Upcoming Seminars:
• Wednesday, February 27, 2019, 3:15pm, 119 Physics, Number Theory Seminar
TBA
Raphael Beuzart-Plessis (Marseille and Columbia)

• Wednesday, March 6, 2019, 3:15pm, 119 Physics, Number Theory Seminar
TBA
Asif Zaman (Stanford)

TBA

• Wednesday, March 27, 2019, 3:15pm, 119 Physics, Number Theory Seminar
TBA
Michael Harris (Columbia)

TBA

• Wednesday, April 3, 2019, 3:15pm, 119 Physics, Number Theory Seminar
TBD
Levent Alpoge (Princeton University)

TBD

• Wednesday, April 17, 2019, 3:15pm, 119 Physics, Number Theory Seminar
TBA
Dan Yasaki

TBA

• Wednesday, April 24, 2019, 3:15pm, 119 Physics, Number Theory Seminar
Extremal primes for elliptic curves without complex multiplication
Ayla Gafni (University of Rochester)

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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