On the refined class number formula of Gross
For a finite abelian extension \(K/k\) of congruence function fields, let \(G:= \text{ Gal}(K/k)\), \(S\) a finite non-empty set of places of \(k\) containing all ramified places in \(K/k\) and \(L _ S (\chi,s)\) the associated \(S\)-truncated Dirichlet \(L\)-function. If \(T\) is any finite non-empty set of places of \(k\) which is disjoint from \(S\), \textit{B. Gross} [J. Fac. Sci., Univ. Tokyo, Sect. I A 35, No. 1, 177--197 (1988; Zbl 0681.12005)] proved that \(\theta _ {K/k, S, T} (0)\) belongs to \({\mathbb Z}[G]\) where \[ \theta _{K/k,S,T}(s) = \biggl( \prod _ {t\in T} (1-N t ^ {1-s} \sigma _ t)\biggr) \sum_ {\chi \in \widehat{G}} e _ \chi L_ S (\chi, s). \] Here \(\widehat {G}\) is the dual of \(G\), \(e_ \chi\) denotes the corresponding idempotent \({{1}\over{| G| }} \sum _ {g\in G} \chi(g^ {-1}) g\in {\mathbb C} [G]\) and \(\sigma _ t\) is the Frobenius automorphism. B. Gross conjectured that \(\theta _ {K/k,S,T}(0)\) is congruent with \(\pm\) the \((S, T)\) class number of \(k\) multiplied by an explicit factor modulo \(I _ G ^ {| S| }\) where \(I_ G\) is the augmentation ideal of \({\mathbb Z}[G]\). In the paper under review it is verified that the conjecture of Gross is valid for any finite group \(G\) of prime exponent.
scientific article; zbMATH DE number 2091711