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

TL;DR

This paper formulates and proves an analogue of the Birch and Swinnerton-Dyer conjecture for Abelian schemes over higher dimensional bases over finite fields, with results depending on certain finiteness conditions.

Contribution

It extends the BSD conjecture to higher dimensional bases over finite fields and proves partial results under specific conditions.

Findings

Proves the prime-to-p part of the conjecture conditionally on Tate-Shafarevich group finiteness.

Establishes the conjecture for constant or isoconstant Abelian schemes over certain bases.

Reduces the conjecture to the case of surfaces as the base.

Abstract

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