# A proof of N\'eron--Ogg--Shafarevich criterion via its archimedean analogue

## TL;DR

This paper derives the classical Néron--Ogg--Shafarevich criterion for good reduction of abelian varieties from an archimedean analogue involving holomorphic families and monodromy, linking complex analysis with arithmetic geometry.

## Contribution

It introduces a novel proof of the Néron--Ogg--Shafarevich criterion using an archimedean analogue and topological monodromy, bridging complex and arithmetic perspectives.

## Key findings

- Extension of holomorphic families corresponds to trivial monodromy.
- Classical criterion deduced from archimedean analogue.
- Provides a new conceptual understanding of good reduction.

## Abstract

In this short note, we deduce the classical N\'eron--Ogg--Shafarevich criterion on good reduction of abelian varieties from its archimedean analogue: a holomorphic family of abelian varieties over a punctured disc extends to the whole unit disc if and only if the topological monodromy representation is trivial.