Construction of normal bases by special values of Hilbert modular functions
Let \(K\) be a cyclic imaginary extension of \(\mathbb{Q}\). In this paper, the author shows how to construct elements \(\theta\) generating a normal basis of certain class fields over \(K\), by virtue of Hilbert modular functions. The details are very intricate, but (as the author points out) the general line becomes quite clear when one looks at earlier work of \textit{T. Okada} [J. Lond. Math. Soc., II. Ser. 22, 221-225 (1980; Zbl 0448.12009)]\ for the case that \(K\) is imaginary quadratic. Quite generally, a given element \(\theta\) generates a normal basis of the abelian \(G\)-Galois extension \(L/K\) iff none of the resolvents \((\theta,\chi)=\sum_{\sigma\in G}\chi(\sigma) \theta^\sigma\) vanishes, where \(\chi\) runs over all complex characters of \(G\). In Okada's work, as well as in the present article, the resolvents which come up can be identified with special values of Hecke L-series, not for \(\chi\) but for a modified character \(\chi\phi\) (note \(\chi\) is an ordinary character, and \(\chi\phi\) will be a true Hecke character). The general theory of Hecke L-series then allows to conclude that these values do not vanish. In Okada's paper, the modular forms which were used are, up to a factor, division values of derivatives of the Weierstrass \(\wp\)-function. In the setting of the paper under review, explicit Hilbert modular forms on the \(m\)-fold power of the upper half plane are used instead (with \(2m=[K:{\mathbb{Q}}]\)). In the main theorem, there are several technical conditions on the class field \(L\) for which a normal basis is being constructed, and these conditions are nontrivial to check. The author concludes with a nice explicit example where \(K={\mathbb{Q}}(\zeta_5)\); \(L\) turns out to be just the ray class field modulo 7, and with some effort the reviewer succeeded in verifying that all required conditions are met in the example. This is a well-written but also a demanding paper.
scientific article; zbMATH DE number 1057073