An alternative construction of equivariant Tamagawa numbers

TL;DR

This paper introduces a new formulation of the equivariant Tamagawa number conjecture for non-commutative coefficients, simplifying the existing framework by removing complex categorical structures and expressing Tamagawa numbers in an idele group.

Contribution

It presents an alternative, simplified formulation of the ETNC that avoids Picard groupoids, determinant functors, virtual objects, and K-groups, and proves its equivalence to the Burns-Flach formulation.

Findings

New formulation of ETNC using idele groups

Equivalence established with Burns-Flach formulation

Simplifies the conceptual framework of Tamagawa numbers

Abstract

We propose a new formulation of the equivariant Tamagawa number conjecture (ETNC) for non-commutative coefficients. We remove Picard groupoids, determinant functors, virtual objects and relative K-groups. Our Tamagawa numbers lie in an idele group instead of any kind of K-group. Our formulation is proven equivalent to the one of Burns-Flach.