Dynamical modular curves of small genus

02/18/2014 - 12:15pm
02/18/2014 - 1:10pm
John Doyle (University of Georgia)

If K is a number field and f(z) is a quadratic polynomial defined over K, then the set of K-rational preperiodic points for f(z) may be endowed with the structure of a (finite!) directed graph G(f,K). For a given finite directed graph G, one can ask for which maps f(z) the graph G(f,K) contains G as a subgraph. To help answer this question, we construct a "dynamical modular curve" for each such graph G. I will define and discuss these dynamical modular curves in general, and then I will focus on those curves with genus at most two. I will end by giving an application in the case that K is a quadratic extension of Q.

Mudd Science Library 126, Pomona College