Modular curves, invariant theory and $E_8$

TL;DR

This paper presents a novel construction of the $E_8$ root lattice using the modular curve $X(13)$ and invariant theory for $ ext{PSL}(2, 13)$, providing explicit descriptions and new insights into their relationship.

Contribution

It introduces a new method to construct the $E_8$ lattice from the modular curve $X(13)$ via invariant theory, linking lattice theory and modular forms.

Findings

Constructed the $E_8$ root lattice from $X(13)$

Provided an explicit description of the modular curve $X(13)$

Established a connection between $E_8$ lattice and modular curve via invariant theory

Abstract

The $E_8$ root lattice can be constructed from the modular curve $X(13)$ by the invariant theory for the simple group $\text{PSL}(2, 13)$. This gives a different construction of the $E_8$ root lattice. It also gives an explicit construction of the modular curve $X(13)$.