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

TL;DR

This paper determines the growth rates of elliptic curves with fixed torsion subgroups over the Gaussian rationals, extending previous results from the rational numbers to $\

Contribution

It computes the exponents $d(G)$ for all torsion groups over $\

Findings

Calculated $d(G)$ for 15 torsion groups over $\

Extended the computation of $d(G)$ to 9 additional torsion groups over $\

Provides a comprehensive understanding of the distribution of elliptic curves with fixed torsion over $\

Abstract

Mazur'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 $\mathbb{Q}[i]$. Furthermore, in a collection of recent papers it was proven that there are 9 more possibilities for the torsion subgroup in the base field $\mathbb{Q}[i]$. We compute the value of $d(G)$ for these new groups.