Equality of Field of Definition and Field of Moduli

In January 2024, I gave a talk at WDC, a peer conference on diverse topics in mathematics. I introduced the problem of establishing equality between the field of moduli and a field of definition for a dynamical system on $\mathbb{P}^1$, and the resolution of J. H. Silverman, involving concepts from various mathematical domains.

Abstract

The dynamics of rational maps on $\mathbb{P}^1$ is a central focus in the theory of arithmetic dynamics. In this talk, we address the problem of establishing equality between the field of moduli and a field of definition for a dynamical system on $\mathbb{P}^1$. The approach to this problem incorporates a range of terminology from various mathematical domains, including Galois cohomology, algebraic curves, and finite subgroups of the projective linear group. This exploration is based on Silverman’s result in 1995.

Lecture Note

These are notes for my talk.

Photo