MATHEMATICS OF COMPUTATION Volume 66, Number 219, July 1997, Pages 1161–1168 S 0025-5718(97)00860-0 UNIVERSAL BINARY HERMITIAN FORMS A. G. EARNEST AND AZAR KHOSRAVANI Abstract. We will determine (up to equivalence) all of the integral positive definite Hermitian lattices in imaginary quadratic fields of class number 1 that represent all positive integers. 1. Introduction The search for positive definite quaternary integral quadratic forms which repre- sent all positive integers has a long and illustrious history, dating back to Lagrange’s proof in 1770 that the form x 2 + y 2 + z 2 + w 2 has this property. Such forms are referred to as universal in the contemporary literature. More generally, a positive integral quadratic form over a totally real number field is said to be universal if every totally positive integer of the field is represented by the form. While no uni- versal positive binary quadratic forms exist, Maass [8] showed that the sum of three squares is universal over Q( √ 5). In 1994, Chan, Kim and Raghavan [1] showed that among the real quadratic fields, only the fields Q( √ 2), Q( √ 3), and Q( √ 5) admit universal ternary classic integral quadratic forms; all such forms are listed by the authors. In this paper, we consider the analogous problem of finding universal positive definite Hermitian forms. It will be shown that over all imaginary quadratic fields, there exist only finitely many classes of universal binary positive definite Hermit- ian forms. All such forms will be determined for the imaginary quadratic fields of class number 1; i.e., the fields Q( √ m) where m = −1, −2, −3, −7, −11, −19, −43, −67, −163. Computational methods were used to produce a list containing all potentially universal binary Hermitian forms, and all classes in their genera, over the nine imaginary quadratic fields of class number 1. We now give a brief outline of the method used. First, an upper bound for the discriminant of a universal binary Hermitian form is determined for each of the fields. Next, inequalities are obtained via reduction theory for the coefficients of such forms. The potentially universal reduced forms are then listed and are screened for possible universality by determin- ing whether the integers 1 through 5 are represented. This rough screening leaves thirteen candidates. Of these, six give rise via the trace mapping of Jacobson [4] to diagonal integral quaternary quadratic forms; their universality is established by ap- pealing to known results for universality of such diagonal forms. For the remaining forms, all binary Hermitian forms of the given discriminant are computer-generated Received by the editor May 15, 1996. 1991 Mathematics Subject Classification. Primary 11E39; Secondary 11E20, 11E41. Research supported in part by a grant from the National Security Agency. c 1997 American Mathematical Society 1161 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use