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.

💡 Research Summary

This paper, titled “Classification of totally real number fields via their zeta function, regulator, and log unit lattice,” establishes several conditional results under the assumption of the Weak Schanuel Conjecture (WSC). It explores profound interconnections between the analytic, algebraic, and geometric invariants of totally real number fields.

The central objects of study are totally real number fields (where all embeddings into the complex numbers are real). The paper addresses three fundamental questions inspired by the known phenomenon that arithmetically equivalent number fields (those sharing the same Dedekind zeta function) can have different class numbers and regulators. Specifically, it asks: (1) Do two totally real fields with the same regulator have to be arithmetically equivalent? (2) Do two totally real fields with isometric or similar log unit lattices have to be isomorphic? (3) Are the residues at s=1 of the Dedekind zeta functions of pairwise non-arithmetically equivalent totally real fields linearly independent over the field of algebraic numbers?

The paper’s first major result (Theorem B) provides a conditional positive answer to question (3). Assuming WSC, it proves that for any collection of distinct, pairwise non-arithmetically equivalent totally real number fields, the sets containing their regulators {reg(K_i)} and the residues of their zeta functions at s=1 {res_{s=1} ζ_{K_i}(s)} are both linearly independent over the field of algebraic numbers Q̄. A key corollary (Theorem A) answers question (1): under WSC, two totally real number fields have the same regulator if and only if they have the same class number and the same Dedekind zeta function. Thus, the regulator alone determines both the zeta function and the class number for totally real fields.

The second major thread of the paper investigates the discriminative power of the log unit lattice. For a number field K, its log unit lattice Λ_K is the lattice in a real vector space formed by taking the real logarithms of the images of a basis of the unit group O_K^x under its archimedean embeddings. Its covolume is essentially the regulator. Corollary C, a direct consequence of Theorem A, states that if two totally real fields have isometric log unit lattices, then their regulators (and hence zeta functions and class numbers) are equal. A more refined result (Theorem D) is proven for real Galois fields of degree at least 4: under WSC, the similarity class of the log unit lattice (i.e., its shape up to rotation and scaling) is a complete invariant. Two such fields are isomorphic if and only if their log unit lattices are similar.

The engine behind all these proofs is a deep technical theorem (Theorem E). For a real Galois number field K with group G = Gal(K/Q), the Gram matrix of its log unit lattice (with respect to a weak Minkowski unit) naturally lives in a certain closed irreducible Q-subvariety V_G of the space of symmetric matrices. Theorem E asserts that, assuming WSC, this Gram matrix is a generic point of V_G. This means it does not lie in any proper Zariski-closed subset of V_G that is defined over Q. This genericity property becomes a powerful tool. The proofs of Theorems B and D proceed by contradiction: assuming a linear dependence between residues or a similarity between lattices from non-isomorphic fields translates into non-trivial algebraic equations satisfied by the Gram matrix. Theorem E then forces these equations to hold for every point in V_G, which is shown to be impossible, leading to the desired conclusions.

The paper situates its work within recent literature on log unit lattices, highlighting connections to results on their geometry and distribution. It also links Theorem B to classical transcendence theory, noting its analogy with Baker’s theorem on the linear independence of values of Dirichlet L-functions at s=1.

In summary, this paper demonstrates that, contingent on the Weak Schanuel Conjecture, the classical invariants of a totally real number field—its zeta function, regulator, and class number—are tightly bound together. Furthermore, it elevates the log unit lattice from a computational device for the regulator to a central geometric object whose isometry or similarity class can carry highly specific information about the isomorphism type of the underlying field, especially in the Galois case. The unifying methodology, leveraging transcendental number theory (WSC) to establish genericity in an algebraic variety, is a significant conceptual contribution.