On a conjecture of Gross on special values of L-functions
The author studies the case \(k={\mathbb{Q}}\) of a conjecture of \textit{B. H. Gross} [J. Fac. Sci., Univ. Tokyo, Sect. I A 35, No.1, 177-197 (1988; Zbl 0681.12005)]: \(\theta_ G\equiv h_ T\cdot \det_ G \lambda\) mod \(I^{n+1}\), where \(\theta_ G\in {\mathbb{Z}}[G]\) is determined by \(\chi (\theta_ G)=L_ T(\chi,0)\) for all \(\chi\in \hat G\), \(L_ T(\chi,s)\) is the Hecke-Tate L-function relative to S and modified at T, S and T are sets of places of k, \(G=Gl(K/k)\), K an Abelian extension unramified outside S, \(\det_ G \lambda \in I^ n/I^{n+1}\) is the regulator defined over S-units, I the augmentation ideal of \({\mathbb{Z}}[G]\), \(h_ T\) is the element number of the group of invertible A-modules with trivialization at T, A the S-integers. He proves the conjecture for \(n=1\) in an elementary way and shows \(\theta_ G\in I^ 2\) for \(n\geq 2\). He also gives an example in case \(n=2\) for which the conjecture holds.
scientific article; zbMATH DE number 4128935