Methods for constructing elliptic and hyperelliptic curves with rational points

TL;DR

This paper introduces methods for constructing elliptic and hyperelliptic curves over various fields with specified rational points, including over number fields and function fields, with applications to large Galois groups.

Contribution

It provides new explicit constructions of elliptic and hyperelliptic curves with rational points over diverse fields, including methods for large Galois groups and points of infinite order.

Findings

Constructed elliptic curves over any number field with rational points of infinite order.

Developed hyperelliptic curves with high genus and large rank of rational points.

Built hyperelliptic curves over rational function fields with points over fields with large Galois groups.

Abstract

I provide methods of constructing elliptic and hyperelliptic curves over global fields with interesting rational points over the given fields or over large field extensions. I also provide a elliptic curves defined over any given number field equipped with a rational point, (resp. with two rational points) of infinite order over the given number field, and elliptic curves over the rationals with two rational points over `simplest cubic fields.' I also provide hyperelliptic curves of genus exceeding any given number over any given number fields with points (over the given number field) which span a subgroup of rank at least $g$ in the group of rational points of the Jacobian of this curve. I also provide a method of constructing hyperelliptic curves over rational function fields with rational points defined over field extensions with large finite simple Galois groups, such as the Mathieu group $M_{24}$.