Tetraelliptic modular curves $X_1(N)$

TL;DR

This paper classifies all tetraelliptic modular curves $X_1(N)$ over $Q$, constructs explicit tetraelliptic maps to elliptic curves, and explores their Galois properties and applications to torsion subgroups over cyclic quartic fields.

Contribution

It provides a complete classification of tetraelliptic $X_1(N)$ over $Q$, explicit rational functions for the maps, and demonstrates their Galois nature and applications to elliptic curve torsion structures.

Findings

All tetraelliptic $X_1(N)$ over $Q$ are determined.

Explicit rational functions for tetraelliptic maps are constructed.

The maps are shown to be Galois, leading to applications in elliptic curve torsion subgroup analysis.

Abstract

In this paper, we determine all tetraelliptic modular curves $X_1(N)$ over $\mathbb Q$, and find some tetraelliptic maps $\phi_N$ from $X_1(N)$ to elliptic curves for those tetraelliptic $X_1(N)$. Also we will construct $\phi_N$ explicitly as rational functions. Moreover, we will show that all $\phi_N$ we found are Galois and find elliptic curves with torsion subgroup $\mathbb Z/17\mathbb Z$ over cyclic quartic number fields by using the cyclic map $\phi_{17}$.