Fontaine and Abrashkin showedd in 1983 that there do not exist abelian varieties over Q that have good reduction everywhere. We extend this result. We show that for l=2,3,5 there do not exist abelian varieties over Q that have good reduction modulo all primes p different from l and are semistable modulo l. The same holds for l=7 under the assumption of certain Generalized Riemann Hypotheses.
For l=11 and l \ge 17, the Jacobian of the modular curve X_0(l) has dimension >0 and has good reduction outside l but is semistable modulo l. So the result is almost best possible. Only for l=13 I do not know what the situation is.