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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)].

MSC:

11R42 Zeta functions and \(L\)-functions of number fields
11R20 Other abelian and metabelian extensions
11R27 Units and factorization
11R37 Class field theory
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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
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