On universal binary Hermitian forms. (English) Zbl 1192.11021

Kominers, Scott Duke

Integers 9, No. 1, Article A02, 9-15 (2009).

Let \(L\) be an \(\mathcal O\)-lattice on a positive definite binary Hermitian space \((V,H)\) over an imaginary quadratic number field \(E\), where \(\mathcal O\) denotes the ring of integers of \(E\). Such a lattice \(L\) is said to be universal if for every positive rational integer \(a\), there exists \(x \in L\) such that \(H(x,x)=a\). All the integral lattices \(L\) with this property have been determined in a series of papers by A.G. Earnest and A. Khosravani [Math. Comput. 66, No. 219, 1161–1168 (1997; Zbl 0877.11028)], H. Iwabuchi [Rocky Mt. J. Math. 30, No. 3, 951–959 (2000; Zbl 0972.11024)], and J.-H. Kim and P.-S. Park [Proc. Am. Math. Soc. 135, No. 1, 47–49 (2007; Zbl 1173.11021)].
In the present paper, the author observes that the 290-theorem, which gives simple criteria for universality of positive definite quadratic forms with rational integer coefficients, can be applied to give a uniform verification for the universality of the binary Hermitian lattices found in the papers cited above. At the time of the publication of the original papers, the validity of the 290-theorem had not yet been established; consequently, the authors relied on a variety of ad hoc methods to verify the universality of the forms identified as candidates for universality.

Reviewer: Andrew G. Earnest (Carbondale)

MSC:

11E39	Bilinear and Hermitian forms
11E12	Quadratic forms over global rings and fields
11E20	General ternary and quaternary quadratic forms; forms of more than two variables

Keywords:

universal binary Hermitian forms; 290-Theorem

Citations:

Zbl 0877.11028; Zbl 0972.11024; Zbl 1173.11021