Semistable abelian Varieties over Q

2003/11/21 by Frank Calegari, Calegari, Frank
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #Polynomial and algebraic computation #math.AG #math.NT #msc:14K15

paper · pdf · doi:10.48550/arxiv.math/0311365

24 pages, to appear in Manuscripta Mathematica

arxiv created 2003/11/21 · arxiv updated 2009/12/01

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 is not in the set 1,2,3,5,6,7,10,13.

Related

• Semistable abelian varieties over Z[1/6] and Z[1/10]
• Non-existence of certain semistable abelian varieties
• Semistable abelian varieties over ℚ with bad reduction at 19 only
• Abelian varieties over Q with bad reduction in one prime only
• Reduction of abelian varieties