On Motohashi’s formula

Abstract

We complement and offer a new perspective of the proof of a Motohashi-type formula relating the fourth moment of L L -functions for G L 1 GL_1 with the third moment of L L -functions for G L 2 GL_2 over number fields, studied earlier by Michel-Venkatesh and Nelson. Our main tool is a new type of pre-trace formula with test functions on M 2 ( A ) M_2(\mathbb {A}) instead of G L 2 ( A ) GL_2(\mathbb {A}) , on whose spectral side the matrix coefficients are replaced by the standard Godement-Jacquet zeta integrals. This is also a generalization of Bruggeman-Motohashi’s other proof of Motohashi’s formula. We give a variation of our method in the case of division quaternion algebras instead of M 2 M_2 , yielding a new spectral reciprocity, for which we are not sure if it is within the period formalism given by Michel-Venkatesh. We also indicate a further possible generalization, which seems to be beyond what the period method can offer.