Without real vector spaces all regulators are rational

Abstract

Every LCA group has a Haar measure unique up to rescaling by a positive scalar. Clausen has shown that the Haar measure describes the universal determinant functor of the category LCA in the sense of Deligne. We show that when only working with LCA groups without allowing real vector spaces, any conceivable determinant functor is unique up to rescaling by at worst rational values. As a result, no transcendental real nor p-adic regulators could ever show up in special L-value conjectures (as in Tamagawa number conjectures or Weil-etale cohomology) if anyone had the, admittedly outlandish and bizarre, idea to try to circumvent incorporating a real (Betti) realization of the motive.