# On the slopes of the lattice of sections of Hermitian line bundles

## TL;DR

This paper investigates the asymptotic distribution of sections of Hermitian line bundles over number fields, providing criteria for the existence of a limiting measure and methods to compute it explicitly, with applications in number theory and capacity theory.

## Contribution

It introduces new criteria for the existence of a limiting measure for the distribution of sections and develops explicit methods to determine this measure.

## Key findings

- Established criteria for the limiting measure of section distributions.
- Developed explicit methods to compute the limiting measure.
- Applied results to Petersson norms and capacity theory.

## Abstract

In this paper we study the distribution of successive minima of global sections of powers of a metrized ample line bundle on a variety over a number field. We develop criteria for there to exist a measure on the real line describing the limiting behavior of this distribution as one considers increasing powers of the bundle. When this measure exists, we develop methods for determining it explicitly. We present applications to the distribution of Petersson norms of cusp forms of increasing weight for SL_2(Z) and to the minimal sup norm of algebraic functions on adelic subsets of curves arising in capacity theory.