Resource type

Thesis type

(Thesis) M.Sc.

Date created

2018-09-18

Authors/Contributors

Author: Wang, Shanzhao

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

Identifier

etd19865

Copyright statement

Copyright is held by the author.

Scholarly level

Supervisor or Senior Supervisor

Thesis advisor: Chen, Imin

Member of collection

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