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

TL;DR

This paper studies how the successive minima of lattices formed by cusp forms' Petersson norms distribute as the weight increases, extending previous results to more general modular curves.

Contribution

It generalizes the distribution analysis of successive minima of cusp form lattices to arbitrary finite index subgroups of PSL(2,Z).

Findings

Distribution of successive minima as weight tends to infinity.

Extension of previous results from PSL(2,Z) to general subgroups.

Asymptotic behavior of cusp form lattices' minima.

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 $\Gamma = \text{PSL}_2(\mathbb{Z})$.