Keller, Timo On an analogue of the conjecture of Birch and Swinnerton-Dyer for abelian schemes over higher dimensional bases over finite fields. (English) Zbl 1443.11126 Doc. Math. 24, 915-993 (2019). In this paper, the author presents an analogue for the Birch and Swinnerton-Dyer conjecture for abelian schemes with everywhere good reduction over higher dimensional bases over finite fields.Let \(p\) be the characteristic of the base field. The author proves the prime-to-\(p\) part of the conjecture assuming either the finiteness of the \(p\)-primary part of the Tate-Shafarevich group or the equality of the analytic and algebraic rank.If the base variety is a product of curves, an abelian variety or a \(K\)3 surface then the author proves the prime-to-\(p\) part for constant or isoconstant abelian schemes.Moreover, the author shows that the general conjecture can be reduced to the surface case. Reviewer: Remke Kloosterman (Padova) Cited in 2 Documents MathOverflow Questions: Picard of the product of two curves Zeta function of Abelian variety over finite field MSC: 11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture 11G50 Heights 19F27 Étale cohomology, higher regulators, zeta and \(L\)-functions (\(K\)-theoretic aspects) 11G10 Abelian varieties of dimension \(> 1\) 14F20 Étale and other Grothendieck topologies and (co)homologies 14K15 Arithmetic ground fields for abelian varieties Keywords:\(L\)-functions of varieties over global field; Birch and Swinnerton-Dyer conjecture; higher regulators; étale and other Grothendieck topologies and cohomologies; arithmetic ground fields Software:MathOverflow × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] Altman, Allen and Kleiman, Steven: Introduction to Grothendieck duality theory. Lecture Notes in Mathematics 146, Berlin-Heidelberg-New York: Springer-Verlag, 1970. 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