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.

quarticrather than biquadratic: quartic definitely means "degree 4", while biquadratic suggests a specific type of quartic extension, namely a composite of two quadratic extensions (so of the form ${\mathbf Q}(\sqrt{a},\sqrt{b})$ with rational $a$ and $b$). $\endgroup$2more comments