (X, Y) satisfying (1), together with ∞ +, ∞ − , which will be distinct points on this non-singular curve. We shall assume that F (X) has nonzero discriminant, which implies that resultant(G(X), H(X)) is also nonzero. Equation (1) is a classical model of a hyperelliptic curve whose Jacobian J has an element of order n define

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