The conformal structure on the cuspidal torus is usually called the "cusp shape."

See Adams, Hildebrand, Weeks [Hyperbolic invariants of knots and links][1] and McReynolds, [Arithmetic cusp shapes are dense][2], for starters.  

See also work on "geometric inflexibility" by Neumann and Reid.

Also check out the work of Marc Lackenby and Jessica Purcell.




  [1]: http://www.jstor.org/pss/2001854
  [2]: http://www.springerlink.com/content/77361p22133714w4/