Modular curves, invariant theory and $E_8$

Lei Yang

Published 2017-04-06Version 1

We find that three different appearances of $E_8$: the equation of $E_8$-singularity (theory of singularities), the exotic del Pezzo surface of degree one (differential topology of $4$-manifolds), and the automorphism group of the configuration of $120$ tritangent planes of Bring's curve (representation theory and classical algebraic geometry), which are well-known to be constructed from the icosahedron, can be constructed from the modular curve $X(13)$ by the theory of invariants for the simple group $\text{PSL}(2, 13)$. This implies that $E_8$ is not uniquely determined by the icosahedron and solves a problem of Brieskorn in his talk at ICM 1970 on the mysterious relation between the icosahedron and $E_8$. As applications, we give an explicit construction of the modular curve $X(13)$, which is a classical problem studied by Klein, and find an explicit example to Serre's question on the finite subgroups in the Cremona group of large rank.