One of the main results of the BSD conjecture in characteristic $p>0$ is the fact that the algebraic rank is less or equal than the analytic rank and that there is an equality iff the $\ell$-torsion ($\ell\neq p$) of the Tate-Shafarevich group is finite.

The proof that I know of this fact (which goes back to Tate) uses the elliptic surface associated to the elliptic curve, proves an analogue of this result for the surface and goes back to the elliptic curve.

Does there exist a proof of this result without using the elliptic surface?