Locating a shortest vector in certain $2$-dimensional lattices

Abstract

Let $a$, $m$ be positive integers, $1<a<m$, $\gcd(a,m)=1$. We determine the location of a shortest vector in the $2$-dimensional lattices $$ Λ(a,m) = \{(x, y)\in\mathbb{Z}\times\mathbb{Z}\mid ax + y\equiv 0~(\bmod\,m)\}. $$ This confirms a conjecture of Han Wu and Guangwu Xu.