Class numbers of multinorm-one tori

We present a formula for the class number of a multinorm one torus $T_{L\mathrm{/}k}$ associated to any étale algebra $L$ over a global field $k$. This is deduced from a formula for analogues of invariants introduced by T.~Ono, which are interpreted as a generalization of Gauss genus theory. This paper includes the variants of Ono’s invariant for arbitrary $S$-ideal class numbers and the narrow version, generalizing results of Katayama, Morishita, Sasaki and Ono.