Arithmetic Theta Lifts and the Arithmetic Gan–gross–prasad Conjecture for Unitary Groups

In 1980s, Gross–Zagier [GZ86] established a formula that relates the Neron–Tate height of Heegner points on modular curves to the central derivative of certain L-functions associated to modular forms. Around the same time, Waldspurger proved a formula, relating toric periods of modular forms to the central value of certain L-functions. Gross put both of these formula in the framework of representation theory in his MSRI lecture in 2002. In this framework, the formula of Waldspurger concerns the toric periods of automorphic forms on quaternion algebras, while the formula of Gross–Zagier maybe viewed as a formula for the “periods” of “automorphic forms” on the incoherent quaternion algebras. The proof of the most general form of the Gross–Zagier formula given in [YZZ13] has been largely inspired by the proof of Waldspurger’s formula. Gross–Prasad [GP92] formulated a conjecture which generalizes the framework of Waldspurger to relate the nonvanishing of SO(n)-periods of automorphic form on SO(n) × SO(n + 1) and the nonvanishing of the central value of certain Rankin–Selberg L-functions, with Waldspurger’s formula being the case n = 2. Gan–Gross–Prasad [GGP12] further generalized this framework to include all classical groups. Parallel to the periods of automorphic forms, a conjectural generalization of the Gross–Zagier formula to higher-dimensional Shimura varieties has been proposed, for instance, in [GGP12,Zha12]. These are generally referred to as the arithmetic Gan–Gross–Prasad conjectures.