The Gross–Zagier–Zhang formula over function fields

We prove the Gross–Zagier–Zhang formula over global function fields of arbitrary characteristics. It is an explicit formula which relates the Néron-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 Zhang. We apply our results to the Birch and Swinnerton–Dyer conjecture for abelian varieties of GL2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm {GL}}_2$$\end{document}-type. In particular, we prove the conjecture for elliptic curves of analytic rank 1.