### Extremal primes of non-CM elliptic curves

Heilbronn Number Theory Seminar

7th March 2018, 4:00 pm – 5:00 pm

Howard House, 4th Floor Seminar Room

Fix an elliptic curve E/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 up to X, and give the first non-trivial upper bound for the number of such primes in the non-CM setting. In order to obtain this bound, we count primes with certain arithmetic characteristics and combine those results with the Chebotarev density theorem. This is joint work with Chantal David, Amita Malik, Neha Prabhu, and Caroline Turnage-Butterbaugh.

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