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Ramified characters of idèle groups of one-class quadratic fields. (English. Russian original) Zbl 0956.11027
Proc. Steklov Inst. Math. 224, 107-114 (1999); translation from Tr. Mat. Inst. Steklova. 224, 122-129 (1999).
The author considers ramified characters \(\theta\) of the idele group \(A_d^\times\) of a one-class quadratic field \(\mathbb Q(\sqrt{d})\). Necessary and sufficient conditions are found for a character \(\theta\) to be trivial on the multiplicative group \(\mathbb Q^\times (\sqrt{d})\). Thus a complete description of characters of the group \(A_d^\times /\mathbb Q^\times (\sqrt{d})\) is found.
Note that in a subsequent paper [V. S. Vladimirov, Tr. Mat. Inst. Steklova 228, 76-89 (2000)] the author used this result in order to obtain adelic formulas for gamma and beta functions of one-class quadratic fields.
For the entire collection see [Zbl 0942.00074].

11R56 Adèle rings and groups
11R11 Quadratic extensions
22B99 Locally compact abelian groups (LCA groups)