The conductors of Eisenstein characters in cyclotomic number fields
To summarize this paper one can do little better than to paraphrase (part of) the author's excellent introduction: Let \({\mathbf k}= \mathbb{F}_q (T)\) be the field of rational functions over the finite field \(\mathbb{F}_q\) and let \(M\in \mathbb{F}_q [T]\) be monic. Let \({\mathbf k} (M)\) be the cyclotomic function field obtained by adjoining the \(M\)-division points of the Carlitz module; thus \({\mathbf k} (M)/{\mathbf k}\) is abelian and shares many properties with \(\mathbb{Q} (\zeta_n)/ \mathbb{Q}\). In ``Hecke characters and Eisenstein reciprocity in function fields'' [J. Number Theory 43, 251-292 (1993; Zbl 0774.11028)]\ the author constructed a Hecke character \(\Phi_K\) for any subextension \(K/{\mathbf k}\) of \({\mathbf k} (M)/{\mathbf k}\) (in fact, the author's construction works quite generally for sub- extensions of cyclotomic function fields over arbitrary base fields). The character \(\Phi_{{\mathbf k} (M)}\) is an analog of the Gauss sum Hecke character of \textit{A. Weil} constructed in ``La cyclotomie jadis et naguére'' [Enseign. Math., II. Ser. 20, 247-263 (1974; Zbl 0352.12006)]\ and ``Sommes de Jacobi et charactères de Hecke'' [Nachr. Akad. Wiss. Göttingen, II. Math.-Phys. Kl. 1974, 1-14 (1974; Zbl 0367.10035)]. (N.B.: In \textit{A. Weil}'s first paper ``Jacobi sums as `Grössencharaktere'\ '' [Trans. Am. Math. Soc. 73, 487-495 (1952; Zbl 0048.270)], Hecke characters were constructed only out the Jacobi sums. In the Göttingen paper (loc. cit.) Weil observed that the Jacobi sum is a product of Gauss sums and so he attempted to find the most general combination of Gauss sums that lie in a given subfield and make a Hecke character. It is found that in \(\mathbb{Q} (\zeta_n)\), the \(n\)-th -- \(2n\)- th if \(n\) is odd -- power of the (normalized) Gauss sum can be made into a Hecke character. This is what the author calls a ``Gauss sum Hecke character''. In general, the Brumer-Stark units in a subfield make a Hecke character that the author calls the ``Eisenstein character''.) The techniques of finite Fourier analysis of Weil do not carry over to the function field setting because \({\mathbf k} (M)\) is not generated by roots of unity but rather division values of rank one Drinfeld modules. Instead, the author's construction of \(\Phi_K\) is based on the axioms of the Brumer-Stark conjecture over \({\mathbf k}\). These axioms alone imply that the Brumer-Stark ``units'' in \(K\) can be aligned into a Hecke character that satisfies Eisenstein reciprocity. This reciprocity law is an analog of the law established by Weil for Gauss sums in his Enseign. paper (loc. cit.). In the paper under review the author applies some techniques from his function field Hecke character paper to the normalized Gauss sum Hecke character \(\varepsilon_K\) (called, as above, the ``Eisenstein character of \(K\)'') taking values in any given subfield \(K\) of a cyclotomic number field. The main result is an exact calculation of the conductor of \(\varepsilon_K\) through a new use of Eisenstein reciprocity.
scientific article; zbMATH DE number 796751