On some special classes of complex elliptic curves

TL;DR

This paper classifies complex elliptic curves with cyclic subgroups leading to isomorphic quotients and explores fixed points of the Fricke involution on modular curves, providing new insights into their structure and symmetries.

Contribution

It introduces a classification of elliptic curves with specific cyclic subgroups that produce isomorphic quotients and analyzes the fixed points of the Fricke involution on modular curves.

Findings

Identified conditions for elliptic curves with isomorphic quotients via cyclic subgroups.

Provided explicit examples of such elliptic curves.

Characterized fixed points of the Fricke involution on modular curves.

Abstract

In this paper we classify the complex elliptic curves $E$ for which there exist cyclic subgroups $C\leq (E,+)$ of order $n$ such that the elliptic curves $E$ and $E/C$ are isomorphic, where $n$ is a positive integer. Important examples are provided in the last section. Moreover, we answer the following question: given a complex elliptic curve E, when can one find a cyclic subgroup $C$ of order $n$ of $(E,+)$ such that $(E,C)\sim(\frac{E}{C},\frac{E[n]}{C})$, $E[n]$ being the $n$-torsion subgroup of $E$, classifying in this way the fixed points of the action of the Fricke involution on the open modular curves $Y_0(n)$