  Following a strategy suggested by Michel--Venkatesh, we study the cubic
moment of automorphic $L$-functions on $\operatorname{PGL}_2$ using regularized
diagonal periods of products of Eisenstein series. Our main innovation is to
produce vectors whose integral transforms achieve arbitrarily weighted moments.
Applications include general Motohashi-type identities and Weyl-type subconvex
bounds for some families of $L$-functions, extending some results of
Conrey--Iwaniec and Petrow--Young to the number field setting. We deduce
improved estimates for representation numbers of ternary quadratic forms over
number fields and for the prime geodesic theorem on arithmetic hyperbolic
$3$-folds.
