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