The Fifteen Theorem for Universal Hermitian Lattices over Imaginary Quadratic Fields

TL;DR

This paper establishes a criterion, called the fifteen theorem, for determining the universality of Hermitian lattices over imaginary quadratic fields, extending classical results to a new algebraic setting.

Contribution

It introduces a universal criterion for Hermitian lattices over all imaginary quadratic fields, including a specific set of numbers that must be represented for universality.

Findings

A new criterion for universality of Hermitian lattices over imaginary quadratic fields.

Identification of a specific set of numbers (1, 2, 3, 5, 6, 7, 10, 13, 14, 15) that determine universality.

Extension of the classical fifteen theorem to Hermitian lattices in imaginary quadratic fields.

Abstract

We will introduce a method to get all universal Hermitian lattices over imaginary quadratic fields over $\mathbb{Q}(\sqrt{-m})$ for all m. For each imaginary quadratic field $\mathbb{Q}(\sqrt{-m})$, 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.