#### Date of Award

5-2017

#### Document Type

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mathematical Sciences

#### Committee Member

Dr. Kevin James, Committee Chair

#### Committee Member

Dr. Jim Brown

#### Committee Member

Dr. Hui Xue

#### Committee Member

Dr. Michael Burr

#### Abstract

Let E be a rational elliptic curve, and let p be a rational prime of good reduction. Let a_p denote the trace of the Frobenius endomorphism of $E$ at the prime $p$, and let N_p be the number of rational points on the reduced curve E modulo p. In this dissertation we investigate two different questions regarding the statistical distribution of these two quantities. We say p is a champion prime of E if when N_p is as large as possible in accordance with the Hasse bound. In a similar vein, we say p is a trailing prime of E when N_p is as small as possible in accordance with the Hasse bound. Together, we say that these primes constitute the extremal primes of E. The first result of this dissertation establishes an asymptotic on the number of elliptic curve champion primes that are less than a real number X in an average sense. As an immediate corollary, we also gain asymptotics on the average number of trailing primes less than and the average number of extremal primes less than X. In 1988, Koblitz made a conjecture on how often N_p is prime for any fixed rational elliptic curve. Balog, Cojocaru, and David have proved that Koblitz's conjecture is true on average for rational elliptic curves. The second result of this dissertation generalizes their average result to elliptic curves over certain higher number fields.

#### Recommended Citation

Giberson, Luke M., "Average Frobenius Distributions for Elliptic Curves: Extremal Primes and Koblitz's Conjecture" (2017). *All Dissertations*. 1900.

https://tigerprints.clemson.edu/all_dissertations/1900