p-adic L-functions via local-global interpolation: the case of GL2 x GU(1)

Let F be a totally real field, and let E/F be a CM quadratic extension. We construct a p-adic L-function attached to Hida families for the group GL(2/F) x Res(E/F)GL(1) . It is characterized by an exact interpolation property for critical Rankin-Selberg L-values, at classical points corresponding to representations pi boxed times chi with the weights of chi smaller than the weights of pi . Our p-adic L-function agrees with previous results of Hida when E/F splits above p or F = Q, and it is new otherwise. Exploring a method that should bear further fruits, we build it as a ratio of families of global and local Waldspurger zeta integrals, the latter constructed using the local Langlands correspondence in families. In an appendix of possibly independent recreational interest, we give a reality-TV-inspired proof of an identity concerning double factorials.