Abelian class groups of reductive group schemes

TL;DR

This paper introduces the abelian class group for reductive group schemes over rings of arithmetic interest and explores its properties, including a surjection from the class set in certain global fields.

Contribution

It defines the abelian class group for reductive group schemes and establishes a surjection from the class set over specific global fields, advancing understanding of their arithmetic properties.

Findings

Existence of a surjection from C(G) to C_{ab}(G) over global fields without real primes.

Properties of the abelian class group C_{ab}(G) for reductive group schemes.

Framework for studying class groups in arithmetic geometry.

Abstract

We introduce the abelian class group C_{ab}(G) of a reductive group scheme G over a ring A of arithmetical interest and study some of its properties. In particular, we show that if the fraction field of A is a global field without real primes, then there exists a surjection C(G)-->> C_{ab}(G), where C(G) is the class set of G.