Abstract

Let E be an elliptic curve defined over 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 minimal or maximal in relation to the Hasse bound, i.e. In the first case, we say that p is a trailing
prime. In the second case, we say that p is a champion prime. The notion of extremal primes was generalized to primes such that ap(E)/2√p lie in a short interval around c ∈ (0,1] in [AHJ+18], who considered the case of curves with complex multiplication.
In this thesis, assuming E does not have complex multiplication, we study the distribution in short intervals for
(a_p (E))/(2√p)∈(c-f(p),c) (1)
where c ∈ (−1,1] and f(x) = x^δ such that −1/2 ≤ δ < 0. The distribution is different if c = 1 or
c ≠ 1, influenced by the Sato-Tate distribution (see Conjecture 1.1). We use the techniques
of David, Gafni, Malik, Prabhu, and Turnage-Butterbaugh [DGM+19], who considered the extremal primes for elliptic curves without complex multiplication to get an upper bound for the number of primes such that (1) holds, under GRH (Theorem 1.4) and unconditionally (Theorem 1.5).