Integral trace forms of cyclic number fields without tameness

Abstract

Let $K/\mathbf{Q}$ be a cyclic number field and let $T_K=\bigl(\mathcal{O}_K,(x,y)\mapsto \operatorname{Tr}_{K/\mathbf{Q}}(xy)\bigr)$ be its integral trace lattice. We prove that two cyclic number fields have isometric integral trace lattices if and only if they have the same degree, signature, and signed discriminant. More generally, an isomorphism between the Dirichlet character groups of two finite abelian extensions that preserves the conductor and parity of every character yields an isometry of their integral trace lattices, equivariant for the dual identification of Galois groups. For cyclic fields, the degree, signature, and discriminant determine the character group together with these two functions. This proves Bolaños's conjecture for wildly ramified cyclic fields and gives the corresponding extension of the tame classification: in general, the signature must be retained as an additional invariant.