## 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.

### MSC:

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

Zbl 0681.12005