This thesis is divided into two parts. One is devoted to integral points on modular curves, and the other concerns pairing-friendly elliptic curves. In the first part, we give some effective upper bounds for the 'j'-invariant of integral points on arbitrary modular curves corresponding to congruence subgroups over arbitrary number fields assuming that the number of cusps is not less than 3. Especially, in the non-split Cartan case we provide much better bounds. As an application, we get similar results for certain modular curves with less than three cusps. In the second part, a new heuristic upper bound for the number of isogeny classes of ordinary pairing-friendly elliptic curves is given. We also heuristically analyze the Cocks-Pinch method to confirm some of its general consensuses. Especially, we present the first known heuristic which suggests that any efficient construction of pairing-friendly elliptic curves can efficiently generate such curves over pairing-friendly fields. Finally, some numerical evidence is given.
