Twin Prime Questions For Elliptic Curves

For an elliptic curve $E$ defined over $\Q$ and for a rational prime $p$ of good reduction, one can define an integer $a_p$ related to the number of $\F_p$-points lying on the reduction of $E$ modulo $p$ as $a_p = p + 1 - \#E(\F_p)$. The integer $a_p$, called the Frobenius trace of $E$ modulo $p$, lies in the interval $(- 2 \sqrt{p}, 2 \sqrt{p})$ and has several other remarkable properties. In this thesis, we study the arithmetic properties of $a_p$, specifically how often $a_p$ is prime. Using heuristical reasoning similar to that used in formulating the Hardy-Littlewood Conjecture regarding the number of twin primes up to a bound $x$, it is natural to formulate a conjecture for the asymptotic growth of the number of primes $p \leq x$ for which $a_p$ is also prime. As evidence in support of this conjecture, we prove two main results, each in the case when $E$ is without complex multiplication and under the $\theta$-quasi Generalized Riemann Hypothesis. First, we establish an upper bound for the number of primes $p \leq x$ for which $a_p$ is prime; this bound has the correct order of magnitude, as predicted by the aforementioned conjecture. Then we prove a lower bound, also with the correct order of magnitude, for the number of primes $p \leq x$ such that $a_p$ is ``almost" prime, in the sense of having at most a certain fixed number of prime factors, distinct or indistinct.