  For a single cusped hyperbolic 3-manifold, Hodgson proved that there are only
finitely many Dehn fillings of it whose trace fields have bounded degree. In
this paper, we conjecture the same for manifolds with more cusps, and give the
first positive results in this direction. For example, in the 2-cusped case, if
a manifold has linearly independent cusp shapes, we show that the manifold has
the desired property.To prove the results, we use the proof of the Bounded
Height Conjecture in arithmetic geometry.
