Fundamental units of imaginary quartic fields Let $F/{\mathbb Q}$ be an imaginary quartic extension (i.e. the degree $[K:{\mathbb Q}]=4$ and no embedding of $K$ in ${\mathbb C}$ has its image inside the real numbers). Then the unit group of the integer ring ${\mathcal O}_K$ is infinite cyclic up to the roots of unity in $K$ and one can pick a generator $\varepsilon_F$ with absolute value $>1$ (say we have chosen an embedding into ${\mathbb C}$). 
I am interested in the asymptotics of $|\varepsilon_F|$; more precisely my question is the following: if we fix an imaginary quadratic extension $F_D={\mathbb Q}(\sqrt{-D})$ of ${\mathbb Q}$ then for any fundamental discriminant $d$ in the ring of integers ${\mathcal O}_D$, $F=F_D(\sqrt d)/{\mathbb Q}$ is an imaginary quartic extension, and it is known that $|\varepsilon_F|$ tends to infinity as $d$ does. I would be interested in knowing whether this convergence is uniform in $D$ or not, that is whether if given $M>1$ there is a $N\ge 0$ such that for any $D\in{\mathbb Z}_{>0}$ there are at most $N$ fundamental discriminants $d\in {\mathcal O}_D$ such that $|\varepsilon_F|\le M$. If this turns out not to be the case then I would be interested in the asymptotics of the numbers of $d$ with $|\varepsilon_F|\le M$ as (square-free) $D\to +\infty$. 
It is well-known that one can reformulate this in terms of Pell-like equations: the units in such a $F$ are given by $1/2(t+u\sqrt d)$ where $(t,u)$ is an integer solution of $t^2-u^2d=4$.
My motivation for asking this question comes from geometry: the norms of fundamental units in quadratic extensions of $F_D$ correspond to the lengths of closed geodesics on the associated Bianchi orbifold, and the question amounts to asking if the number of such lengths which are less than $e^M$ is bounded when $D$ varies. 
 A: There's certainly some uniform bound, as a special case of the
theorem that for each $n$ and $M$ there are only finitely many
algebraic integers $\epsilon$ of degree $n$ each of whose conjugates
has absolute value at most $M$.  Here $n=4$, and since $\epsilon$
is a unit conjugate to $\pm\epsilon^{-1}$ (once $D \lt -4$), the proof
leads to the estimate $N = O(M^2)$ with an effective (and reasonably small)
implied constant.
Write the minimal equation of $\epsilon$ as
$0 = (x-\epsilon) (x\mp\epsilon^{-1}) = x^2 + ax \pm 1$
when $\epsilon$ is an algebraic number of degree $2$ with norm $\pm 1$,
and as
$$
0 = (x-\epsilon) (x-\bar\epsilon) (x\mp\epsilon^{-1}) (x\mp\bar\epsilon^{-1})
$$ $$
= (x^2 - 2{\rm Re}(\epsilon) + 1) (x^2 \mp 2{\rm Re}(\epsilon^{-1}) + 1)
$$ $$
= x^4 + ax^3 + bx^2 \pm ax + 1
$$
when $\epsilon$ has degree $4$.
In each case the coefficients $a$ or $a,b$ are integers of size $O(M)$.
Thus there are $O(M^2)$ possibilities in all, so you can take $N = O(M^2)$
as claimed.
As with Pell equations, we expect that the actual count is asymptotically smaller,
but $N$ must still grow as some power of $M$ because of families such as
$d = 4(t^2 \mp 1)$, $\epsilon = t + \sqrt{t^2 \mp 1}$.
