Fermat curves and explicit class field theory for real quadratic number fields

We use Fermat curves with real multiplication and classical CM theory to construct an explicit class field theory for arbitrary real quadratic number fields. The underlying strategy is well-established in computational class field theory, our main contribution is a geometric model to realize this strategy explicitly. This approach complements the recent breakthrough works solving conjectures by Brumer, Stark and Shintani and gives an alternative constructive solution to Hilbert's 12th problem for real quadratic fields.