Computing the Mordell-weil Rank of Jacobians of Curves of Genus Two

We derive the equations necessary to perform a two-descent on the Jacobians of curves of genus two with rational Weierstrass points. We compute the Mordell-Weil rank of the Jacobian of some genus two curves defined over the rationals, and discuss the practicality of using this method. Introduction Let C he a curve of genus two defined over a number field K, and J its Jacobian variety. The Mordell-Weil theorem states that J(K) is a finitelygenerated Abelian group, but except in a few special cases, it has never been explicitly determined. Recent work of Vojta [24], Faltings [9], and Bombieri [3] relates the number of rational points on C to the rank of J(K), increasing the interest in computing the latter. Further conjectures and results relating C(K) and J(K) can be found in [17 and 23]. In addition, elliptic curves have recently been applied to many computational problems, such as primality testing and factorization [15], and cryptography [14]. There are indications that curves of higher genus have similar uses. They have been proposed for better primality tests [1] and new cryptosystems [13]. However, the lack of explicit knowledge of the properties of these curves have slowed their widespread use. The recent formulations of the group law on the Jacobian by Cantor in [4] and the second author in [11] are a beginning, but more remains to be done. In this paper we show how to compute the rank of J(K) for a wide class of genus two curves, namely those which have all their Weierstrass points defined over K, and whose Jacobians have no two-torsion in their Tate-Shafarevich groups. The former constraint could be removed by performing a Galois descent from K(J[2]) to K. This would take us too far afield, so we refrain from making this descent now. The latter, however, is a serious constraint, for what we actually compute is the two-Selmer group of /. In principle, it is well known how to compute the Selmer group: much of this work was done by Cassels [6]. One missing ingredient there was the defining equations for /, which have now been worked out by Flynn in [10] and the second author in [11]. For computational reasons, more work has to be done Received by the editors October 11, 1990 and, in revised form, March 1, 1991. 1980 Mathematics Subject Classification (1985 Revision). Primary 11Y50; Secondary 14K15. The first author partially supported by a University of Georgia Faculty Research Grant. The second author supported by a NATO Postdoctoral Fellowship. ©1993 American Mathematical Society 0002-9947/93 $1.00 + 5.25 per page