On an Analogue of the Conjecture of Birch and Swinnerton-Dyer for Abelian Schemes over Higher Dimensional Bases over Finite Fields

  • Timo Keller

    Mathematisches Institut, Universität Bayreuth, 95440 Bayreuth, Germany
On an Analogue of the Conjecture of Birch and Swinnerton-Dyer for Abelian Schemes over Higher Dimensional Bases over Finite Fields cover
Download PDF

This article is published open access.

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 pp. We prove the prime-to-pp part conditionally on the finiteness of the pp-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-pp part of the conjecture for constant or isoconstant Abelian schemes, in particular the prime-to-pp part for (1) relative elliptic curves with good reduction or (2) Abelian schemes with constant isomorphism type of A[p]\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.

Cite this article

Timo Keller, On an Analogue of the Conjecture of Birch and Swinnerton-Dyer for Abelian Schemes over Higher Dimensional Bases over Finite Fields. Doc. Math. 24 (2019), pp. 915–993

DOI 10.4171/DM/697