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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:
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