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Zbl 1158.11320
Oukhaba, Hassan
Sign functions of imaginary quadratic fields and applications. (English)
[J] Ann. Inst. Fourier 55, No. 3, 753-772 (2005). ISSN 0373-0956; ISSN 1777-5310/e

From the introduction: In this paper we introduce the concept of a sign function of an imaginary quadratic field. As is proved in this article this concept is very helpful in the study of some arithmetical problems. Classically by sign we mean the extension to $\Bbb R^\times$ of the continuous homomorphism $s:\Bbb Q^\times\to \{-1,1\}$ satisfying $s(-1)=-1$. Here $A^\times$ is the multiplicative group of the ring $A$. In 1985 {\it D. R. Hayes} introduced the concept of a sign function of a global function field and used this notion to normalize Drinfeld modules of rank one [cf. Compos. Math. 55, 209--239 (1985; Zbl 0569.12008)]. The torsion points of these modules have many important arithmetical properties. They are essential in the construction of Stickelberger elements, Stark units, Euler systems, groups of cyclotomic units in characteristic $p$, etc. To recall this definition we let $K$ be a global function field. We denote by $\infty$ a fixed place of $K$, and by $\widehat{K}$ the completion of $K$ at $\infty$. Let us also denote by $\Bbb F_q$ both the finite field of $q$ elements and the constant field of $\widehat{K}$. Then a sign function, with respect to $(K,\infty)$, is a continuous homomorphism $s:\widehat{K}^\times\to\Bbb F_q^\times$ satisfying $s(a)=a$ for all $a\in\Bbb F_q^\times$. Our definition of a sign function in the case of an imaginary quadratic field $k\subset\Bbb C$ is as follows. Let $H\subset\Bbb C$ be the Hilbert class field of $k$. Then a sign function of $k$ is a surjective group homomorphism $s:\Lambda_6\to\mu_H$ satisfying $s({\cal U}_{12})=1$ and such that $s(\xi)=\xi$ for all $\xi\in\mu_k$ (see the notation below). As one may check the homomorphism $\widetilde{\kappa}:\Lambda_6\to\mu_H$ induced by $\kappa^{-1}$, where $\kappa$ is the character defined by {\it F. Hajir} and {\it F. R. Villegas} [Duke Math. J. 90, No.~3, 495--521 (1997; Zbl 0898.11025)] satisfies all these properties. Hence $\widetilde{\kappa}$ is a sign function of $k$. In Section 2 we associate to each couple $(s,{\germ m})$, where $s$ is a sign function and ${\germ m}$ is a nonzero integral ideal of $k$ prime to 6, a finite abelian extension $k_{{\germ m},s}\subset\Bbb C$ of $k$. The field $k_{{\germ m},s}$ is well described by class field theory. In particular $k_{{\germ m},s}$ contains the ray class field modulo ${\germ m}$, which we denote by $k_{{\germ m}}$. The extension $k_{{\germ m},s}/k_{{\germ m}}$ is cyclic of degree $w_H$ (resp. $w_H/w_k$) if ${\germ m}\ne(1)$ (resp. ${\germ m}=(1)$). The properties of the ramification in the extension $k_{{\germ m},s}/H_s$, where $H_s= k_{(1),s}$ lead us to consider $k_{{\germ m},s}$ as the analog of a cyclotomic number field or a cyclotomic function field as well. In Section 3 we associate to each integral ideal ${\germ c}$ of $k$ prime to $6N({\germ m})$ an algebraic integer $\Gamma_{{\germ m}}({\germ c})$, which is a root of the Ramachandra invariant. The construction of $\Gamma_{{\germ m}}({\germ c})$ involves the Klein function and the eta function of Hajir-Villegas. First we describe the Galois action on $\Gamma_{{\germ m}}({\germ c})$. This is essentially done by using the Shimura reciprocity law. In particular we prove that $\Gamma_{{\germ m}}({\germ c})\in k_{m,s}$, where $m= n({\germ m})$ and $s$ is the sign. Then we describe the behavior under the norm map of a certain power of $\Gamma_{{\germ m}}(1)$. In this we use the distribution law of the Siegel function stated in [{\it D. S. Kubert}, Invent. Math. 117, No. 2, 227--273 (1994; Zbl 0834.14016), \S2]. The result we get is a refinement of the well-known Theorem 2 of {\it G. Robert} [Bull. Soc. Math. Fr., Suppl., Mém. 36, 77 p. (1973; Zbl 0314.12006)] that gives the norm formulas satisfied by the Ramachandra invariants. In Section 4 we define the level ${\germ m}$ universal ordinary $s$-distribution $U_s({\germ m})$, in the spirit of those considered in Kubert, Anderson and Yin. We give the structure of $U_s({\germ m})$ as an abelian group and compute the Tate cohomology groups $\widehat{H}^n(J,U_s({\germ m}))$, where $J=\text{Gal}(k_{{\germ m},s} /k_{{\germ m}})$. In this we follow the method of {\it Y. Ouyang} [Proc. Am. Math. Soc. 130, No.~8, 2203--2213 (2002; Zbl 0997.11089)], which essentially uses Anderson's resolution and related spectral sequences. These cohomology groups naturally appear in many settings. See for instance Anderson's theory of epsilon extensions and its analog for function fields [{\it G. W. Anderson}, Duke Math. J. 114, No.~3, 439--475 (2002; Zbl 1056.11060) and {\it S. Bae} and {\it L. Yin}, Manuscr. Math. 110, No.~3, 313--324 (2003; Zbl 1098.11058)]. Let us remark that $U_s({\germ m})$ is naturally a $\text{Gal}(k_{{\germ m}/s}/k)$-module. Its Galois module structure is closely related to a certain group of elliptic units. This connexion will be made clear in a forthcoming paper in which we extend some results of Ouyang's paper (loc. cit.) to our case and use them to improve Theorem B of [{\it H. Oukhaba}, Compos. Math. 137, No.~1, 1--22 (2003; Zbl 1045.11043)].

MSC 2000:: *11G16 Elliptic and modular units
14K22 Complex multiplication (abelian varieties)
11R21 Other number fields

Keywords: sign function; narrow ray class field; Shimura reciprocity law; ordinary $s$-distributions; Anderson's resolution; spectral sequences

Citations: Zbl 0569.12008; Zbl 0898.11025; Zbl 0834.14016; Zbl 0314.12006; Zbl 0997.11089; Zbl 1056.11060; Zbl 1098.11058; Zbl 1045.11043

Cited in: Zbl 1161.11356

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