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arXiv · 2607.00232

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

Frank Lu
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Abstract

In this paper, we prove that the \'{e}tale fundamental group of the N\'{e}ron model of an abelian variety over a number field $K$ is the semidirect product of a finite group with the \'{e}tale 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 \'{e}tale covers of its N\'{e}ron 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 \'{e}tale fundamental groups for the N\'{e}ron model of an elliptic curve over $\mathbb{Q}.$

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Frank Lu. 2026-06-30. Finiteness for \'{E}tale Fundamental Groups of N\'{e}ron Models. https://arxiv.org/abs/2607.00232

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