# 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.