The Frequency of Elliptic Curves Over $\mathbb{Q}[i]$ with Fixed Torsion

Authors

Alan S. Zhao, Columbia University in the City of New YorkFollow

Abstract

Mazur\textsc{\char13}s Theorem states that there are precisely 15 possibilities for the torsion subgroup of an elliptic curve defined over the rational numbers. It was previously shown by Harron and Snowden that the number of isomorphism classes of elliptic curves of height up to $X$ that have a specific torsion subgroup $G$ is on the order of $X^{1/{d(G)}}$, for some positive $d(G)$ depending on $G$. We compute $d(G)$ for these groups over $\Qi$. Furthermore, in a collection of recent papers it was proven that there are 9 more possibilities for the torsion subgroup in the base field $\Qi$. We compute the value of $d(G)$ for these new groups.

Author Bio

Alan Zhao is currently a third-year undergraduate mathematics researcher at Columbia University in the City of New York. This paper was written while he was working towards high school diploma.†

Faculty Sponsor

Stefan Patrikis

Recommended Citation

Zhao, Alan S. (2024) "The Frequency of Elliptic Curves Over $\mathbb{Q}[i]$ with Fixed Torsion," Rose-Hulman Undergraduate Mathematics Journal: Vol. 25: Iss. 2, Article 5.
Available at: https://scholar.rose-hulman.edu/rhumj/vol25/iss2/5