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

TL;DR

This paper demonstrates the modularity and explicit uniformization of a genus 50 algebraic space curve using theta constants, linking it to modular curves, Galois coverings, and singularity theory, providing new insights into higher genus curves and modular equations.

Contribution

It introduces a new explicit modular parametrization of a high-genus algebraic space curve, connecting it to modular curves, Galois theory, and singularity classifications, with novel modular equations and realizations.

Findings

Proves modularity of a genus 50 space curve in P^5.

Provides explicit modular equations of order 13.

Establishes isomorphism with the modular curve X(13).

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.