Abstract: In this talk I'll explain my recent results on the spinor genus of the integral trace form of a number field. I'll show how from them one can decide in terms of finitely many ramification invariants, and under some restrictions, whether or not a pair of number fields have isometric integral trace forms. Inspired by the work of R. Perlis on number fields with the same zeta function, I'll define the notion of weak arithmetic equivalence, and I'll show that under certain hypothesis this equivalence determines the local root numbers of the number field, and the isometry class of the integral trace form.