On a ring of modular forms related to the Theta gradients map in genus 2

TL;DR

This paper studies a specific modular forms ring related to Theta gradients in genus 2, introducing a subgroup to achieve an injective map and analyzing algebraic structures and compactifications.

Contribution

It identifies a subgroup between known levels to make the Theta gradients map injective and provides an algebraic description of the associated modular forms ring.

Findings

A subgroup between a72(4,8) and a72(2,4) is found.

The map becomes injective on the new level moduli space.

Structural theorems for the ring of modular forms and cusp forms are established.

Abstract

The level moduli space $A_g^{4,8}$ is mapped to the projective space by means of gradients of odd Theta functions, such a map turning out no to be injective in the genus 2 case. In this work a congruence subgroup $\Gamma$ is located between $\Gamma_2(4,8)$ and $\Gamma_2(2,4)$ in such a way the map factors on the related level moduli space $A_{\Gamma}$, the new map being injective on $A_{\Gamma}$. Satake's compactification $\text{Proj}A(\Gamma)$ and the desingularization $\text{Proj}S(\Gamma)$ are also due to be investigated, since the map does not extend to the boundary of the compactification; to aim at this, an algebraic description is provided, by proving a structure theorem both for the ring of modular forms $A(\Gamma)$ and the ideal of cusp forms $S(\Gamma)$