Lethbridge Number Theory and Combinatorics Seminar: James Parks

Let $E$ be an elliptic curve defined over $\mathbb{Q}$. A pair $(p,q)$ of distinct prime numbers is called an \textit{amicable pair} of $E$ if $E$ has good reduction at $p$ and $q$ and $\#{E}_{p}(\mathbb{F}_{p})=q$ and $\#{E}_{q}(\mathbb{F}_q)=p$. In this talk we show a non-trivial upper bound for the number of amicable pairs on average over a family of elliptic curves.