BSD and generalisation of Gross-Zagier formula

Question

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 Gross-Zagier formula gives something on the BSD 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 point of infinite 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 (so BSD conjecture is true for analytic rank 0). So an analogue of the Gross-Zagier formula in the function field case, gives the BSD conjecture for analytic rank 1.

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 ?

Answer 1

The BSD conjecture for an abelian variety $A$ over a function field holds if Ш$(A)[\ell^\infty]$ is finite for some prime $\ell$ ($\ell = p$ allowed). This is a theorem by Schneider, Bauer and Kato-Trihan. If $A$ is a constant abelian variety, Ш$(A)$ is finite by Milne's PhD thesis.

Edit: Since the analytic rank $\rho$ is always greater or equal than the algebraic rank, one has BSD if $\rho = 0$ (by the equivalence of weak BSD and the finiteness of an $\ell$-primary component of Sha). I show this inequality even for Abelian schemes over higher dimensional bases over finite fields in http://kellertimo.name/Height.pdf, Lemma 2.17.

Answer 2

Note that the Gross-Zagier theorem only yields the rank inequality for rank $0$ and $1$.

But the full rank inequality is alredy a theorem (of Tate and Milne) in the function fields case.

So perhaps the Yun and Zhang's formula, together with modularity (which is also known in positive characteristic), gives an alternative prove of the inequality in the particular case of low ranks, but it definitely doesn't say anything new about Birch and Swinnerton-Dyer for function fields.