Abstract: In the 1990's, Morton and Silverman proposed a dynamical analogue of the strong uniform boundedness conjecture (now a theorem of Merel) for torsion points on elliptic curves over number fields. We will discuss recent progress on the function field version of the dynamical uniform boundedness conjecture, relying on the geometry of certain dynamical analogues of classical modular curves. This includes joint work with Xander Faber and Bjorn Poonen.
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