Implementing 2-descent for Jacobians of hyperelliptic curves

. This paper gives a fairly detailed description of an algorithm that computes (the size of) the 2-Selmer group of the Jacobian of a hyperellitptic curve over Q. The curve is assumed to have even genus or to possess a Q-rational Weierstra point. 1. Introduction Given some curve C over Q , one would like to determine as much as possible of its arithmetical properties. One of the more important invariants is the MordellWeil rank of its Jacobian J , i.e., the free abelian rank of J(Q ) (finite by the Mordell-Weil Theorem). There is no algorithm so far that provably determines this rank, but it is possible (at least in theory) to bound it from above by computing the size of a suitable Selmer group. It is also fairly easy to find lower bounds by looking for independent rational points on the Jacobian. (It can be difficult, however, to find the right number of independent points, when some of the generators are large.) With some luck, both bounds coincide, and the rank is determined. In...