**Availbale Spring 2015**

Math 5010.351, Elliptic curves and dynamical systems, taught by Professor Fili:

We will study some classical problems in the arithmetic of elliptic curves and modern generalizations in the arithmetic of dynamical systems. Our goal will be to introduce the classical questions---many of which have been solved---and then discuss the analogous questions in arithmetic dynamics---many of which are completely open. We will prove several basic results for elliptic curves, such as the Mordell-Weil theorem. As we do this we will set up a running analogy between rational torsion points on elliptic curves and preperiodic points in dynamical systems. We will also introduce the notion of height---essentially, of arithmetic or dynamical complexity---and use this notion to prove some interesting (and useful) equidistribution results in each case. Students should be familiar with ring and field theory as in the 5000-level abstract algebra sequence, which may be taken concurrently.