Abelian varieties over Q with bad reduction in one prime only

TL;DR

This paper classifies semi-stable abelian varieties over Q with good reduction outside a single prime, identifying specific primes where such varieties exist or are isogenous to modular Jacobians, and establishing non-existence results for others.

Contribution

It provides a complete classification of semi-stable abelian varieties over Q with good reduction outside one prime, including explicit prime conditions and isogeny relations.

Findings

No non-zero semi-stable abelian varieties over Q with good reduction outside l for l=2,3,5,7,13.

Any such variety with l=11 is isogenous to a power of the Jacobian of X_0(11).

No non-zero abelian varieties over Q with good reduction outside l acquiring semi-stable reduction at l over tamely ramified extension for l ≤ 5.

Abstract

Let l be a prime. We show that there do not exist any non-zero semi-stable abelian varieties over Q with good reduction outside l if and only if l=2, 3, 5, 7 or 13. We show that any semi-stable abelian variety over Q with good reduction outside l=11 is isogenous to a power of the Jacobian variety of the modular curve X_0(11). In addition we show that there do not exist any non-zero abelian varieties over Q with good reduction outside l acquiring semi-stable reduction at l over a tamely ramified extension if and only if l \le 5.