# Spring 2020 Mathematics Colloquium

## Michelle Manes

NSF – University of Hawaii at Manoa

### Complex multiplication in arithmetic geometry and arithmetic dynamics

The theory of complex multiplication in arithmetic geometry involves both the arithmetic of elliptic curves and orders in imaginary quadratic fields. It has a distinguished history, and in fact was the subject of Dick Gross’ AMS Colloquium Lectures at JMM in 2019. His talked started with Euler’s work on elliptic integrals in 1751 and continued through research from the past 20 years. The field of arithmetic dynamics is the study of number theoretic properties of iterated functions. The field draws inspiration from dynamical analogues of theorems and conjectures in classical arithmetic geometry. In this talk I will give a bit of background in the ideas from arithmetic geometry that inspire much of the current work in arithmetic dynamics, with a focus on attempts (by myself and others) to develop a “dynamical” theory of complex multiplication.