• Heegner point
• quadratic twist
• l-function
• congruent number
• elliptic curves
• modular forms
• rational point

In the 10th century, Arab scholars studied 3-term arithmetic progressions of rational squares x^2, y^2, z^2 with integral common difference n, posing the question, now known as the Congruent Number Problem (CNP), of which positive integers values, called congruent numbers, n could take. This seemingly arbitrary problem was rooted in elementary geometry, since it was known from the time of Diophantus that such arithmetic progressions are in bijection with right triangles having rational sides and integral area. Easy to state as it is, this question is yet to be completely answered, and it has been linked to deep subjects in modern Number Theory like the theory of elliptic curves and modular forms. The purpose of this thesis is twofold: while we describe the connections between the CNP and these objects, starting from the definitions and building up to the partial results obtained in the last few decades, we also focus on a specific aspect of the arithmetic theory of elliptic curves which is intimately connected to the problem: the behaviour of ranks in quadratic twist families. Here, the plural "ranks" does not just indicate that the rank varies inside the family, but also that we have two different ranks, algebraic and analytic, which, although famously conjectured to be equal, are treated very differently. We apply some techniques for controlling the algebraic rank to the CNP, but most of the results presented here concern the analytic rank, including a novel partial progress towards the existence of infinitely many elliptic curves with fixed "high" analytic rank.