2-Descent on the Jacobians of Hyperelliptic Curves Original Research Article

Let J be the Jacobian of the hyperelliptic curve Y2 = ƒ(X) over a field K of characteristic 0, where ƒ has odd degree. We shall present an embedding of the group J(K)/2J(K) into the group L*/L*2 where L = K[T]/ƒ(T). Since this embedding is derived from the coboundary map of Galois cohomology, it can be used to compute a 2-descent for the Jacobian. We will use this embedding to compute J(Q)/2J(Q) for a rank-2 Jacobian of a hyperelliptic curve of genus 3.