Extremal primes for elliptic curves without complex multiplication


Ayla Gafni


University of Mississippi


Mon, 13/01/2020 - 12:00pm to 1:00pm


RC-4082, The Red Centre, UNSW


Fix an elliptic curve $E$ over $\mathbb{Q}$.  An ``extremal prime'' for $E$ is a prime $p$ of good reduction such that the number of rational points on $E$ modulo $p$ is maximal or minimal in relation to the Hasse bound.  In this talk, I will discuss what is known and conjectured about the number of extremal primes $p\le X$, and give the first non-trivial upper bound for the number of such primes when $E$ is a curve without complex multiplication.  The result is conditional on the hypothesis that all the symmetric power $L$-functions associated to $E$ are automorphic and satisfy the Generalized Riemann Hypothesis.  In order to obtain this bound, we use explicit equidistribution for the Sato-Tate measure as in recent work of Rouse and Thorner, and refine certain intermediate estimates taking advantage of the fact that extremal primes have a very small Sato-Tate measure.

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