The classical Gross-Zagier formula and the modularity theorem leads to a proof of half-BSD (i.e. an inequality and not equality) for elliptic curves of analytic rank 0.
The classical Gross-Zagier formula andgives something on the modularity theorem leads toBSD conjecture for elliptic curves over $\mathbb{Q}$. In particular the something that interest me is that if an elliptic curve as analytic rank 1, then the elliptic curve as a proofpoint of half-BSDinfinite order so as algebraic rank at least 1.
As said in the answers, in the function field case a theorem of Tate and Milne, says that the algebraic rank is smaller than the analytic rank (i.eso BSD conjecture is true for analytic rank 0). So an inequality and not equality) for elliptic curvesanalogue of the Gross-Zagier formula in the function field case, gives the BSD conjecture for analytic rank 01.
This article of Yun-Zhang pretends :
This identity can be viewed as a function-field analog of the Waldspurger and Gross–Zagier formula for higher derivatives of L-functions.
Does this article as any consequence on the analogue of BSD in function fields ?