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Improvement of a congruence for certain quadratic Stickelberger elements. (Amélioration d’une congruence pour certains éléments de Stickelberger quadratiques.) (French) Zbl 0895.11021
In an earlier paper, A. Bayad, W. Bley, and P. Cassou-Noguès [J. Algebra 179, 145-190 (1996; Zbl 0863.11074)] had found ideal factorizations of certain elliptic resolvents \(\tilde T_p(P,Q)\) in the spirit of the proof of Stickelberger’s theorem, thereby obtaining certain elements \(\theta_2(p)\) which annihilate parts of the classgroup of \(N=F(\zeta_l+\zeta_l^{-1})\). (For the exact conditions on the field \(F\) and the primes \(p\) and \(l\) we have to refer to the paper.) These elements \(\theta_2(p)\) were aptly called “quadratic Stickelberger elements” for the following reason: they lie in a rational group ring \(Q\Gamma\) with \(\Gamma= \text{Gal}(N/F)\), and they are congruent to \(c\sum t^2 \sigma_t^{-1}\) modulo \(Z\Gamma\), for an explicit rational \(c\).
The present paper consists of two parts: in the second part, it is shown that the elliptic resolvents in the earlier paper are \((p^3-p^2)\)-th powers. This allows to eliminate a factor of \(p^3-p^2\) from the previously constructed annihilators (which in a way is the main point). The \((p^3-p^2)\)-roots are constructed with the aid of elliptic functions of the second kind \(D_\Omega\) which are studied in the first part. The authors (re)prove a nice distribution relation for values of these functions which was first found by Frobenius. For readers not so familiar with the earlier work, the second part of the paper under review demands some effort. But this is more than compensated by the highly polished and very careful exposition of the whole paper.
Reviewer: C.Greither (Laval)

MSC:
11G05 Elliptic curves over global fields
11R33 Integral representations related to algebraic numbers; Galois module structure of rings of integers
11R29 Class numbers, class groups, discriminants
33E05 Elliptic functions and integrals
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References:
[1] BAYAD (A.) . - Résolvantes elliptiques et éléments de Stickelberger , Université de Bordeaux I, thèse soutenue le 24 avril 1992 .
[2] BAYAD (A.) . - Loi de réciprocité quadratique dans les corps quadratiques imaginaires , Ann. Inst. Fourier, t. 45 (5), 1995 , p. 1223-1237. Numdam | MR 96j:11139 | Zbl 0843.11047 · Zbl 0843.11047
[3] BAYAD (A.) , BLEY (W.) , CASSOU-NOGUÈS (Ph.) . - Sommes arithmétiques et éléments de Stickelberger , J. Algebra, t. 179 (1), 1996 , p. 145-190. MR 96k:11131 | Zbl 0863.11074 · Zbl 0863.11074
[4] CASSOU-NOGUÈS (Ph.) , TAYLOR (M.J.) . - Un élément de Stickelberger quadratique , J. Number Th., t. 37 (3), 1991 , p. 307-342. MR 92e:11125 | Zbl 0719.11075 · Zbl 0719.11075
[5] SHIH-PING CHAN . - Modular functions, elliptic functions and Galois module structure , J. Reine angew. Math., t. 375, 1987 , p. 67-82. MR 88j:11077 | Zbl 0613.12007 · Zbl 0613.12007
[6] EGAMI (Sh.) . - An elliptic analogue of the multiple Dedekind sums , Comp. Math., t. 99, 1995 , p. 99-103. Numdam | MR 96g:11040 | Zbl 0838.11029 · Zbl 0838.11029
[7] FROBENIUS (F.G.) . - Über die elliptischen Functionen zweiter Art , Ges. Abhand. b. II, p. 81-96 ; J. reine angew. Math., t. 93, 1882 , p. 53-68. JFM 14.0389.01 · JFM 14.0389.01
[8] HERMITE (Ch.) . - Sur quelques applications des fonctions elliptiques, Œuvres , t. III, p. 266 ; C. R. Acad. Sci. Paris, t. 85-94, 1877 - 1882 . JFM 14.0402.02 · JFM 14.0402.02
[9] HUSEMÖLLER (D.) . - Elliptic curves , Graduate texts in Math. 111, Springer-Verlag, 1986 . Zbl 0605.14032 · Zbl 0605.14032
[10] ITO (H.) . - On a product related to the cubic Gauss sums , J. reine angew. Math., t. 395, 1989 , p. 202-213. MR 90b:11080 | Zbl 0662.10027 · Zbl 0662.10027
[11] KUBERT (D.) . - Product formulae on elliptic curves , Invent. Math., t. 117, 1994 , p. 227-273. MR 95d:11075 | Zbl 0834.14016 · Zbl 0834.14016
[12] KUBERT (D.) , LANG (S.) . - Modular units , Grundlehren der math. Wiss. 244. - Springer-Verlag, 1981 . MR 84h:12009 | Zbl 0492.12002 · Zbl 0492.12002
[13] LANG (S.) . - Elliptic functions . - Addison-Wesley, 1973 . MR 53 #13117 | Zbl 0316.14001 · Zbl 0316.14001
[14] MUMFORD (D.) . - Abelian varieties . - Tata Institute of fundamental Research, Bombay, vol. 5, Oxford Univ. Press, 1970 . MR 44 #219 | Zbl 0223.14022 · Zbl 0223.14022
[15] MUMFORD (D.) . - Tata lectures on theta I , Progress in Math., vol. 28, Birkhäuser, 1983 . MR 85h:14026 | Zbl 0509.14049 · Zbl 0509.14049
[16] ROBERT (G.) . - Unités elliptiques , Bull. Soc. Math. France, Mémoire 36, 1973 . Numdam | MR 57 #9669 | Zbl 0314.12006 · Zbl 0314.12006
[17] ROBERT (G.) . - Concernant la relation de distribution satisfaite par la fonction \varphi associée à un réseau complexe , Invent. Math., t. 100, 1990 , p. 231-257. MR 91j:11049 | Zbl 0729.11029 · Zbl 0729.11029
[18] SRIVASTAV (A.) , TAYLOR (M.J.) . - Elliptic curves with complex multiplication and Galois module structure , Inv. Math., t. 99, 1990 , p. 165-184. MR 91b:11127 | Zbl 0705.14031 · Zbl 0705.14031
[19] WEIL (A.) . - Variétés kählériennes , Publication de l’Institut de Math. de l’Université de Nancago, VI, Hermann, Paris, 1958 . Zbl 0137.41103 · Zbl 0137.41103
[20] WEIL (A.) . - Elliptic functions according to Eisenstein and Kronecker , Ergeb. der Math. 88, Springer-Verlag, 1976 . MR 58 #27769a | Zbl 0318.33004 · Zbl 0318.33004
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