It is a classical theorem of Griffiths that homological and algebraic equivalence do not coincide in general for algebraic cycles of codimension two or larger on a smooth projective complex variety. The situation over number fields is more delicate and is related to a higher dimensional generalization of the Birch-Swinnerton-Dyer conjecture, namely the Bloch-Beilinson conjecture. I will outline some of the history of this problem and explain some recent work (joint with Bertolini/Darmon) that uses p-adic L-functions to construct an infinite family of examples of this phenomenon over number fields. The motives in question are CM motives and this leads to a simple (to state) conjecture about non-vanishing of derivatives of L-functions of certain Hecke characters of imaginary quadratic fields.