Semistable abelian varieties over $\mathbb{Q}$ with bad reduction at 19 only: an overview of the Fontaine--Schoof strategy

Abstract

In this paper we provide an overview of a strategy pioneered by Fontaine and heavily refined by Schoof to classify abelian varieties with prescribed bad reduction. Throughout the overview, we prove various non-trivial background results turning it into an introduction for readers unacquainted with this topic. With the overview completed, we provide explicit examples of the strategy in action. At first we give introductory examples, classifying semistable abelian varieties over $\mathbb{Q}$ with bad reduction at exactly one of 3 or 5 up to isogeny over $\mathbb{Q}$. We then move onto a harder example, proving the analogous result for 19, which is new.