Computing a Selmer group of a Jacobian using functions on the curve

Abstract

1In general, algorithms for computing the Selmer group of the Jacobian of a curve have relied on either homogeneous spaces or functions on the curve. We present a theoretical analysis of algorithms which use functions on the curve, and show how to exploit special properties of curves to generate new Selmer group computation algorithms. The success of such an algorithm will be based on two criteria that we discuss. To illustrate the types of properties which can be exploited, we develop a (1 − ζ p )-Selmer group computation algorithm for the Jacobian of a curve of the form y p = f (x) where p is a prime not dividing the degree of f .We compute Mordell-Weil ranks of the Jacobians of three curves of this form. We also compute a 2-Selmer group for the Jacobian of a smooth plane quartic curve using bitangents of that curve, and use it to compute a Mordell-Weil rank.

Non-trivial \Sha in the Jacobian of an infinite family of curves of genus 2

“…First, we outline the method of complete 2-descent [8], [11], [12]; we shall do this for the quintic case, but note that there are also algorithms described for the general sextic case, for example, in [4], [9], [14]. We let C : y 2 = F (x) denote a curve of genus 2 defined over Q and assume that deg(F (x)) = 5.…”

Generalized Explicit Descent and Its Application to Curves of Genus 3

“…Let A be the dual abelian variety of A, and let φ : A → J be the dual of φ. (It was first explained in [36] that the generality in which one relates the functions in a true descent setup to an isogeny is for an isogeny whose image is a Jacobian.) PROPOSITION 6.14.…”

Rational points on curves

“…To achieve the former, we can use n-descent again, but this time on the Jacobian J. This is feasible for hyperelliptic curves when n = 2 and in a few other rather special cases, see [40,37,48,38]. As with elliptic curves, large generators can be a problem, however.…”