Arithmetic aspects of the Burkhardt quartic threefold

Author: 
File(s): 
Date created: 
2016-08-09
Identifier: 
etd9738
Supervisor(s): 
Nils Bruin
Department: 
Science: Department of Mathematics
Keywords: 
Level three structure
Genus 2 curves
Moduli of curves
Arithmetic geometry
Burkhardt quartic
Abelian surfaces
Abstract: 

The Burkhardt quartic is a 3-dimensional projective hypersurface defined over the rational numbers. It is known that sufficiently general points on the Burkhardt quartic parameterize abelian surfaces with a full level 3 structure. Furthermore, it is classical that the Burkhardt quartic is birational to 3-dimensional projective space after adjoining a cube root of unity. In this thesis we will show that the Burkhardt quartic is birational to 3-dimensional projective space over the rational numbers, and describe a geometric method of constructing a generic family of hyperelliptic curves corresponding to points on the Burkhardt quartic, whose Jacobians have a full level 3 structure. Specifically, we give an explicit family of hyperelliptic curves which contain almost all complex genus 2 curves with a full level 3 structure.

Thesis type: 
(Thesis) M.Sc.
Document type: 
Thesis
Rights: 
This thesis may be printed or downloaded for non-commercial research and scholarly purposes. Copyright remains with the author.
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