Spectral Moment Formulae for $GL(3)\times GL(2)$ $L$-functions II: The Eisenstein Case

Abstract: This is a sequel to Kwan [Kw23]. We continue our investigations on the spectral moments of $L$-functions of $GL(3)\times GL(2)$-type from the perspective of period integrals. By manipulating an equality between two different periods for the group $GL(3)$, we establish an exact identity of Motohashi type connecting the shifted cubic moment of $GL(2)$ $L$-functions to the shifted fourth moment of $GL(1)$ $L$-functions. Furthermore, we are able to provide an intrinsic, automorphic explanation for the sources and symmetries of the full sets of main terms for both moments. They are shown to be in agreement with the CFKRS Moment Conjectures. This work presents a strategy for accomplishing the aforementioned tasks, distinct from the previous ones in the literature.