Four-variable p-adic triple product L-functions and the trivial zero conjecture

We construct the four-variable primitive p-adic L-functions associated with the triple product of Hida families and prove the explicit interpolation formulae at all critical values in the balanced range. Our construction is to carry out the p-adic interpolation of Garrett's integral representation of triple product L-functions via the p-adic Rankin-Selberg convolution method. As an application, we obtain the cyclotomic p-adic L-function for the motive associated with the triple product of elliptic curves and prove the trivial zero conjecture for this motive. In particular, this proves the first cases of the Greenberg-Benois trivial zero conjecture where multiple trivial zeros are present and the Galois representation is not of GL(2)-type.