Minkowski theory and rational points on hyperelliptic curves

Author: 
Date created: 
2018-09-18
Identifier: 
etd19865
Keywords: 
Rational points
Hyperelliptic curves
Bhargava theory
Minkowski theory
Abstract: 

A remarkable recent result of M.\ Bhargava shows in a certain precise sense that `most' hyperelliptic curves over $\q$ have no rational points. An object central to his proof is a certain representation $\Z^2 \otimes \Sym_2 \Z^2$ of $\GL_n(\Z)$. Elements in the set $\Z^2 \otimes \Sym_2 \Z^2$ can be viewed as a pair of $n \times n$ symmetric matrices with entires in $\z$ up to a $\GL_n(\Z)$ equivalence. Alternatively, $\Z^2 \otimes \Sym_2 \Z^2$ has an algebraic number theoretic description. By taking advantage of this property, in this thesis, we investigate $\Z^2 \otimes \Sym_2 \Z^2$ from the point of view of Minkowski theory. In particular, by assuming the hyperelliptic curve $C$ is given by $z^2 = f(x,y)$, where $f(x,y)$ is irreducible over $\q$, we gave a direct `Minkowski' style proof that a certain part of the set $\Z^2 \otimes \Sym_2 \Z^2$, which contains the elements arising from the rational points on $C$, is finite. Although, our main principle of proof mirrors the classical proof of finiteness for the class number of a number field, we develop new arguments when there exist some notable differences, and we strive to give self-contained proofs of some of the components of Bhargava's paper which we utilize.

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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Senior supervisor: 
Imin Chen
Department: 
Science: Department of Mathematics
Thesis type: 
(Thesis) M.Sc.
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