Classification of totally real number fields via their zeta function, regulator, and log unit lattice

TL;DR

This paper demonstrates, assuming the weak Schanuel Conjecture, that invariants like the zeta function residues, regulator, and log unit lattice uniquely identify totally real number fields and their properties.

Contribution

It establishes new criteria under WSC for classifying totally real number fields using their zeta functions, regulators, and log unit lattices.

Findings

Residues of Dedekind zeta functions are linearly independent for non-equivalent fields.

Same regulator implies same class number and zeta function under WSC.

Log unit lattice properties characterize the field's isomorphism class.

Abstract

In this paper, assuming the weak Schanuel Conjecture (WSC), we prove that for any collection of pairwise non-arithmetically equivalent totally real number fields, the residues at $s=1$ of their Dedekind zeta functions form a linearly independent set over the field of algebraic numbers. As a corollary, we obtain that, under WSC, two totally real number fields have the same regulator if and only if they have the same class number and Dedekind zeta function. We also prove that, under WSC, the isometry and similarity classes of the log unit lattice of a real Galois number field of degree $[K:\Q]\geq 4$, characterize the isomorphism class of said field. All of our results follow from establishing that, under WSC, any Gram matrix of the log unit lattice of a real Galois number field yields a generic point of certain closed irreducible $\Q$-subvariety of the space of symmetric matrices of appropriate size.