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

Abstract

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