## On the special values of abelian L-functions

J. Math. Sci. Univ. Tokyo
Vol. 1 (1994), No. 2, Page 305--319.

Tan, Ki-Seng
On the special values of abelian L-functions
Here we give a proof of the $p$-portion of a conjecture of Gross over the global function fields of characteristic $p$. In this case, the conjecture is in fact a refinement of the class number formula. Here the classic Dedekind Zeta function is generalized by a $p$-adic measure which interpolates the special values of abelian L-functions, and the regulator of the units group is generalized by a $p$-adic regulator. The L-functions are associated to the characters of the maximal abelian extension of the given global field unramified outside a finite set {$v_0, v_1, \dots, v_r,$} of places of the field. The case that $r=1$ has been proved by Hayes. Gross also proved some congruence of the formula.