Abstract: A dynamical modular curve is an algebraic curve whose points parametrize quadratic polynomial functions together with a collection of marked preperiodic points. (This is analogous to classical modular curves, whose points correspond to isomorphism classes of elliptic curves with marked torsion points.) Among all such curves, we determine those that have infinitely many points defined over cubic extensions of $\mathbb Q$. This is joint work with Alex Galarraga (graduate student at University of Washington).
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