Finiteness for \'{E}tale Fundamental Groups of N\'{e}ron Models

In this paper, we prove that the étale fundamental group of the Néron model of an abelian variety over a number field $K$ is the semidirect product of a finite group with the étale fundamental group of the ring of integers of $K.$ We prove this by studying how the Faltings height of an abelian variety changes under covers that spread out to finite étale covers of its Néron model.

We then strengthen this result for elliptic curves.

Using Merel's torsion theorem, we show the size of this finite group can be uniformly bounded for a fixed number field.

We conclude by giving the list of all possible étale fundamental groups for the Néron model of an elliptic curve over $\mathbb{Q}.$