Harder's denominator problem for ${\mathrm{S}\mathrm{L}}_2(\mathbb{Z})$ and its applications

The aim of this paper is to give a full detail of the proof given by Harder of a theorem on the denominator of the Eisenstein class for ${\mathrm{S}\mathrm{L}}_2(\mathbb{Z})$ and to show that the theorem has some interesting applications including the proof of a recent conjecture by Duke on the integrality of the higher Rademacher symbols. We also present a sharp universal upper bound for the denominators of the values of partial zeta functions associated with narrow ideal classes of real quadratic fields in terms of the denominator of the values of the Riemann zeta function.