Explicit arithmetic of modular curves

Samir Siksek

The arithmetic of modular curves is the key to many celebrated theorems in arithmetic geometry, such as Mazur's isogeny theorem, Merel's uniform boundedness theorem, and the split Cartan case of Serre's uniformity conjecture due to Bilu, Parent and Rebolledo. In the last few years the subject has caught the imagination of computational number theorists who can prove concrete theorems about torsion subgroups of elliptic curves through calculations. The purpose of this minicourse is to convey the ideas and methods used in making the arithmetic and geometry of modular curves explicit. We will start by sketching the background on modular curves, their Jacobians, and their moduli interpretation. We will survey explicit methods for writing down equations for low genus modular curves, their degeneracy maps and \(j\)-maps, computing Mordell-Weil groups of modular Jacobians, and enumerating the rational points and low degree points in certain simple cases.