Counting flags of primitive lattices

TL;DR

This paper develops a counting method for flags of primitive lattices in ^n, providing new proofs for the Manin conjecture and establishing equidistribution results for rational points and lattice shapes in flag varieties.

Contribution

It introduces a novel counting approach for primitive lattice flags, leading to new proofs of the Manin conjecture and equidistribution results in flag varieties.

Findings

Established a new proof of the Manin conjecture for flag varieties.

Proved equidistribution of rational points in flag varieties.

Demonstrated equidistribution of shapes of quotient lattices.

Abstract

We count flags of primitive lattices, which are objects of the form ${0}=\Lambda^{(0)}<\Lambda^{(1)}< \cdots <\Lambda^{(\ell)}= \mathbb{Z}^n$, where every $\Lambda^{(i)}$ is a primitive lattice in $\mathbb{Z}^n$. The counting is with respect to two different natural height functions, allowing us to give a new proof of the Manin conjecture for flag varieties over rational numbers. We deduce the equidistribution of rational points in flag varieties, as well as the equidistribution of the shapes of the successive quotient lattices, $\Lambda^{(i)}/\Lambda^{(i-1)}$. In doing so, we generalize previous work of Schmidt, as well as our own, on counting primitive lattices of rank $d<n$.