Without real vector spaces all regulators are rational
Oliver Braunling
TL;DR
The paper proves that in the category of locally compact abelian groups without real line summands ($\mathsf{LCA}_{\operatorname{vf}}$), any determinant functor is determined up to a rational rescaling, implying that transcendental regulators cannot arise from determinant data in this setting. It achieves this by computing the universal determinant functor via Deligne’s framework and $K$-theory, showing that the vector-free Haar determinant $Ha^{\mathbb{Q}}$ is universal and lands in $\mathsf{Tors}(\mathbb{Q}_{>0}^{\times})$. The key technical steps include constructing $Ha^{\mathbb{Q}}$ from a rationalized root measure, leveraging Verdier localization sequences to compute $K_{1}(\mathsf{LCA}_{\operatorname{vf}})$, and identifying the universal determinant using decorated vector-free LCA categories. The results constrain the appearance of regulators in special $L$-value conjectures when real realizations are excluded, with implications for Tamagawa-number type conjectures and Weil–étale cohomology in purely LCA contexts.
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.