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On the quasi-group of a cubic surface over a finite field

Titelangaben

Elsenhans, Andreas-Stephan ; Jahnel, Jörg:
On the quasi-group of a cubic surface over a finite field.
In: Journal of Number Theory. Bd. 132 (Juli 2012) Heft 7 . - S. 1554-1571.
ISSN 0022-314X
DOI: 10.1016/j.jnt.2012.01.010

Abstract

We study the Mordell–Weil group MW(V) for cubic surfaces V over finite fields that are not necessarily irreducible and smooth. We construct a surjective map from MW(V) to a group that can be computed explicitly. For #MW(V), this yields a lower bound, which is (often but) not always trivial. To distinguish cases, we follow the classification of cubic surfaces, originally due to Schläfli and Cayley. On the other hand, we describe an algorithm that a priori gives an upper bound for MW(V). We report on our experiments for “randomly” chosen surfaces of the various types, showing that in all but one case lower and upper bounds agree. Finally, we give two applications to the number field case. First, we prove that the number of generators of MW(V) is unbounded. A second application explains why, for many reduction types, the Brauer–Manin obstruction may not distinguish points reducing to the smooth part.

Weitere Angaben

Publikationsform: Artikel in einer Zeitschrift
Begutachteter Beitrag: Ja
Fachklassifikationen: Mathematics Subject Classification Code: 11G25 (11D25 11G35 14G15)
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: 11 Apr 2016 06:31
Letzte Änderung: 11 Apr 2016 06:31
URI: https://eref.uni-bayreuth.de/id/eprint/32135