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 which relates Néron-Tate heights CM points on abelian varieties and central derivatives associated quadratic base change L-functions. Our proof based arithmetic variant a relative trace identity Jacquet. This approach proposed by Zhang. apply our results to Birch Swinnerton–Dyer conjecture for $${\mathrm {GL}}_2$$ -type. In particular, we elliptic curves analytic rank 1.