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This is a companion paper to our previous work, where we proved thefiniteness of the Tate-Shafarevich group for an arbitrary torus $T$ over afinitely generated field $K$ with respect to any divisorial set $V$ of placesof $K$. Here, we extend this result to any $K$-group $D$ whose connectedcomponent is a torus (for the same $V$), and as a consequence obtain afiniteness result for the local-to-global conjugacy of maximal tori inreductive groups over finitely generated fields. Moreover, we prove thefiniteness of the Tate-Shafarevich group for tori over function fields $K$ ofnormal varieties defined over base fields of characteristic zero and satisfyingSerre's condition (F), in which case $V$ consists of the discrete valuationsassociated with the prime divisors on the variety (geometric places). In thissituation, we also establish the finiteness of the number of $K$-isomorphismclasses 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 positivecharacteristic. Finally, we prove the finiteness of the number of isomorphismclasses of forms of an absolutely almost simple group defined over the functionfield of a complex surface that have good reduction at all geometric places.

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