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# Texas A&M Number Theory Seminar

##
Department of Mathematics

Blocker 220

Wednesdays, 1:45–2:45 PM

### John Doyle

Louisiana Tech University

#### Wednesday, November 1, 2017

#### Blocker 220, 1:45PM

**Title:** *Dynamical modular curves and strong uniform boundedness
*

**Abstract:**
In an unpublished 1996 preprint, Nguyen and Saito proved the strong uniform boundedness conjecture for torsion points on elliptic curves over function fields by reducing the problem to that of showing that the gonalities of the modular curves Y_1(n) tend to infinity. By studying the geometry of dynamical modular curves, we have recently proven uniform boundedness of preperiodic points for certain interesting families of polynomial maps over function fields. I will discuss this result as well as a consequence for the dynamical uniform boundedness conjecture over number fields, originally posed by Morton and Silverman. This is joint work with Bjorn Poonen.