METHODS FOR CONSTRUCTING ELLIPTIC AND HYPERELLIPTIC CURVES WITH RATIONAL POINTS

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 M₂₄.

11G05, 11G10, 11G30, 14H52