Algebraic points on curves

Abstract: To begin, I will introduce rational and degree $d>1$ points on curves, describe how their behavior differs, and define what it means for a degree $d>1$ point to be ``unexpected/sporadic''. Then, I will talk about joint with with J.~Gunther where we prove, under a technical assumption, that for each positive integer $d>1$, there exists a number $B_d$ such that for each $g > d$, a positive proportion of odd hyperelliptic curves of genus $g$ over $\mathbb{Q}$ have at most $B_d$ ``unexpected'' points of degree $d$; furthermore, I will briefly say how one may take $B_2 = 24$ and $B_3 = 114$. After this, I will discuss joint work with A.~Etropolski, M.~Derickx, M.~van Hoeij, and D.~Zureick-Brown where we use the explicit determination of ``unexpected/sporadic" cubic points on modular curves to classify torsion subgroups of elliptic curves over cubic number fields.

number theory

Audience: researchers in the topic

Front Range Number Theory Day

Series comments: Description: Short research conference on number theory

The FRNTD is a twice-yearly conference that brings together number theorists working on the Front Range. We are excited to be able to open our virtual doors to number theorists around the world this semester. Please see the website for the schedule, further details, and registration.