The equidistribution of lattice shapes of rings of integers in cubic, quartic, and quintic number fields

TL;DR

This paper proves that the lattice shapes of rings of integers in degree 3, 4, and 5 number fields become uniformly distributed as their discriminants grow, revealing a deep statistical regularity in their geometric structure.

Contribution

It establishes the equidistribution of lattice shapes of rings of integers in cubic, quartic, and quintic number fields when ordered by discriminant, extending understanding of their geometric distribution.

Findings

Lattice shapes become equidistributed in the space of lattices.

Results hold for degree 3, 4, and 5 number fields.

Distribution aligns with predictions from random lattice models.

Abstract

For $n=3$, 4, and 5, we prove that, when $S_n$-number fields of degree $n$ are ordered by their absolute discriminants, the lattice shapes of the rings of integers in these fields become equidistributed in the space of lattices.