$g$-invariant on unary Hermitian lattices over imaginary quadratic fields with class number $2$ or $3$

TL;DR

This paper investigates the minimal universal unary Hermitian lattices over imaginary quadratic fields with class number 2 or 3, explicitly determining the set of representable lattices and the smallest universal rank.

Contribution

It explicitly characterizes the set of unary Hermitian lattices representable by certain standard lattices and computes the minimal universal rank for fields with class number 2 or 3.

Findings

Explicit forms of representable unary Hermitian lattices are determined.

Exact values of the minimal universal rank g_d(1) are computed.

Results are specific to imaginary quadratic fields with class number 2 or 3.

Abstract

In this paper, we study the unary Hermitian lattices over imaginary quadratic fields. Let $E=\mathbb{Q}\big(\sqrt{-d}\big)$ be an imaginary quadratic field for a square-free positive integer $d$, and let $\mathcal{O}$ be its ring of integers. For each positive integer $m$, let $I_m$ be the free Hermitian lattice over $\mathcal{O}$ with an orthonormal basis, let $\mathfrak{S}_d(1)$ be the set consisting of all positive definite integral unary Hermitian lattices over $\mathcal{O}$ that can be represented by some $I_m$, and let $g_d(1)$ be the smallest positive integer such that all Hermitian lattices in $\mathfrak{S}_d(1)$ can be represented by $I_{g_d(1)}$ uniformly. The main results of this paper determine the explicit form of $\mathfrak{S}_d(1)$ and the exact value of $g_d(1)$ for every imaginary quadratic field $E$ with class number $2$ or $3$.