Characterization of global fields by Dirichlet \(L\)-series. (English) Zbl 1460.11137
Using only classical methods from number theory (class field theory, Chebotarev, Grunwald-Wang, and inverse Galois theory) and work of K. Uchida [J. Math. Soc. Japan 28, 617–620 (1976; Zbl 0329.12013)] and
Y. Hoshi [Publ. Res. Inst. Math. Sci. 50, No. 2, 269–285 (2014; Zbl 1297.11135)], the authors prove that two global fields are isomorphic if and only if there is an isomorphism of groups of Dirichlet characters that preserves \(L\)-series. In the final sections of this paper, the authors deal with number fields, and use representation theory to prove some stronger results.
Reviewer: Abdelmalek Azizi (Oujda)
MSC:
| 11R37 | Class field theory |
| 11R42 | Zeta functions and \(L\)-functions of number fields |
| 11R56 | Adèle rings and groups |
| 14H30 | Coverings of curves, fundamental group |
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