Near universal property of a quadratic form of Hessian discriminant 4 (Q6548311)

Let \(K\) be a totally real number field. A positive definite quadratic form \(Q\) with rational integer coefficients that represents all totally positive integers in \(K\) is called universal over \(K\). For example, \(x^2+y^2+z^2\) over \(K=\mathbb{Q}(\sqrt{5})\) [\textit{H. Maaß}, Abh. Math. Semin. Univ. Hamb. 14, 185--191 (1941; JFM 67.0103.02)], and \(x^2+y^2+z^2+w^2+xy-xz-xw\) is universal over \(K=\mathbb{Q}(\cos(2\pi/7))\) [\textit{V. Kala} and \textit{P. Yatsyna}, Adv. Math. 377, Article ID 107497, 25 p. (2021; Zbl 1462.11028)]. However, these are nearly all the known examples, and it is established that only a finite number of totally real number fields of a given degree can admit such forms [\textit{V. Kala} and \textit{P. Yatsyna}, Bull. Lond. Math. Soc. 55, No. 2, 854--864 (2023; Zbl 1536.11067)].\N\NThis paper proves a more general result for the representation of quadratic form \(x^2+y^2+z^2+w^2+xy-xz-xw\) over \(\mathbb{Q}(\sqrt{2})\). Specifically, this form represents almost all totally positive integers. Additionally, the paper describes the exceptional integers that are not represented. The proof builds on the author's earlier work using quaternions to address this problem [\textit{J. I. Deutsch}, J. Number Theory 104, No. 2, 263--278 (2004; Zbl 1047.11034)].