Abstract: Merel's uniform boundedness theorem states that the torsion subgroup of an elliptic curve defined over a number field can be bounded by a constant depending only on the degree of that number field. Motivated by an analogy between torsion points on elliptic curves and preperiodic points for rational functions, Morton and Silverman posed an analogous conjecture for dynamical systems over defined over number fields. While the Morton-Silverman conjecture is still far from being proven, we discuss a weak form of the conjecture for quadratic polynomial functions. This is ongoing joint work with David Krumm.
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