Modular Curves, Rational Points and Diophantine Equations

Exploring solutions to Diophantine equations over a number field stands as a key challenge in number theory. Using the modular techniques employed by Wiles in proving Fermat’s last theorem and its extensions, we can extend our ability to solve various Diophantine equations. Additionally, understanding points on the classical modular curve contributes significantly to this methodology. In this presentation, I will address specific questions and share results that have emerged from this area of study.