Universal binary Hermitian forms

Abstract

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.

The fifteen theorem for universal Hermitian lattices over imaginary quadratic fields

“…These Hermitian lattices are all universal by [6] and [17], and we know that there are no other new Hermitian universal lattices over Q( √ −m) for m = 1, 2; 7. Hence we assume that m = 1, 2; 7.…”

“…In 1997, Earnest and Khosravani [6] found 13 universal binary Hermitian forms over imaginary quadratic fields of class number 1. If the quadratic field over Q has a class number greater than 1, Iwabuchi [10] determined all universal binary Hermitian lattices over this field.…”

“…Recently, Bhargava and Hanke enunciated that they proved the 290-conjecture which characterizes the universality of (nonclassical) quadratic forms. That is, if a (nonclassical) quadratic form represents the 29 numbers, 1, 2, 3, 5,6,7,10,13,14,15,17,19,21,22,23,26,29,30,31,34,35,37,42,58,93,110,145,203,290, then it is universal (see [9]). One can use this theorem to prove the universality of a Hermitian lattice.…”

Even universal binary Hermitian lattices over imaginary quadratic fields

“…A positive definite Hermitian lattice over an imaginary quadratic field is called universal if it represents all positive integers. There are many contributions to list all universal binary Hermitian lattices over imaginary quadratic fields (see [4], [6], [11]). In the previous article [9], the authors provided a list of universal Hermitian lattices over imaginary quadratic field Q( √ −m) for all positive square-free integers m, and the Conway-Schneeberger-Bhargava type criterion on universality, so called the fifteen theorem for universal Hermitian lattices.…”

A few uncaught universal Hermitian forms

“…Recently, several people studied the problem analogous to universal forms over the imaginary quadratic fields. Earnest and Khosravani [1] defined a universal Hermitian form as a positive definite one representing all positive integers, and they found 13 universal binary Hermitian forms over the imaginary quadratic fields of class number 1. More generally, Iwabuchi [2] investigated Hermitian lattices and found 9 lattices over the imaginary quadratic fields of class number bigger than 1.…”