# The Néron-Ogg-Shafarevich theorem over Q_p and over C

# The Néron-Ogg-Shafarevich theorem over Q_p and over C

Via the number field-function field analogy, a variety over $\mathbb{Q}_p$ can be thought of as analogous to a family of varieties over a punctured disk, with the action of the Galois group of $\mathbb{Q}_p$ on the curve corresponding to the monodromy of the family. The Néron-Ogg-Shafarevich criterion states that an elliptic curve over $\mathbb{Q}_p$ reduces to a smooth curve modulo p if and only if the inertia group of $\mathbb{Q}_p$ acts trivially; the analogous statement over the complex numbers is that a family of elliptic curves over a punctured disk can be filled in to a smooth family over the disk if and only if the fundamental group of the punctured disk acts trivially on the fibers. I will prove both of these statements, discuss the analogy between them, and apply them to deduce facts about CM elliptic curves.