Simple Proofs for Universal Binary Hermitian Lattices

It has been a central problem in the theory of quadratic forms to find integers represented by quadratic forms. The celebrated Four Square Theorem by Lagrange [10] was an outstanding result in this study. Ramanujan generalized this theorem and found 54 positive definite quaternary quadratic forms which represent all positive integers [13]. We call a positive definite quadratic form universal, if it represents all positive integers. The classification of nondiagonal universal classical quadratic forms was completed by Conway and Schneeberger using their Fifteen Theorem in 2000 [4], [1]. The theorem states that if a positive definite classical quadratic form (with four or more variables) represents up to 15, it is universal. In 1997 Earnest and Khosravani defined universal Hermitian forms and they sought 13 positive definite binary Hermitian forms over imaginary quadratic fields of class number one [5]. Iwabuchi extended the result to imaginary quadratic fields of class number bigger than one and he found 9 binary Hermitian lattices (as a generalization of Hermitian forms) [6]. Jae-Heon Kim and the author complete the list by appending 3 universal binary Hermitian forms [9]. Moreover, Kim, Kim and the author found an analogous result to Fifteen Theorem: If a positive definite Hermitian lattice represents up to 15, then it represents all positive integers [8]. The proof was more complicated than that of the Conway-Scheeberger Theorem for it contains nonclassical quadratic forms. The criterion, 290-Theorem, for universal nonclassical quadratic forms was recently proved by Bhargava and Hanke [2]. In the present article we give simple and unified proofs for universal binary Hermitian lattices. Although the three papers ([5], [6], [9]) proposed proofs, they were complicated and used local properties of Hermitian forms. But, here, we use merely well known results about quadratic forms.