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

Abstract

Let $\Gamma \subseteq \text{PSL}_2(\mathbb{Z})$ be a finite index subgroup. Let $\mathscr{X}(\Gamma)$ be a regular proper model of the modular curve associated with $\Gamma$, and let $\overline{\mathscr{L}}^{\otimes k}$ be the logarithmically singular metrized line bundle on $\mathscr{X}(\Gamma)$ associated to modular forms of level $\Gamma$ and weight $12k$, endowed with the Petersson metric. For each $k \geq 1$, the sub-lattice $\mathscr{S}_k \subseteq H^0(\mathscr{X}(\Gamma), \mathscr{L}^{\otimes k})$ of integral cusp forms of level $\Gamma$ 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 $\mathscr{S}_k$ as $k \to \infty$, generalizing the work of Chinburg, Guignard, and Soul\'{e} which addressed the case $\Gamma = \text{PSL}_2(\mathbb{Z})$.