Coincidence of \(L\)-functions. (English) Zbl 1509.11107
Let \(K/\mathbb Q\) be a Galois extension with Galois group \(G\) and assume that for some prime \(p\) there is a normal abelian group \(H\subset G\) of index \(p\), having a character \(\psi\) whose lift \(\psi^G\) to \(G\) is irreducible. The authors show that there exists a normal abelian subgroup \(H_1\ne H\) of \(G\) having an irreducible character \(\xi\) satisfying \(\xi^G = \psi^G\) if and only if \(G\) is isoclinic to the Heisenberg group over the \(p\)-element field. This makes more precise a special case of an earlier result of H. Ishii [Jpn. J. Math., New Ser. 12, 37–44 (1986; Zbl 0607.12005)].
Reviewer: Władysław Narkiewicz (Wrocław)
MSC:
| 11R42 | Zeta functions and \(L\)-functions of number fields |
| 11R32 | Galois theory |
| 11R21 | Other number fields |
Keywords:
Artin \(L\)-function; Hecke \(L\)-function; isoclinism; Heisenberg group; Shintani-Stark unitCitations:
Zbl 0607.12005Software:
MagmaReferences:
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