This work is a continuation of the author's earlier papers on Hilbert modular functions and arithmetic Hilbert modular forms. Here a q- expansion principle is established necessary to obtain reciprocity laws for special values of arithmetic Hilbert modular functions (associated to a totally real algebraic number field k) which subsumes the classical results on complex multiplication. After developing the action of the idele group I(k) on the 'special points' of an adelic double coset space arising from \(GL_ 2(k)\), the author studies the relevant Eisenstein series, their Fourier expansions and behaviour under \(Gal({\mathbb{Q}}_{ab}/{\mathbb{Q}})\) and the structure of the graded algebra of arithmetic Hilbert modular forms. This is followed by a careful analysis of the splitting of class polynomials (attached to orders in a complex quadratic extension of k) yielding a generalization of Hecke's result (for real quadratic k) and finally a q-extension principle in the case of k-arithmetic Hilbert modular functions. The author's objective to develop a theory independent of abelian varieties and based on the analytic and arithmetic properties of the modular functions themselves and on (Kronecker-type) congruence relations has indeed been realized in his recent 'elementary' proof of Shimura's reciprocity law (for the case of Hilbert modular functions). [For part I see Prog. Math. 46, 49-92 (1984; Zbl 0549.10022).]