Deformations of Theta Integrals and A Conjecture of Gross-Zagier

TL;DR

This paper proves a conjecture by Gross and Zagier on the algebraicity of higher Green functions at CM points, introducing a novel incoherent Eisenstein series constructed via theta lifts over real quadratic fields.

Contribution

It provides a complete proof of the Gross-Zagier conjecture using a new incoherent Eisenstein series related to deformed theta integrals.

Findings

Proof of the Gross-Zagier conjecture completed.

Construction of an incoherent Eisenstein series over a real quadratic field.

Connection established between theta lifts and algebraic properties of Green functions.

Abstract

In this paper, we complete the proof of the conjecture of Gross and Zagier 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.