Icosahedron, exceptional singularities and modular forms

TL;DR

This paper reveals two distinct symmetry groups and modular parametrizations for the $E_8$-singularity, connecting it to icosahedral and $ ext{PSL}(2, 13)$ symmetries, and extends these ideas to other exceptional singularities.

Contribution

It demonstrates that $E_8$ is not uniquely determined by the icosahedron, providing counterexamples to longstanding problems in singularity theory and Lie algebra relations.

Findings

$E_8$ has two symmetry groups: icosahedral and $ ext{PSL}(2, 13)$.

Provides modular parametrizations for $Q_{18}$, $E_{20}$, and a specific singularity.

Offers new solutions related to the Fermat-Catalan conjecture.

Abstract

We find that the equation of $E_8$-singularity possesses two distinct symmetry groups and modular parametrizations. One is the classical icosahedral equation with icosahedral symmetry, the associated modular forms are theta constants of order five. The other is given by the group $\text{PSL}(2, 13)$, the associated modular forms are theta constants of order $13$. As a consequence, we show that $E_8$ is not uniquely determined by the icosahedron. This solves a problem of Brieskorn in his ICM 1970 talk on the mysterious relation between exotic spheres, the icosahedron and $E_8$. Simultaneously, it gives a counterexample to Arnold's $A, D, E$ problem, and this also solves the other related problem on the relation between simple Lie algebras and Platonic solids. Moreover, we give modular parametrizations for the exceptional singularities $Q_{18}$, $E_{20}$ and $x^7+x^2 y^3+z^2=0$ by theta constants of order $13$, the second singularity provides a new analytic construction of solutions for the Fermat-Catalan conjecture and gives an answer to a problem dating back to the works of Klein.