Modular curves, rational points, and automorphism groups

Modular curves have a very rich arithmetic, algebraic, and analytic structure. They have been important in the recent history of number theory in particular for their link with Fermat's Last Theorem. We start defining what a modular curve is and giving some examples. Then, we introduce and state Serre's uniformity conjecture, a conjecture about rational points on modular curves. Finally, we illustrate some conjectures and results on the automorphism groups of modular curves.