When lattice is fractional ideal?

Question

Let $K$ be a imaginary quadratic number field of class number $1$ and $O_K$ be ring of integers of $K$.

Fractional ideal of $O_K$ can be always regarded as lattice (discrete subgroup of $ \Bbb{C}$ which spans $ \Bbb{C}$ as $ \Bbb{R}$vector space).

But I heard lattice is not always fractional ideal.

In which situation lattice is not fractional ideal, and what kind of condition ensures lattice is fractional ideal ?

Thank you for your kind help.

Answer 1

You are using the term "fractional ideal" only relative to $\mathcal O_K$, but there are also fractional ideals for orders in $K$ (subrings of $\mathcal O_K$ with finite index). Here is how the definitions go.

Let $K$ be a number field (not necessarily a quadratic field) of degree $n$ over $\mathbf Q$.

1. A lattice $L$ in $K$ is a free $\mathbf Z$-module of rank $n$ in $K$.

2. An order in $K$ is a subring of $\mathcal O_K$ with finite index (or, more abstractly, a subring of $K$ that is a free $\mathbf Z$-module of rank $n$).

3. A fractional ideal for an order $O$ is a nonzero $O$-module $M$ in $K$ with a common denominator: there is a nonzero $d$ in $O$ such that $dM$ is an ideal in $O$. For example, nonzero ideals in $O$ are fractional ideals for $O$.

We can now ask if a lattice in $K$ is a fractional ideal for some order even if that order is not a fractional ideal for $\mathcal O_K$.

To each lattice $L$ in $K$ there is an associated ring: its ring of multipliers $R(L) = \{\alpha \in K : \alpha L \subset L\}$. This ring an order in $K$, and if $L$ is going to be a fractional ideal for some order, then it must be a fractional ideal for the order $R(L)$. By only considering fractional ideals for $\mathcal O_K$, you are missing the possibility that a lattice in $K$ that you care about could be a fractional ideal for an order smaller than $\mathcal O_K$.

A possibly surprising fact: when $K$ is a quadratic field, it turns out that every lattice $L$ in $K$ is a fractional ideal for $R(L)$, even if $R(L) \not= \mathcal O_K$. So all lattices in quadratic fields are fractional ideals (for orders). In number fields of degree greater than $2$, there are lattices $L$ that are not a fractional ideal for $R(L)$, and thus are not a fractional ideal for any order at all.