We present a formula for the class number of a multinorm one torus $T_{L/k}$ 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, Ono and Sasaki.

中文翻译：

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多范一环面的类数

我们提出一个多范数环面的类数公式${\mathrm{时间}}_{L/k}$与全局域k上的任何代数L相关联。这是从 T. Ono 引入的不变量类似公式推导出来的，该公式被解释为高斯亏格理论的推广。本文包括小野不变量对于任意S理想类数的变体和狭义版本，概括了 Katayama、Morishita、Ono 和 Sasaki 的结果。