Average Cyclicity for Elliptic Curves in Torsion Families

Let $E/\QQ$ be an elliptic curve; for all but finitely many primes $p$, reduction modulo $p$ yields an elliptic curve over the finite field $\mathbb{F}_p$, and it is natural to ask about the properties of these reductions for varying primes. The purpose of this dissertation is to study one such question, namely, how frequently the reductions result in an elliptic curve with cyclic group structure. To be precise, we let $\pi_E^{cyc}(x) $ denote the number of primes less than $x$ for which the reduction of $E$ modulo $p$ is cyclic. The asymptotic behavior of this function has been established by Serre conditional on Generalized Riemann Hypothesis. Furthermore, Banks and Shparlinski showed that this asymptotic holds unconditionally on average over the family of elliptic curves given by short Weierstrass equations with coefficients taken in a `box.' Inspired by the work of Battista, Bayless, Ivanov and James on the Lang-Trotter conjecture, we study the average asymptotic behavior of the functions $\pi_E^{cyc}$ where the average is taken over certain thin families of elliptic curves: elliptic curves with a rational point of order $m$ defined over $\mathbb{Q}$. The results we obtain are again in agreement with the conditional asymptotic. We also extend the study of cyclicity from elliptic curves defined over the rational numbers to elliptic curves defined over a quadratic extension of $\QQ$ and obtain partial results in that case.As a key tool, we prove an analogue of a result of Vl\u{a}du\c{t} that estimates the number of elliptic curves over a finite field which have some specified torsion and which have group structure that is as cyclic as possible