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. In this paper we give a determinant formula for congruentzeta functions of real Abelian function elds. We also give some examplesof congruent zeta functions when the conductor of real Abelian function eld is monic irreducible. 1. IntroductionLet k = F q ( T ) be the rational function eld over the nite eld F q andA = F q [ T ]. Let 1 be the place of k associated to 1 =T , which is called thein nite one of k. Write A + = f 1 = M 2 A : M is monic g and A +irr = fP 2 A + : P is irreducible g . For any M 2 A + , write K M for the M th cyclotomicfunction eld and K + M for the maximal real sub eld of K M . In this paper, byan Abelian function eld , we always mean a nite Abelian extension F of kwhich is contained in a cyclotyomic function eld K M , and F is said to be real if 1 splits completely in F . Let N = N ( F ) 2 A + be the conductor of F , thatis, K N is the smallest cyclotomic function eld containing F . For such a eld F , there exists a polynomial P F ( X ) 2 Z[

A DETERMINANT FORMULA FOR CONGRUENT ZETA FUNCTIONS OF REAL ABELIAN FUNCTION FIELDS