Abstract
We prove that for N=6 and N=10, there do not exist any non-zero semistable abelian varieties over ℚ with good reduction outside primes dividing N. Our results are contingent on the GRH discriminant bounds of Odlyzko. Combined with recent results of Brumer-Kramer and of Schoof, this result is best possible: if N is squarefree, there exists a non-zero semistable abelian variety over ℚ with good reduction outside primes dividing N precisely when N∉{1,2,3,5,6,7,10,13}.
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