James Parks

Let E be an elliptic curve over the field Q of rational numbers. In this talk we consider several open conjectures about the distribution of local invariants associated with the reductions of E modulo p as p varies over the primes. In order to gain...

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...

Let $E$ be an elliptic curve defined over $\mathbb{Q}$. Let $M_E(N)$ be the function that counts the number of primes $p$ of good reduction such that $\#E_p(\mathbb{F}_p) = N$ where $N$ is a fixed integer and $E_p(\mathbb{F}_p)$ denotes the group of...