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

Abstract

This work is the second in a series, following Part I (Algebra Number Theory 18.10 (2024)) and preceding Part III (Math. Ann. 391.1 (2025)). We continue our investigation of spectral moments of $\hbox{GL}(3)\times \hbox{GL}(2)$ $\hbox{L}$-functions from the perspective of period integrals. Using an identity between two distinct periods for the $\hbox{GL}(3)$ Eisenstein series, we establish an exact Motohashi-type identity linking the shifted cubic moment of $\hbox{GL}(2)$ $\hbox{L}$-functions to the shifted fourth moment of $\hbox{GL}(1)$ $\hbox{L}$-functions. In addition, we offer a novel, intrinsic and automorphic account for the sources and symmetries of the full set of main terms for both moments, in agreement with the CFKRS Moment Conjectures (Proc. Lond. Math. Soc.(3) 91 (2005)).