Arithmetic Properties of the Frobenius Indices Associated to the Reductions of an Elliptic Curve

For an elliptic curve $E$ defined over $\Q$ without complex multiplication and for a rational prime $p$ of good reduction, one can consider the subring of the $\F_p$-endomorphism ring of $E$ generated by multiplication by an integer and the Frobenius endomorphism, $\phi_p$. Defining $a_p=p+1-\#E(\F_p)$, the Frobenius endomorphism satisfies the polynomial $x^2-a_px+p$. We can associate $\phi_p$ with a $\ol\Q$-root of this polynomial, $\pi$. Realizing the endomorphism ring and its subring as orders in a quadratic field, we get the chain of orders $$\Z[\pi_p]\subseteq \End_{\F_p}(E)\subseteq \mc{O}_{\Q\left(a_p^2-4p\right)}.$$ If we define $\Delta_p$ as the discriminant of $\End_{\F_p}(E)$ and $b_p$ as the index of $\Z[\pi_p]$ inside $\End_{\F_p}(E)$, we see that $$b_p^2\Delta_p=a_p^2-4p.$$ In this thesis, we study properties of $b_p$, specifically how many divisors it has on average as $p$ varies. The relationship between $b_p$ and $\Delta_p$ allows us to use properties of $b_p$ to better understand $\Delta_p$. Assuming the generalized Riemann hypothesis, Cojocaru and Duke proved that the number of $p\le x$ such that $b_p=1$ is asymptotic to a constant multiple of $\pi(x)$. We build on this by proving that the sum of the number of divisors of $b_p$ is asymptotic to a constant multiple of $\pi(x)$ under the same assumption. We then use the relation between $b_p$ and $\Delta_p$ to prove that for almost all $p$, $|\Delta_p|$ is close to $4p-a_p^2$ when the generalized Riemann hypothesis is assumed. We also explore the constants in Cojocaru and Duke's theorem and our theorem on the number of divisors of $b_p$. We prove formulas for each constant and then prove that on average over elliptic curves without complex multiplication, these constants tend to idealized constants. Finally, we look at computational data which provides evidence for our theorems unconditionally.