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

Title data

Elsenhans, Andreas-Stephan ; Jahnel, Jörg:
On the quasi-group of a cubic surface over a finite field.
In: Journal of Number Theory. Vol. 132 (July 2012) Issue 7 . - pp. 1554-1571.
ISSN 0022-314X
DOI: https://doi.org/10.1016/j.jnt.2012.01.010

Abstract in another language

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.

Further data

Item Type: Article in a journal
Refereed: Yes
Subject classification: Mathematics Subject Classification Code: 11G25 (11D25 11G35 14G15)
Institutions of the University: Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Chair Mathematics II
Faculties
Faculties > Faculty of Mathematics, Physics und Computer Science
Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics
Result of work at the UBT: Yes
DDC Subjects: 500 Science > 510 Mathematics
Date Deposited: 11 Apr 2016 06:31
Last Modified: 11 Apr 2016 06:31
URI: https://eref.uni-bayreuth.de/id/eprint/32135