Properness of the global-to-local map for algebraic groups with toric connected component and other finiteness properties

TL;DR

This paper extends finiteness results for Tate-Shafarevich groups and related structures from tori to more general algebraic groups with toric components over finitely generated fields, with applications to conjugacy and reduction properties.

Contribution

It generalizes previous finiteness results for Tate-Shafarevich groups to algebraic groups with toric connected components and explores implications for reductive groups and algebraic tori.

Findings

Finiteness of Tate-Shafarevich groups for certain algebraic groups over function fields.

Finiteness of isomorphism classes of algebraic tori with good reduction.

Finiteness of forms of almost simple groups with good reduction.

Abstract

This is a companion paper to our previous work, where we proved the finiteness of the Tate-Shafarevich group for an arbitrary torus $T$ over a finitely generated field $K$ with respect to any divisorial set $V$ of places of $K$. Here, we extend this result to any $K$-group $D$ whose connected component is a torus (for the same $V$), and as a consequence obtain a finiteness result for the local-to-global conjugacy of maximal tori in reductive groups over finitely generated fields. Moreover, we prove the finiteness of the Tate-Shafarevich group for tori over function fields $K$ of normal varieties defined over base fields of characteristic zero and satisfying Serre's condition (F), in which case $V$ consists of the discrete valuations associated with the prime divisors on the variety (geometric places). In this situation, we also establish the finiteness of the number of $K$-isomorphism classes of algebraic $K$-tori of a given dimension having good reduction at all $v \in V$, and then discuss ways of extending this result to positive characteristic. Finally, we prove the finiteness of the number of isomorphism classes of forms of an absolutely almost simple group defined over the function field of a complex surface that have good reduction at all geometric places.