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.

© 2020 Nandita Sahajpal