Stark's conjecture and Abelian \(L\)-functions with higher order zeros at \(s=0\)
The author proposes improvements of Stark's conjectures on Artin \(L\)-functions. Let \(K/k\) be an abelian extension of number fields, with Galois group \(G\); let \(T\subsetneq S\) be finite sets of places of \(k\) such that \(S\) contains the ramified ones in \(K/k\) and \(T\) splits completely in \(K/k\) (and contains the unramified infinite places). For \[ L_S(s,\chi)=\prod_{q\not\in S} (1-\chi (\text{Frob}(q))Nq^{-s})^{-1},\quad \chi \in \hat G,\;\text{Re}(s)>1, \] the following conjecture is proposed (where \(r=| T|\), \(w_K=| \text{tor}(K^\times)|\), \(c\in\mathbb Z)\): \[ (2^{-c}w_K/r!) L_S(0,\chi)=\theta (\chi) R(\chi,E),\text{ for\;all } \chi \in \hat G, \] where \(\theta (\chi)=<\chi,\mu >\in\mathbb Z[\chi]\), for a suitable \(\mu\in \mathbb Q[G]\), \(R(\chi,E)\) is a suitable \(\chi\)-regulator built ``essentially'' on independent \(T\)-units of \(K\). For \(r=1\), \(c=0\), \(\mu =1\), these are the Stark or Brumer-Stark conjectures [see \textit{J. Tate}, ``Les conjectures de Stark sur les fonctions \(L\) d'Artin en \(s=0\)'', Prog. Math. 47. Boston etc.: Birkhäuser (1984; Zbl 0545.12009)]. The author gives many remarks and observations to justify the conjecture, proves some essential functional properties in varying \(S\), \(k\). Finally, he proves some interesting cases of his conjecture, especially when the classical Stark's one is yet proved. Some aspects of \(S\)-genera theory are needed.
scientific article; zbMATH DE number 4025537
scientific article