  We elaborate an explicit version of the relative trace formula on $\PGL(2)$
over a totally real number field for the toral periods of Hilbert cusp forms
along the diagonal split torus. As an application, we prove (i) a spectral
equidistribution result in the level aspect for Satake parameters of
holomorphic Hilbert cusp forms weighted by central $L$-values, and (ii) a bound
of quadratic base change $L$-functions for Hilbert cusp forms with a subconvex
exponent in the weight aspect.

Number Theory