Exploring the arithmetic of generalised modular Jacobians and of power sums

This thesis presents a series of results related to understanding rational points on curves, all different in flavour. In the first part of this dissertation we explore the group of rational torsion points on (generalised) Jacobians of certain modular curves. In Chapter 2 we study the classical modular curve X0(N) over Q while in Chapter 3 we move to function fields and consider the Drinfeld modular curve X0(n) defined over Fq(T). In the second half we take a look at questions related to the Diophantine equation (x1)^(m1) +... +(xk)^(mk) = y^(n), which equates sum of k different powers to a perfect power. In Chapter 4 we fix k=2 and take a close look at arithmetic invariants of certain hyperelliptic curves related to the equation via the so-called Darmon's program. In Chapter 5, we relax the condition on k, but ask for the variables x1,...., xk to take values in an arithmetic progression.