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We formulate an analogue of the conjecture of Birch and Swinnerton-Dyer forAbelian schemes with everywhere good reduction over higher dimensional basesover finite fields of characteristic $p$. We prove the prime-to-$p$ partconditionally on the finiteness of the $p$-primary part of the Tate-Shafarevichgroup or the equality of the analytic and the algebraic rank. If the base is aproduct of curves, Abelian varieties and K3 surfaces, we prove the prime-to-$p$part of the conjecture for constant or isoconstant Abelian schemes, inparticular the prime-to-$p$ part for (1) relative elliptic curves with goodreduction or (2) Abelian schemes with constant isomorphism type of$\mathscr{A}[p]$ or (3) Abelian schemes with supersingular generic fibre, andthe full conjecture for relative elliptic curves with good reduction overcurves and for constant Abelian schemes over arbitrary bases. We also reducethe conjecture to the case of surfaces as the basis.

On an analogue of the conjecture of Birch and Swinnerton-Dyer for Abelian schemes over higher dimensional bases over finite fields