×

A topological ’proof’ of a theorem of Ribet. (English) Zbl 0543.12008

Current trends in algebraic topology, Semin. London/Ont. 1981, CMS Conf. Proc. 2, 1, 43-47 (1982).
Let \(A\) denote the \(p\)-Sylow subgroup of the ideal class group of the cyclotomic field \(\mathbb{Q}(\xi_p)\); it is classical that there is a direct decomposition \(A=\oplus_{i \bmod p-1}A^{(i)}\), and that if \(0\ne A^{(1-k)}\) for \(k\) even, then \(p\mid B_k,\) the \(k\)-th Bernoulli number. The theorem of K. A. Ribet [Invent. Math. 34, 151–162 (1976; Zbl 0338.12003)] establishes the converse.
The author’s aim is to show that Ribet’s theorem can also be derived by an application of the machinery of algebraic \(K\)-theory with coefficients. He points out that the application involves a circular argument, since it is based on deep applications of number theory to \(K\)-theory, but that it illustrates how \(K\)-theory might be used in future.
[For the entire collection see Zbl 0538.00016.]

MSC:

11R70 \(K\)-theory of global fields
11R18 Cyclotomic extensions
11R29 Class numbers, class groups, discriminants
18F25 Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects)
11B68 Bernoulli and Euler numbers and polynomials