Distribution of the successive minima of the Petersson norm on cusp forms

Let $\mathrm{\Gamma }\subseteq {\text{PSL}}_2(\mathbb{Z})$ be a finite index subgroup. Let $\mathcal{X}(\mathrm{\Gamma })$ be a regular proper model of the modular curve associated with $\mathrm{\Gamma }$, and let ${\bar{\mathcal{L}}}^{\otimes k}$ be the logarithmically singular metrized line bundle on $\mathcal{X}(\mathrm{\Gamma })$ associated to modular forms of level $\mathrm{\Gamma }$ and weight $12k$, endowed with the Petersson metric. For each $k\ge 1$, the sub-lattice ${\mathcal{S}}_k\subseteq H^0(\mathcal{X}(\mathrm{\Gamma }),{\mathcal{L}}^{\otimes k})$ of integral cusp forms of level $\mathrm{\Gamma }$ and weight $12k$ is a euclidean lattice with respect to the Petersson norm. In this paper, we describe the distribution of the successive minima of the ${\mathcal{S}}_k$ as $k\to \mathrm{\infty }$, generalizing the work of Chinburg, Guignard, and Soulé which addressed the case $\mathrm{\Gamma }={\text{PSL}}_2(\mathbb{Z})$.