Recently, H.Darmon and J.Vonk have introduced a construction of “p-adic singular moduli” for real quadratic fields.

This provides a conjectural p-adic analogue to the theory of complex multiplication in the context of real quadratic fields. In this talk, I will first review their constructions, and then explain ongoing work towards finding generalizations to other fields. This is work in progress with
Xavier Guitart (U. Barcelona) and Xavier Xarles (U. Autònoma de Barcelona).