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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

Titelangaben

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. Bd. 24 (2019) . - S. 915-993.
ISSN 1431-0635
DOI: https://doi.org/10.25537/dm.2019v24.915-993

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Projektfinanzierung: Studienstiftung des deutschen Volkes

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 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.

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Publikationsform: Artikel in einer Zeitschrift
Begutachteter Beitrag: Ja
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
Fachklassifikationen: Mathematics Subject Classification Code: 11G40, 11G50, 19F27, 11G10, 14F20, 14K15
Institutionen der Universität: Fakultäten > Fakultät für Mathematik, Physik und Informatik > Mathematisches Institut > Lehrstuhl Mathematik II (Computeralgebra)
Fakultäten
Fakultäten > Fakultät für Mathematik, Physik und Informatik
Fakultäten > Fakultät für Mathematik, Physik und Informatik > Mathematisches Institut
Titel an der UBT entstanden: Ja
Themengebiete aus DDC: 500 Naturwissenschaften und Mathematik > 510 Mathematik
Eingestellt am: 15 Okt 2019 07:59
Letzte Änderung: 15 Okt 2019 07:59
URI: https://eref.uni-bayreuth.de/id/eprint/52764