An explicit trace formula of Jacquet-Zagier type for Hilbert modular forms

We give an exact formula of the average of adjoint L-functions of holomorphic Hilbert cusp forms with a fixed weight and a square-free level, which is a generalization of Zagier's formula known for the case of elliptic cusp forms on SL2(Z). As an application, we prove that the Satake parameters of Hilbert cusp forms with a fixed weight and with growing square-free levels are equidistributed in an ensemble constructed by values of the adjoint L-functions.