An algorithm for g-invariant on unary Hermitian lattices over imaginary quadratic fields

Abstract

Let $E=Q\left(\sqrt{-d}\right)$ be an imaginary quadratic field for a square-free positive integer d, and let $O$ be its ring of integers. For every positive integer m, let $I_m$ be the free Hermitian lattice over $O$ with an orthonormal basis, let $S_d\left(1\right)$ be the set consisting of all the positive definite integral unary Hermitian lattices over $O$ which can be represented by some $I_m$, and let $g_d\left(1\right)$ be the smallest positive integer such that all the lattices in $S_d\left(1\right)$ can be uniformly represented by $I_{g_d\left(1\right)}$. In this work, I provide an algorithm to compute the explicit form of $S_d\left(1\right)$ and the exact value of $g_d\left(1\right)$ for every imaginary quadratic field E, which may be viewed as a natural extension of the Pythagoras number in the lattice setting.