The Library
Cubic surfaces over finite fields
Tools
Cooley, Jenny (2014) Cubic surfaces over finite fields. PhD thesis, University of Warwick.
|
PDF
WRAP_THESIS_Cooley_2014.pdf - Submitted Version - Requires a PDF viewer. Download (1037Kb) | Preview |
Official URL: http://webcat.warwick.ac.uk/record=b2751208~S1
Abstract
It is well-known that the set of rational points on an elliptic curve forms an abelian group. When the curve is given as a plane cubic in Weierstrass form the group operation is defined via tangent and secant operations. Let S be a smooth cubic surface over a field K. Again one can define tangent and secant operations on S. These do not give S(K) a group structure, but one can still ask for the size of a minimal generating set.
In Chapter 2 of the thesis I show that if S is a smooth cubic surface over a field K with at least 4 elements, and if S contains a skew pair of lines defined over K, then any non-Eckardt K-point on either line generates S(K). This strengthens a result of Siksek [20].
In Chapter 3, I show that if S is a smooth cubic surface over a finite field K = Fq with at least 8 elements, and if S contains at least one K-line, then there is some point P > S(K) that generates S(K).
In Chapter 4, I consider cubic surfaces S over finite fields K = Fq that contain no K-lines. I find a lower bound for the proportion of points generated when starting with a non-Eckardt point P > S(K) and show that this lower bound tends to 1/6 as q tends to infinity.
In Chapter 5, I define c-invariants of cubic surfaces over a finite field K = Fq with respect to a given K-line contained in S, give several results regarding these c-invariants and relate them to the number of points SS(K)S.
In Chapter 6, I consider the problem of enumerating cubic surfaces over a finite field, K = Fq, with a given point, P > S(K), up to an explicit equivalence relation.
Item Type: | Thesis (PhD) | ||||
---|---|---|---|---|---|
Subjects: | Q Science > QA Mathematics | ||||
Library of Congress Subject Headings (LCSH): | Surfaces, Cubic, Finite fields (Algebra) | ||||
Official Date: | June 2014 | ||||
Dates: |
|
||||
Institution: | University of Warwick | ||||
Theses Department: | Mathematics Institute | ||||
Thesis Type: | PhD | ||||
Publication Status: | Unpublished | ||||
Supervisor(s)/Advisor: | Siksek, Samir | ||||
Extent: | vii, 95 leaves | ||||
Language: | eng |
Request changes or add full text files to a record
Repository staff actions (login required)
View Item |
Downloads
Downloads per month over past year