Average Frobenius Distribution in Families of Elliptic Curves

Abstract

An elliptic curve $E$ over a field $\mathbb{F}$ can be defined by the equation $$y^2 = x^3 + ax+ b,$$ where $a, \, b \in \mathbb{F}.$ For any $r \geq 1$, let $a_E{(p^r)}$ denote the trace of the Frobenius endomorphism of $E$ over the field $\mathbb{F}_{p^r}$, $p$ being a prime. For a natural number $k$, let $\kappa$ denote the set of all $k$-th powers of natural numbers. James and Yu in their work computed the distribution of $$\{a_E{(p)}:\,a_E{(p)}\in \kappa\}$$ as the primes $p \to \infty$ by averaging over suitable families of elliptic curves. In this thesis, we review the work of James and Yu. In an effort to obtain a smooth analogue of the main result proved by James-Yu, we present a methodology for the same and explain the technical problems encountered. At the end of this thesis, we provide a result about the distribution of ${a_E}{(p^2)}$ by taking the average over a family of elliptic curves.