# Bayou Arithmetic Research Days

Number theory mini-conferences in Louisiana

## BARD 1 Schedule

BARD 1 will take place at LSU in Baton Rouge during the afternoon of November 15, 2022, with a pre-talk taking place over Zoom the previous Friday. All talks will take place in Lockett Hall. Times are central time.

There is an option to attend any of the talks on Zoom. Just click the word “Zoom” by the talk.

3:30pm–5:00pm, Friday, Nov 11: Pre-talk (Zoom)

1:30pm–1:40pm, Tuesday, Nov 15: Registration (241 Lockett Hall)

1:40pm–2:30pm: Plenary lecture by Abbey Bourdon (241 Lockett Hall and Zoom)

Sporadic Torsion on Elliptic Curves

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 $F/\mathbb{Q}$ 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 $F/\mathbb{Q}$ 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.

2:30pm–3:10pm: Coffee, refreshments, and discussions (James E. Keisler Lounge, 3rd floor of Lockett Hall)

3:10pm–4:00pm: Plenary lecture by Nicole Looper (232 Lockett Hall and Zoom)

Diophantine Techniques in Arithmetic Dynamics

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.

4:00pm–4:15pm: Break

4:15pm–5:45pm: Lightning talks (232 Lockett Hall and Zoom)

6:30pm: Dinner at The Chimes