Andrew Herring (Math) - PhD Public Lecture

"Genus bounds for some dynatomic modular curves"

We show that for n=11 and every n at least 13 the following holds: there are at most finitely many rational numbers c such that the quadratic polynomial x^2 + c has a point of period n in the field of rational numbers. As we will discuss, to prove this result it suffices to show that every member in a family of "dynatomic modular curves" has genus at least two. For each n, the maximal subgroups of the "nth dynatomic galois group" come in two flavors which we call "vanilla" and "chocolate." We prove the genus bounds for dynatomic modular curves corresponding to vanilla maximal subgroups using ramification theory and the Riemann-Hurwitz genus formula; the genus bounds for dynatomic modular curves corresponding to chocolate maximal subgroups follow from a theorem of Guralnick and Shareshian.