Popescu, Cristian D. Stark’s question and a refinement of Brumer’s conjecture extrapolated to the function field case. (English) Zbl 1059.11069 Compos. Math. 140, No. 3, 631-646 (2004). Let \(K/k\) denote a finite Galois extension of global fields. In the case that these are number fields, a well-known general conjecture by H. M. Stark predicts that the leading term at \(s=0\) of the Artin \(L\)-function for \(K/k\), with the Euler factors in a set \(S\) removed, has an arithmetic interpretation involving the \(S\)-units in \(K\). In [Abstr. Am. Math. Soc. 1, 28 (1980)] H. M. Stark posed a detailed question specifically for the case of second order zeros when \(K/k\) is abelian. The present author formulates a natural extrapolation of Stark’s question to the case of function fields. He shows that this question then has a negative answer, even in a weakened form.Secondly, the author considers the Brumer-Stark conjecture, more exactly, its “Brumer part” which in a certain sense generalizes the classical Stickelberger theorem; see chapter IV of J. Tate’s monograph “Les conjectures de Stark sur les fonctions L d’Artin en \(s=0\)” [Prog. Math. 47, Birkhäuser, Basel (1984; Zbl 0545.12009)]. He states a refinement of this conjecture, with the annihilator of the modified ideal class group of \(K\) replaced by the Fitting ideal of this group (regarded as a Galois module over \(\mathbb Z\)). He gives some motivation for this refinement and provides links between it and various versions of Stark’s question. As a consequence from the first part of the present work it then follows that this refined conjecture is, in general, false in characteristic \(p>0\). Reviewer: Tauno Metsänkylä (Turku) Cited in 1 ReviewCited in 5 Documents MSC: 11R42 Zeta functions and \(L\)-functions of number fields 11R27 Units and factorization 11R58 Arithmetic theory of algebraic function fields Keywords:Artin \(L\)-functions; units; arithmetic theory of function fields Citations:Zbl 0545.12009 PDFBibTeX XMLCite \textit{C. D. Popescu}, Compos. Math. 140, No. 3, 631--646 (2004; Zbl 1059.11069) Full Text: DOI