In this talk, I plan to explain a construction of a five-variable $p$-adic $L$-function associated with Hida families on $U(2, 1) \times U(1, 1)$ and explain a possible connection with the p-adic Ichino-Ikeda formula. This construction consists of two steps: (i) the p-adic interpolation of the GGP period integrals for Hida families on $U(2, 1) \times U(1, 1)$; (ii) the explicit evaluation of GGP period integrals via the the Ichino-Ikeda formula.

I will primarily focus on some key ingredients in the step (i), in particular, the use of $p$-integrality of the Taylor expansion of CM theta functions due to K. Bannai and S. Kobayashi. This is a joint work with M. Harris and S. Yamana.