On the Irreducibility and Distribution of Arithmetic Divisors

TL;DR

This paper introduces the concept of epsilon-irreducibility for arithmetic cycles, demonstrating that high tensor powers of arithmetically ample line bundles can be represented by such divisors and studying their distribution properties.

Contribution

It establishes the existence of epsilon-irreducible divisors for high tensor powers and analyzes their distribution, including convergence to the first Chern form and applications to polynomial zero sets.

Findings

High tensor powers admit epsilon-irreducible divisors

Divisors of small sections converge to the curvature form

New equidistribution results for polynomial zeros

Abstract

We introduce the notion of $\epsilon$-irreducibility for arithmetic cycles meaning that the degree of its analytic part is small compared to the degree of its irreducible classical part. We will show that for every $\epsilon>0$ any sufficiently high tensor power of an arithmetically ample hermitian line bundle can be represented by an $\epsilon$-irreducible arithmetic divisor. Our methods of proof also allow us to study the distribution of divisors of small sections of an arithmetically ample hermitian line bundle $\overline{\mathcal{L}}$. We will prove that for increasing tensor powers $\overline{\mathcal{L}}^{\otimes n}$ the normalized Dirac measures of these divisors almost always converge to $c_1(\overline{\mathcal{L}})$ in the weak sense. Using geometry of numbers we will deduce this result from a distribution result on divisors of random sections of positive line bundles in complex analysis. As an application, we will give a new equidistribution result for the zero sets of integer polynomials. Finally, we will express the arithmetic intersection number of arithmetically ample hermitian line bundles as a limit of classical geometric intersection numbers over the finite fibers.