Bayou Arithmetic Research Days

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 , there do not exist points of order 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 be an elliptic curve defined over a field extension of degree , and let be a point of order with coordinates in . Such a point is called “rational” since it is defined over the same field as . We say is sporadic if, as one ranges over all fields of degree at most and all elliptic curves , there are only finitely many elliptic curves which possess a rational point of order . Sporadic pairs correspond to exceptional points on modular curves, which are points whose existence is not explained by standard geometric constructions.

This talk will explore some of the most important relationships between Diophantine geometry and arithmetic dynamics. Many questions in arithmetic dynamics are inspired by classical problems in arithmetic geometry, and many dynamical consequences follow from well-known Diophantine inputs such as the abc conjecture. Moreover, ideas drawn from dynamics are often useful in tackling number-theoretic questions. I will give an overview of these links, and then will discuss some concrete illustrative examples. I will also point out some areas of difficulty that appear key to future progress.