Number Theory Seminar

Date: May 6, 2024

Time: 11:00AM - 12:00PM

Location: BLOC 302

Speaker: John Sung Min Lee, University of Illinois at Chicago

Title: On the distribution in arithmetic progressions of primes of various properties related to elliptic curves

Abstract: Given an elliptic curve $E/\mathbb{Q}$ and a prime $p$ of good reduction for $E$, let $\widetilde{E}_p(\mathbb{F}_p)$ denote the group of $\mathbb{F}_p$-rational points of the reduction of $E$ modulo $p$. One can define primes with various properties associated with $E$ based on the structure of $\widetilde{E}_p(\mathbb{F}_p)$. For instance, we call $p$ a prime of cyclic reduction, $m$-divisibility, and Koblitz reduction for $E$ if $\widetilde{E}_p(\mathbb{F}_p)$ is cyclic, has an order divisible by $m$, and has a prime order, respectively. In this talk, we study how the aforementioned primes, for an individual elliptic curve or on average, are distributed across arithmetic progressions. Furthermore, we analyze whether these primes are equally distributed or biased over congruence classes modulo $n$. This work is a partial collaboration with Nathan Jones, Jacob Mayle, and Tian Wang.