Deformations of Theta Integrals and A Conjecture of Gross-Zagier

Abstract

In this paper, we complete the proof of Gross-Zagier's conjecture concerning algebraicity of higher Green functions at a single CM point on the product of modular curves. The new ingredient is an analogue of the incoherent Eisenstein series over a real quadratic field, which is constructed as the Doi-Naganuma theta lift of a deformed theta integral on hyperbolic space.