Seminar in Algebraic Geometry and Number Theory

Abstract: Darmon and Vonk's theory of rigid meromorphic cocycles gives a conjectural description of abelian extensions of real quadratic fields akin to the classical theory of complex multiplication. Special cases of these conjectures have been proved by Darmon-Pozzi-Vonk using p-adic families of Hilbert modular forms. I will explain how a reinterpretation in terms of the so-called Kudla programme suggests an approach to extending their results. This point of view also gives a new proof of a theorem of Rickards on intersections of geodesics on Shimura curves, which is related to forthcoming work of Darmon-Vonk.