Jul 08, 2022|
Universal quadratic forms, positive definite forms that represent all natural numbers, have been of particular interest in number theory, following the first example of the sum of four squares. One natural generalization has been the study of universal quadratic forms over totally real number fields, involving the work of Siegel in 1945 that the sum of any number of squares is universal only over the number field K=ℚ, ℚ(√5). This suggests the following “Lifting problem”: when is it possible for a positive definite quadratic form with ℤ-coefficients to be universal over the ring of integers of a totally real number field? In this seminar, we overview related history and discuss recent results on this lifting problem given by Kala and Yatsyna.