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

Abstract

In this work, we shall study the unary Hermitian lattices over imaginary quadratic fields. Let E=Q(−d) be an imaginary quadratic field for a square-free positive integer d, and denote by O its ring of integers. For each positive integer m, let Im be the free Hermitian lattice over O with an orthonormal basis, let Sd(1) be the set consisting of all positive definite integral unary Hermitian lattices over O that can be represented by some Im, and let gd(1) be the least positive integer such that all Hermitian lattices in Sd(1) can be uniformly represented by Igd(1). The main results here determine the explicit form of Sd(1) and the exact value of gd(1) for all imaginary quadratic fields E with class number 2 or 3, which naturally generalize the Lagrange's four-square theorem.