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

Title data

Keller, Timo:
On an analogue of the conjecture of Birch and Swinnerton-Dyer for Abelian schemes over higher dimensional bases over finite fields.
In: Documenta Mathematica. Vol. 24 (2019) . - pp. 915-993.
ISSN 1431-0635
DOI: https://doi.org/10.25537/dm.2019v24.915-993

Project information

Project financing: Studienstiftung des deutschen Volkes

Abstract in another language

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 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.

Further data

Item Type: Article in a journal
Refereed: Yes
Keywords: L-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture; heights; étale cohomology; higher regulators; zeta and L-functions; abelian varieties of dimension >1; étale and other Grothendieck topologies and cohomologies; arithmetic ground fields
Subject classification: Mathematics Subject Classification Code: 11G40, 11G50, 19F27, 11G10, 14F20, 14K15
Institutions of the University: Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Chair Mathematics II
Result of work at the UBT: Yes
DDC Subjects: 500 Science > 510 Mathematics
Date Deposited: 15 Oct 2019 07:59
Last Modified: 15 Oct 2019 07:59
URI: https://eref.uni-bayreuth.de/id/eprint/52764