A Local-global Summation Formula for Abelian Varieties

Let $K$ be a field finitely generated over $\text{Q}$, and $A$ an Abelian variety defined over $K$. Then by the Mordell-Weil Theorem, the set of rational points $A(K)$ is a finitely-generated Abelian group. In this paper, assuming Tate’s Conjecture on algebraic cycles, we prove a limit formula for the Mordell-Weil rank of an arbitrary family of Abelian varieties $A$ over a number field $k$; this is the Abelian fibration analogue of the Nagao formula for elliptic surfaces $E$, originally conjectured by Nagao, and proven by Rosen and Silverman to be equivalent to Tate’s Conjecture for $E$. We also give a short exact sequence relating the Picard Varieties of the family $A$, the parameter space, and the generic fiber, and use this to obtain an isomorphism (modulo torsion) relating the Neron-Severi group of $A$ to the Mordell-Weil group of $A$.