It can be used to give a short proof of the weak Mordell-Weil theorem, see my answer in Proofs of Mordell-Weil theorem.

Another application: Proving the ($p'$-part of the) conjecture of Birch and Swinnerton-Dyer for Abelian schemes over $1$- or higher dimensional bases over finite fields under the assumption that the algebraic rank equals the analytic rank or one $\ell$-primary subgroup of the Tate-Shafarevich group is finite.

It can be used to give a short proof of the weak Mordell-Weil theorem, see my answer in Proofs of Mordell-Weil theorem.

Another application: Proving the ($p'$-part of the) conjecture of Birch and Swinnerton-Dyer for Abelian schemes over $1$- or higher dimensional bases over finite fields under the assumption that the algebraic rank equals the analytic rank or one $\ell$-primary subgroup of the Tate-Shafarevich group is finite.

It can be used to give a short proof of the weak Mordell-Weil theorem, see my answer in Proofs of Mordell-Weil theorem.

It can be used to give a short proof of the weak Mordell-Weil theorem, see my answer in Proofs of Mordell-Weil theorem.

Another application: Proving the ($p'$-part of the) conjecture of Birch and Swinnerton-Dyer for Abelian schemes over $1$- or higher dimensional bases over finite fields under the assumption that the algebraic rank equals the analytic rank or one $\ell$-primary subgroup of the Tate-Shafarevich group is finite.