For an elliptic curve $E$ over $\mathbb{Q}$, the distribution of the number of points on $E$ mod $p$ has been well-studied over the last few decades. A relatively recent study is that of extremal primes for a given curve $E$. These are the primes $p$ of good reduction for which the number of points on $E$ mod $p$ is either maximal or minimal. If $E$ is a curve with CM, an asymptotic for the number of extremal primes was determined by James and Pollack. The talk will discuss the non-CM case and focus on obtaining upper bounds. This is joint work with C. David, A. Gafni, A. Malik and C. Turnage-Butterbaugh.

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Last updated: 20 Aug 2019