The fifteen theorem for universal Hermitian lattices over imaginary quadratic fields

Abstract

We will introduce a method to get all universal Hermitian lattices over imaginary quadratic fields Q(√-m) for all m. For each imaginary quadratic field Q(√-m), we obtain a criterion on universality of Hermitian lattices: if a Hermitian lattice L represents 1, 2, 3, 5, 6, 7, 10, 13, 14 and 15, then L is universal. We call this the fifteen theorem for universal Hermitian lattices. Note that the difference between Conway-Schneeberger's fifteen theorem and ours is the number 13. In addition, we determine the minimal rank of universal Hermitian lattices for all imaginary quadratic fields. © 2009 American Mathematical Society.