Events

Abbey Bourdon

Wake Forest University
Colloquia

When

Date: Friday, October 14, 2022
Time: 3:30 pm - 5:00 pm
Location: Petty 150
Dr. Abbey Bourdon is currently an Assistant Professor at Wake Forest University. She received her Ph.D. from Wesleyan University in 2014. Following this she was a Postdoctoral Associate at the University of Georgia. Her research area is arithmetic geometry. Dr. Bourdon is currently supported by an NSF CAREER award and an NSF LEAPS award. In 2021, she was the recipient of the MAA Southeastern Section Award for Distinguished Teaching by a Beginning Faculty Member.

An elliptic curve is a curve in projective space whose points can be
given the structure of an abelian group. In this talk, we will focus
on torsion points, which are points having finite order under this
group law. While we can generally determine the torsion points of
a fixed elliptic curve defined over a number field, there are several
open problems which require controlling the existence of torsion
points within infinite families of elliptic curves. Success stories
include Merel’s Uniform Boundedness Theorem, which states that
the order of a torsion point can be bounded by the degree of its
field of definition. On the other hand, a proof of Serre’s Uniformity
Conjecture –which has been open for 50 years– would in particular
imply that for sufficiently large primes p, there do not exist points
of order p^2 arising on elliptic curves defined over field extensions
of “unusually low degree.” In this talk, I will give a brief
introduction to the arithmetic of elliptic curves before addressing
the problem of identifying elliptic curves producing a point of large
order in usually low degree, i.e., those possessing a sporadic
torsion point. More precisely, let E be an elliptic curve defined over
a field extension F/Q of degree d, and let P be a point of order N
with coordinates in F. Such a point is called “rational” since it is
defined over the same field as E. We say P is sporadic if, as one
ranges over all fields F/Q of degree at most d and all elliptic curves
E/F, there are only finitely many elliptic curves which possess a
rational point of order N. Sporadic pairs (E,P) correspond to
exceptional points on modular curves, which are points whose
existence is not explained by standard geometric constructions.