An l≠p-interpolation of genuine p-adic L-functions

Let F be a totally real field, l and p distinct odd prime unramified in F and l a prime above l . Let K / F be a p -ordinary CM quadratic extension and λ an arithmetic Hecke character over K . Hida constructed a measure on the l -anticyclotomic class group of K interpolating the normalised Hecke L -values L alg , l ( 0 , λ ν ) , as ν varies over the finite order l -power conductor anticyclotomic characters. In this article, we interpolate the measures as λ varies in a p -adic family. In particular, this gives p -adic deformation of the measures. An analogue holds in the case of self-dual Rankin–Selberg convolution of a Hilbert modular form and a theta series. In the case of root number - 1 , we describe an upcoming analogous interpolation of the p -adic Abel–Jacobi image of generalised Heegner cycles associated with the convolution.