Even universal binary Hermitian lattices over imaginary quadratic fields

TL;DR

This paper introduces a method to classify all even universal binary Hermitian lattices over imaginary quadratic fields and provides criteria for their universality.

Contribution

It develops a systematic approach to identify all such lattices over various imaginary quadratic fields and establishes optimal criteria for their universality.

Findings

Classified all even universal binary Hermitian lattices over imaginary quadratic fields.

Established optimal criteria for even universality of these lattices.

Provided a comprehensive list of such lattices for different fields.

Abstract

A positive definite even Hermitian lattice is called \emph{even universal} if it represents all even positive integers. We introduce a method to get all even universal binary Hermitian lattices over imaginary quadratic fields $\Q{-m}$ for all positive square-free integers $m$ and we list optimal criterions on even universality of Hermitian lattices over $\Q{-m}$ which admits even universal binary Hermitian lattices.