Class fields of abelian extensions of \(\mathbb Q\). (English) Zbl 0545.12005

It has been a frequent hope to relate the zeros of various zeta functions to the eigenvalues of appropriate operators. For example, this was accomplished for the zeta functions of curves over finite fields by A. Weil in the 1940’s. The authors solve this problem for the Kubota-Leopoldt \(p\)-adic \(L\)-functions by relating these functions to the characteristic polynomials obtained from Galois actions on certain Iwasawa modules (inverse limits of ideal class groups). This relationship had been conjectured by K. Iwasawa and came to be known as the ”Main Conjecture” of Iwasawa’s theory of \(\mathbb Z_ p\)-extensions.
The initial step towards the present result perhaps was that of K. A. Ribet [ibid. 34, 151–162 (1976; Zbl 0338.12003)], who used modular curves to prove the converse of Herbrand’s theorem. Let \(p\) be an odd prime and consider the \(p\)th cyclotomic field \(K_ p\). The action of \(\text{Gal}(K_ p/\mathbb Q)\) breaks the \(p\)-part of the ideal class group of \(K_ p\) into various eigenspaces. Herbrand showed that if an eigenspace is non-trivial then a corresponding Bernoulli number is divisible by \(p\), and Ribet proved the converse, which amounted to constructing certain unramified extensions of \(K_ p\). The second author strengthened and extended Ribet’s results in [ibid. 58, 1–35 (1980; Zbl 0436.12004)], which finally led to the present paper.
It would be pointless to say more about the result and its proof. The authors include an excellent 8-page introduction at the beginning of their paper, outlining the result, the ideas of the proof, and the consequences (see also pp. 214–225). There are also the Bourbaki talk of J. Coates [Sémin. Bourbaki, 33e année, Vol. 1980/81, Exp. No. 575, Lect. Notes Math. 901, 220–241 (1981; Zbl 0506.12001)] and the AMS talk of S. Lang [Bull. Am. Math. Soc., New Ser. 6, 253–316 (1982; Zbl 0482.12002)], both of which can be consulted profitably by the interested reader.
Reviewer: L.Washington


11R23 Iwasawa theory
11G05 Elliptic curves over global fields
Full Text: DOI EuDML


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