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On the special values of abelian \(L\)-functions. (English) Zbl 0820.11069
The author proves the \(p\)-portion of a conjecture of Gross over the global function fields \(L\supset K\) of characteristic \(p\), i.e., \(\theta_ G\equiv \pm h_ T \text{ det}_ G \lambda \pmod {I_ G^{r+1}}\); where \(L\) is the maximal abelian pro-\(p\)-extension of \(K\) unramified outside \(S\) with Galois group \(G\), \(S\) is a fixed set of \(r\) primes of \(K\), \(h_ T\) is the modified class number of \(S\)-integers of \(K\) with an auxiliary set \(T\not\subset S\), \(\text{det}_ G \lambda\) is the \(G\)-regulator, \(\theta_ G\) is an element of the ring \(\mathbb{Z} [[ G]]\) of integral measures of \(G\) defined as the projective limit of \(\theta_{G'}\) over finite quotients \(G'\) of \(G\), and \(\theta_{G'}\in \mathbb{Z}[ G']\) is defined by \(\chi( \theta_{G'} )= L_ T (\chi, 0)\) for a complex character \(\chi\) of \(G'\) and modified \(L\)-function \(L_ T (\chi, s)\), \(I_ G \subset \mathbb{Z}[[ G]]\) is the ideal of measures of volume zero.
Refining the class number formula, the above conjecture was made by B. H. Gross [J. Fac. Sci., Univ. Tokyo, Sect. I A 35, No. 1, 177-197 (1988; Zbl 0681.12005)], and was proved in some cases for \(r=1\) and cyclic \(G\) by Hayes and Gross.

11R58 Arithmetic theory of algebraic function fields
11S40 Zeta functions and \(L\)-functions
11R20 Other abelian and metabelian extensions