Author ORCID Identifier
Year of Publication
Doctor of Philosophy (PhD)
Arts and Sciences
Dr. David B. Leep
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:
- 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?
- What is the relation between u-invariants of a pair of quadratic forms over any global field and the local fields associated with it?
- 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)
Sahajpal, Nandita, "SIMULTANEOUS ZEROS OF A SYSTEM OF TWO QUADRATIC FORMS" (2020). Theses and Dissertations--Mathematics. 76.