Class numbers of cyclic 2-extensions and Gross conjecture over \(\mathbb Q\)
In [``Gross' conjecture on the special values of Abelian \(L\)-functions at \(s=0\)'', Comment. Math. Univ. St. Pauli 40, No. 1, 101--124 (1991; Zbl 0742.11055)], \textit{N. Aoki} announced a proof of the Gross conjecture over \(\mathbb Q\). This article contained serious mistakes, and correct proofs were later given by \textit{D. Burns} [``Equivariant Tamagawa numbers and refined abelian Stark conjectures'', J. Math. Sci., Tokyo 10, No. 2, 225--259 (2003; Zbl 1037.11072)] and \textit{N. Aoki} himself [``On Tate's refinement for a conjecture of Gross and its generalization'', J. Théor. Nombres Bordeaux 16, No. 3, 457--486 (2004; Zbl 1071.11064)]. In this article, the authors study Aoki's first proof; they succeed in showing that Aoki's proof can be saved by proving the following interesting conjecture on class groups of cyclic extensions of number fields. Assume for simplicity that \(m = p_1 \ldots p_n\) is a product of primes of the form \(p_r \equiv 1 \bmod 4\). Let \(L\) be a maximal cyclic \(2\)-extension contained in the field of \(m\)-th roots of unity such that all primes \(p_j\) are completely ramified. Define a tower of quadratic extensions \(E_j/E_{j-1}\) by \(\mathbb Q = E_0 \subseteq E_1 \subseteq \cdots \subseteq E_t = L\), and let \(C_j\) denote the \(2\)-part of the narrow class group of \(E_j\) and \(h_j\) its order. Since the extensions \(E_j/E_{j-1}\) are totally ramified, class field theory shows that the norm map \(C_j \longrightarrow C_{j+1}\) is surjective. Thus \(h_j^* = h_j/h_{j-1}\) is an integer, which by genus theory is divisible by \(2^{n-1}\). Then the authors conjecture that the following statements are equivalent: (1) \(h_1 = 2^{n-1}\); (2) \(h_j^* = 2^{n-1}\) for some \(j\); (3) \(h_j^* = 2^{n-1}\) for all \(j\). The reduction of the Gross conjecture to this one involves Sinnott's class number formula and the Stickelberger ideal.
scientific article; zbMATH DE number 5857692
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