On zeta functions of modular representations of a discrete group

It is proved that the generating function defined by the numbers of isomorphism classes of absolutely irreducible representations of a finitely generated group G into GLd(Fqn) for n⩾1 is a rational function. It is also proved that the generating function defined by weighted sums over isomorphism classes of representations of G into GLd(Fqn) for n⩾1 is meromorphic over both the complex numbers and the p-adic complex numbers.