Semistable abelian varieties over $\mathbb{Q}$ with bad reduction at 19 only

TL;DR

The paper classifies semistable abelian varieties over $\mathbb{Q}$ with bad reduction at exactly one prime, $p=19$, proving that such varieties are isogenous to a power of $J_0(19)$. It develops and applies the Fontaine–Schoof strategy, focusing on finite flat group schemes and the category $\mathcal{C}_{19}$ at $\ell=2$, to classify simple objects, their prolongations, and extensions, and then analyzes the resulting $2$-divisible groups via Tate modules. The key outcome is the identification of the simple objects $\underline{\mathbb{Z}/2\mathbb{Z}}$, $\mu_2$, and $E=X_0(19)[2]$, with $E$ arising from $X_0(19)$ and having $\mathbb{Q}(E)=\mathbb{Q}(\sqrt{-19},\alpha)$; the extension analysis shows that the allowable $\ell$-divisible structures force the variety to lie on the isogeny class of a power of $X_0(19)$. Together with local–global ramification and class-field-theoretic arguments, this yields the main theorem and confirms the effectiveness of the Fontaine–Schoof framework for $N=19$, hinting at extensions to $N=29$ in future work.

Abstract

We classify semistable abelian varieties over $\mathbb{Q}$ with bad reduction at exactly 19 up to isogeny over $\mathbb{Q}$. The general strategy goes back to Fontaine and has been heavily refined by Schoof. In the beginning of this paper we include an overview of this strategy, proving various non-trivial background results along the way, as an introduction for readers unacquainted with this topic.

Theorems & Definitions (70)

• Theorem 1.1: = Theorem \ref{['thm:19_classification']}
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Remark 2.5
• Lemma 2.6
• proof
• Example 2.7
• Remark 2.8