2-descent on the Jacobians of hyperelliptic curves. (English) Zbl 0832.14016

Schaefer, Edward F.

J. Number Theory 51, No. 2, 219-232 (1995).

Let \(K\) be an algebraic number field of characteristic zero and \(f \in K(X)\) a monic, separable polynomial of odd degree \(d\). The equation \(Y^2 = f(X)\) defines a hyperelliptic curve \(C\) over \(K\) of genus \({d - 1 \over 2}\). By the Mordell-Weil theorem the group of \(K\)-rational points \(J(K)\) of the Jacobian variety of \(C\) is a finitely generated abelian group. The paper in question represents a new method for computing the rank of the free part of \(J(K)\), the Mordell-Weil rank of \(J\) over \(K\), or to be more precise the rank of \(J(K)/2J (K)\). The method is applied to compute \(J(\mathbb{Q})/2J (\mathbb{Q})\) for the hyperelliptic curve \(Y^2 = X(X - 2) (X - 3) (X - 4) (X - 5) (X - 7) (X - 10)\).

Reviewer: Ch.Birkenhake (Erlangen)

MSC:

14G05	Rational points
14H40	Jacobians, Prym varieties
14H52	Elliptic curves

Keywords:

rank of group of rational points; Jacobian; Mordell-Weil rank; hyperelliptic curve