$E_8$-singularity, invariant theory and modular forms

Lei Yang

Published 2020-09-07Version 1

As an algebraic surface, the equation of $E_8$-singularity $x^5+y^3+z^2=0$ can be obtained as a quotient $C_Y/\text{SL}(2, 13)$ over the modular curve $X(13)$, where $Y \subset \mathbb{CP}^5$ is an algebraic curve given by a system of $\text{SL}(2, 13)$-invariant polynomials and $C_Y$ is a cone over $Y$. It is different from the Kleinian singularity $\mathbb{C}^2/\Gamma$, where $\Gamma$ is the binary icosahedral group. This gives a negative answer to Arnol'd and Brieskorn's questions about the mysterious relation between the icosahedron and $E_8$, i.e., the $E_8$-singularity is not necessarily the Kleinian icosahedral singularity. Hence, the equation of $E_8$-singularity possesses two distinct modular parametrizations. Moreover, three different algebraic surfaces, the equations of $E_8$, $Q_{18}$ and $E_{20}$-singularities can be realized from the same quotients $C_Y/\text{SL}(2, 13)$ over the modular curve $X(13)$ and have the same modular parametrizations.