Modularity and uniformization of a higher genus algebraic space curve, its distinct arithmetical realizations by cohomology groups and $E_6$, $E_7$, $E_8$-singularities

Abstract

We prove the modularity for an algebraic space curve $Y$ of genus $50$ in $\mathbb{P}^5$, which consists of $21$ quartic polynomials in six variables, by means of an explicit modular parametrization by theta constants of order $13$. This provides an example of modularity, explicit uniformization and hyperbolic uniformization of arithmetic type for a higher genus algebraic space curve. In particular, it gives a new example for Hilbert's 22nd problem. This gives $21$ modular equations of order $13$, which greatly improve the result of Ramanujan and Evans on the construction of modular equations of order $13$. We show that $Y$ is isomorphic to the modular curve $X(13)$. The corresponding ideal $I(Y)$ is invariant under the action of $\text{SL}(2, 13)$, which leads to a $21$-dimensional reducible representation of $\text{SL}(2, 13)$, whose decomposition as the direct sum of $1$, $7$ and $13$-dimensional representations gives two distinct arithmetical realizations of $X(13)$ by character fields $\mathbb{Q}(\chi)=\mathbb{Q}(\zeta_7+\zeta_7^{-1})$ or $\mathbb{Q}(\chi)=\mathbb{Q}(\sqrt{13})$ of irreducible representations of $\text{SL}(2, 13)$ corresponding to the decompositions of cohomology groups of a projective or affine variety with values in a coherent algebraic sheaf on $X(13)$ as well as the geometric construction of $Y$, the geometric realization of the degenerate principal series and the Steinberg representation of $\text{SL}(2, 13)$. The projection $Y \rightarrow Y/\text{SL}(2, 13)$ (identified with $\mathbb{CP}^1$) is a Galois covering whose generic fibre is interpreted as the Galois resolvent of the modular equation $\Phi_{13}(\cdot, j)=0$ of level $13$. The ring of invariant polynomials $(\mathbb{C}[z_1, z_2, z_3, z_4, z_5, z_6]/I(Y))^{\text{SL}(2, 13)}$ over $X(13)$ leads to a new perspective on the theory of $E_6$, $E_7$ and $E_8$-singularities.