The Gross-Zagier-Zhang formula over function fields

TL;DR

This paper proves the Gross-Zagier-Zhang formula over global function fields, linking heights of CM points on abelian varieties with derivatives of quadratic base change L-functions, using an arithmetic trace identity approach.

Contribution

It extends the Gross-Zagier-Zhang formula to arbitrary characteristic function fields using a novel arithmetic trace identity method.

Findings

Established the formula over all characteristics of global function fields.

Connected heights of CM points with derivatives of L-functions explicitly.

Introduced an arithmetic variant of a relative trace identity for the proof.

Abstract

We prove the Gross-Zagier-Zhang formula over global function fields of arbitrary characteristics. It is an explicit formula which relates the Neron-Tate heights of CM points on abelian varieties and central derivatives of associated quadratic base change $L$-functions. Our proof is based on an arithmetic variant of a relative trace identity of Jacquet. This approach is proposed by W. Zhang.