2-universal Hermitian Lattices over Imaginary Quadratic Fields

We call a positive definite integral quadratic form universal if it represents all positive integers. Then Lagrange’s Four Square Theorem means that the sum of four squares is universal. In 1930, Mordell [M] generalized this notion to a 2-universal quadratic form: a positive definite integral quadratic form that represents all binary positive definite integral quadratic forms, and showed that the sum of five squares is 2-universal. In this direction, we refer the readers to [K] and [KKO1, KKO2]. As another generalization of universal quadratic forms, universal Hermitian forms have been studied. This was initiated by Earnest and Khosravani. They defined a universal Hermitian form as the one representing all positive integers, and found 13 universal binary Hermitian forms over imaginary quadratic fields of class number 1 [EK]. The list of binary universal Hermitian forms has been completed by Iwabuchi [I], Jae-Heon Kim and the second author [KP]. The simple and unified proofs was recently obtained by the second author [P]. In this paper, we study 2-universal Hermitian forms. We prove that there are finitely many 2-universal ternary and quaternary Hermitian forms over imaginary quadratic fields, and find them all (sections 4 and 5). A notable recent progress in the representation theory of quadratic forms is the so called Fifteen Theorem of Conway-Schneeberger [C], which states: a positive definite quadratic form is universal if it represents positive integers up to 15. This fascinating result was improved by Bhargava [B], who proved analogies for other infinite subsets of positive integers like the set of all primes, the set of all positive odd integers and so on. Kim et al. [KKO1, KKO2] recently proved the finiteness theorem for representability and provided a 2-universal analogy of the Fifteen Theorem. Recently Kim, Kim and the second author proved