Abelian varieties over $\mathbb{Q}$ with bad reduction in one prime only

Abstract

We show that for the primes l = 2, 3, 5, 7 or 13, there do not exist any non-zero abelian varieties over $\mathbb{Q}$ that have good reduction at every prime different from l and are semi-stable at l. We show that any semi-stable abelian variety over $\mathbb{Q}$ with good reduction outside l = 11 is isogenous to a power of the Jacobian variety of the modular curve X₀(11). In addition, we show that for l = 2, 3 and 5, there do not exist any non-zero abelian varieties over $\mathbb{Q}$ with good reduction outside l that acquire semi-stable reduction at l over a tamely ramified extension.