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

Abstract

In 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.