I have some problems on the Fundamental mesh of ideals in Minkwoski space.
For simplicity, Let $F$ be a totally real number field with $\mathfrak o_F$ the integral ring and $h_F$ the class number. Let $\mathfrak N_1$ and $\mathfrak N_2$ be two co-prime integral ideals of $F$.
Let $\mathcal M:=\prod_{v\mid \infty}F_v$ be the Minkwoski space. We know $\mathfrak o_F$, $\mathfrak N_1$ and $\mathfrak N_2$ are all complete lattices in $\mathcal M$. Let $\mathcal M(\mathfrak o_F)$, $\mathcal M(\mathfrak N_1)$ and $\mathcal M(\mathfrak N_2)$ be the canonical fundamental meshes in $\mathcal M$, respectively.
My problems are the following.
Assume $\mathfrak N_1=(\alpha_1,\alpha_2)$. Do we have $$ \mathcal M(\mathfrak N_1)=\mathcal M((\alpha_1))\bigcap \mathcal M((\alpha_2)) ? $$
For $\mathfrak N_1$ and $\mathfrak N_2$ are co-prime integral ideals. Under what condition does $$ \mathcal M(\mathfrak N_1)\subsetneq\mathcal M(\mathfrak N_2)?$$
I am not sure if the following argument is true.
Since $\mathfrak N_1$ is a free $\mathbb Z$-module, let $\{\alpha_1,\cdots \alpha_n\}$ be the $\mathbb Z$-basis. For $j:F\rightarrow F_\mathbb R$ the injectiion from $F$ into Minkowski space $F_\mathbb R$, we know
$$
\Gamma_1=\mathbb Z j(\alpha_1)+\cdots +\mathbb Z j(\alpha_n)
$$
is a complete lattice in $F_\mathbb R$, and
$$
X=\{c_1 j(\alpha_1)+\cdots c_n j(\alpha_n),\quad -1 The image of $\mathfrak N_2$ in $F_\mathbb R$ is also a complete lattice $\Gamma_2=j(\mathfrak N_2)$. By Minkowski's lattice theorem, if
$$
\mathrm{vol}(X)>2^n\mathrm{vol}(\mathfrak N_2),
$$there exists $0\neq\alpha\in\mathfrak N_2$ which is alos in $X$. Therefore, a sufficient condition that $\mathcal M(\mathfrak N_2)\subsetneq \mathcal M(\mathfrak N_1)$, is
$$|\mathfrak N_1|>|\mathfrak N_2|.
$$