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arXiv · 1711.06242

Methods for constructing elliptic and hyperelliptic curves with rational points

Kirti Joshi
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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 $M_{24}$.

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Kirti Joshi. 2017-11-16. Methods for constructing elliptic and hyperelliptic curves with rational points. https://arxiv.org/abs/1711.06242

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