Author ORCID Identifier

https://orcid.org/0000-0002-6564-2122

Year of Publication

2020

Degree Name

Doctor of Philosophy (PhD)

Document Type

Doctoral Dissertation

College

Arts and Sciences

Department/School/Program

Mathematics

First Advisor

Dr. David B. Leep

Abstract

In this dissertation we investigate the existence of a nontrivial solution to a system of two quadratic forms over local fields and global fields. We specifically study a system of two quadratic forms over an arbitrary number field. The questions that are of particular interest are:

  1. How many variables are necessary to guarantee a nontrivial zero to a system of two quadratic forms over a global field or a local field? In other words, what is the u-invariant of a pair of quadratic forms over any global or local field?
  2. What is the relation between u-invariants of a pair of quadratic forms over any global field and the local fields associated with it?
  3. How is the u-invariant of a pair of quadratic forms over any global field related to the u-invariant of its residue field?

There are many known results that address 1, 2, and 3:

(A) In the context of p-adic fields, a classical result by Dem'yanov states that two homogeneous quadratic forms over a p-adic field have a common nontrivial p- adic zero, provided that the number of variables is at least 9. In 1962, Birch- Lewis-Murphy gave an alternative proof to this result by Dem'yanov.

(B) In a 1964 paper, Swinnerton-Dyer showed that a system of two quadratic forms over the field of rational numbers in 11 variables, satisfying certain number- theoretic conditions, has a nontrivial rational zero

(C) An even more remarkable result proven by Colliot-Thélène, Sansuc, and Swinnerton-Dyer extends Dem'yanov's result to an imaginary number field and also to an arbitrary number field if certain number-theoretic conditions are satisfied.

Our work in this dissertation is motivated by the work on the results stated above.

  • With respect to (A), we generalize the result as well as the proof techniques to prove an analogous result over a complete discretely valued field with characteristic not 2.
  • With respect to (B), we demonstrate that this result, and the techniques used in the proof can be extended to a system of two quadratic forms in at least 11 variables over an arbitrary number field.
  • With respect to (C), we give a more comprehensible and self-contained proof of this result over an arbitrary number field using primarily number-theoretic arguments.

Digital Object Identifier (DOI)

https://doi.org/10.13023/etd.2020.343

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