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

Hyperbolic three manifolds of bounded volume and trace field degree