On the denominators of the special values of the partial zeta functions of real quadratic fields

It is classically known that the special values of the partial zeta functions of real quadratic fields, or more generally, of totally real fields at negative integers are rational numbers.
In this talk, I would like to discuss the denominators of these rational numbers in the case of real quadratic fields.
More precisely, Duke recently presented a conjecture which gives a universal upper bound for the denominators of these special values of the partial zeta functions of real quadratic fields.
I would like to explain that by using Harder's theory on the denominator of the Eisenstein class for SL(2,Z), we can prove the conjecture of Duke and moreover the sharpness of his upper
bound. This is a joint work with Ryotaro Sakamoto.