## 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

### Citations:

Zbl 0521.12009; Zbl 0554.12006
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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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