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

TL;DR

This paper introduces new algorithms for computing Selmer groups of Jacobians by leveraging special properties of curves, including explicit methods for certain curve types, and demonstrates their effectiveness through concrete examples.

Contribution

It develops novel algorithms for Selmer group computation using functions on curves, exploiting specific properties to improve efficiency and applicability.

Findings

Successfully computed Mordell-Weil ranks for specific Jacobians

Developed a $(1- ext{zeta}_p)$-Selmer group algorithm for certain curves

Computed 2-Selmer group for a plane quartic curve using bitangents

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 $(1-\zeta_{p})$-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.