##
**Stark’s conjecture in multi-quadratic extensions, revisited.**
*(English)*
Zbl 1047.11108

Let \(L/F\) be an Abelian multi-quadratic extension of number fields with Gal\((L/F)\simeq ({\mathbb Z}/2{\mathbb Z})^m\). Let \(\mathfrak v\) be a distinguished prime of \(F\) which splits completely in \(L\). The authors prove the existence of an \({\mathfrak v}\)-unit of \(L\), the so-called Stark unit, associated to the Artin \(L\)-function \(L_S(s,\chi)\), where \(\chi\) is a character of Gal\((L/F)\) and \(S\) is a finite set of primes of \(F\) containing \({\mathfrak v}\), all finite primes which ramify in \(L\), and all infinite primes. (One may assume that \(| S| >2\).) This is a part of Stark’s refined Abelian conjecture.

Moreover, they prove this conjecture in full for \(L/F\) in the following three cases: (1) \(\,\,| S| >m+1-r_F(S)\), where \(r_F(S)\) denotes the 2-rank of the \(S_{\text{fin}}\)-class group of \(F\), \(S_{\text{fin}}\) being the set of the finite primes in \(S\); (2) \(\,\, {\mathfrak v}\) is a real infinite prime or a finite prime, except possibly in the case that \(L\) is the maximal multi-quadratic extension of \(F\) which is unramified outside of \(S\) and in which \({\mathfrak v}\) splits completely; (3) \(\,\,m=2\). Previously, the second author proved Stark’s conjecture for \(L/F\) under the assumption that either \(| S| >m+1\) or no prime above 2 is ramified in \(L/F\) [J. Reine Angew. Math. 349, 129–135 (1984; Zbl 0521.12009); Math. Ann. 272, 349–359 (1985; Zbl 0554.12006)].

Moreover, they prove this conjecture in full for \(L/F\) in the following three cases: (1) \(\,\,| S| >m+1-r_F(S)\), where \(r_F(S)\) denotes the 2-rank of the \(S_{\text{fin}}\)-class group of \(F\), \(S_{\text{fin}}\) being the set of the finite primes in \(S\); (2) \(\,\, {\mathfrak v}\) is a real infinite prime or a finite prime, except possibly in the case that \(L\) is the maximal multi-quadratic extension of \(F\) which is unramified outside of \(S\) and in which \({\mathfrak v}\) splits completely; (3) \(\,\,m=2\). Previously, the second author proved Stark’s conjecture for \(L/F\) under the assumption that either \(| S| >m+1\) or no prime above 2 is ramified in \(L/F\) [J. Reine Angew. Math. 349, 129–135 (1984; Zbl 0521.12009); Math. Ann. 272, 349–359 (1985; Zbl 0554.12006)].

Reviewer: Tauno Metsänkylä (Turku)

### MSC:

11R42 | Zeta functions and \(L\)-functions of number fields |

11R20 | Other abelian and metabelian extensions |

11R27 | Units and factorization |

11R37 | Class field theory |

PDF
BibTeX
XML
Cite

\textit{D. S. Dummit} et al., J. Théor. Nombres Bordx. 15, No. 1, 83--97 (2003; Zbl 1047.11108)

### References:

[1] | Janusz, G., Algebraic number fields. Academic Press, New York, 1973. · Zbl 0307.12001 |

[2] | Sands, J.W., Galois groups of exponent two and the Brumer-Stark conjecture. J. Reine Angew. Math.349 (1984), 129-135. · Zbl 0521.12009 |

[3] | Sands, J.W., Two cases of Stark’s conjecture. Math. Ann.272 (1985), 349-359. · Zbl 0554.12006 |

[4] | Stark, H.M., L-functions at s = 1 IV. First derivatives at s = 0. Advances in Math.35 (1980), 197-235. · Zbl 0475.12018 |

[5] | Tate, J.T., Les conjectures de Stark sur les fonctions L d’Artin en s = 0. Birkhäuser, Boston, 1984. · Zbl 0545.12009 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.