On extended interpolation properties of the Katz $p$-adic $L$-function

We describe extended interpolation properties of the Katz $p$-adic $L$-function over an imaginary quadratic field. We have three applications. Firstly, we give an elliptic analogue of the Coleman-Ihara formula. Secondly, we extend Kato's explicit reciprocity law for elliptic units and give a proof of the Tamagawa number conjecture for critical Hecke motives. Finally, we prove a large part of the $p$-adic Beilinson conjecture for the Katz $p$-adic $L$-function.