ASU Electronic Theses and Dissertations
This collection includes most of the ASU Theses and Dissertations from 2011 to present. ASU Theses and Dissertations are available in downloadable PDF format; however, a small percentage of items are under embargo. Information about the dissertations/theses includes degree information, committee members, an abstract, supporting data or media.
In addition to the electronic theses found in the ASU Digital Repository, ASU Theses and Dissertations can be found in the ASU Library Catalog.
Dissertations and Theses granted by Arizona State University are archived and made available through a joint effort of the ASU Graduate College and the ASU Libraries. For more information or questions about this collection contact or visit the Digital Repository ETD Library Guide or contact the ASU Graduate College at gradformat@asu.edu.
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- All Subjects: Mathematics
- Creators: Quigg, John
for integer or rational solutions of algebraic equations. Pythagorean triangles are
an early instance. Diophantus of Alexandria wrote the first related treatise in the
fourth century; it was an area extensively studied by the great mathematicians of the seventeenth century, including Euler and Fermat.
The modern approach is to treat the equations as defining geometric objects, curves, surfaces, etc. The theory of elliptic curves (or curves of genus 1, which are much used in modern cryptography) was developed extensively in the twentieth century, and has had great application to Diophantine equations. This theory is used in application to the problems studied in this thesis. This thesis studies some curves of high genus, and possible solutions in both rationals and in algebraic number fields, generalizes some old results and gives answers to some open problems in the literature. The methods involve known techniques together with some ingenious tricks. For example, the equations $y^2=x^6+k$, $k=-39,\,-47$, the two previously unsolved cases for $|k|<50$, are solved using algebraic number theory and the ‘elliptic Chabauty’ method. The thesis also studies the genus three quartic curves $F(x^2,y^2,z^2)=0$ where F is a homogeneous quadratic form, and extend old results of Cassels, and Bremner. It is a very delicate matter to find such curves that have no rational points, yet which do have points in odd-degree extension fields of the rationals.
The principal results of the thesis are related to surfaces where the theory is much less well known. In particular, the thesis studies some specific families of surfaces, and give a negative answer to a question in the literature regarding representation of integers n in the form $n=(x+y+z+w)(1/x+1/y+1/z+1/w).$ Further, an example, the first such known, of a quartic surface $x^4+7y^4=14z^4+18w^4$ is given with remarkable properties: it is everywhere locally solvable, yet has no non-zero rational point, despite having a point in (non-trivial) odd-degree extension fields of the rationals. The ideas here involve manipulation of the Hilbert symbol, together with the theory of elliptic curves.