Good Reduction of Abelian Varieties

1968 ◽  
Vol 88 (3) ◽  
pp. 492 ◽  
Author(s):  
Jean-Pierre Serre ◽  
John Tate
Keyword(s):  
Abelian Varieties ◽  
Good Reduction

The N�ron fiber of abelian varieties with potential good reduction

1983 ◽  
Vol 264 (1) ◽  
pp. 1-3 ◽  
Author(s):  
Joseph H. Silverman
Keyword(s):  
Abelian Varieties ◽  
Good Reduction

Local Bounds for Torsion Points on Abelian Varieties

2008 ◽  
Vol 60 (3) ◽  
pp. 532-555 ◽  
Author(s):  
Pete L. Clark ◽  
Xavier Xarles
Keyword(s):  
Abelian Varieties ◽  
Residue Field ◽  
Rational Numbers ◽  
Abelian Surface ◽  
Torsion Subgroup ◽  
Good Reduction ◽  
Torsion Points ◽  
Ramification Index ◽  
Numerical Invariants ◽  
Split Torus

AbstractWe say that an abelian variety over a p-adic field K has anisotropic reduction (AR) if the special fiber of its Néronminimal model does not contain a nontrivial split torus. This includes all abelian varieties with potentially good reduction and, in particular, those with complex or quaternionic multiplication. We give a bound for the size of the K-rational torsion subgroup of a g-dimensional AR variety depending only on g and the numerical invariants of K (the absolute ramification index and the cardinality of the residue field). Applying these bounds to abelian varieties over a number field with everywhere locally anisotropic reduction, we get bounds which, as a function of g, are close to optimal. In particular, we determine the possible cardinalities of the torsion subgroup of an AR abelian surface over the rational numbers, up to a set of 11 values which are not known to occur. The largest such value is 72.

scholarly journals Average of the first invariant factor of the reductions of abelian varieties of CM type

2016 ◽  
Vol 12 (02) ◽  
pp. 445-463 ◽  
Author(s):  
Sungjin Kim
Keyword(s):  
Abelian Group ◽  
Asymptotic Formula ◽  
Prime Ideal ◽  
Elliptic Curves ◽  
Short Range ◽  
Abelian Varieties ◽  
Prime Ideals ◽  
Good Reduction ◽  
Invariant Factor ◽  
Field Of Definition

For a field of definition [Formula: see text] of an abelian variety [Formula: see text] and prime ideal [Formula: see text] of [Formula: see text] which is of a good reduction for [Formula: see text], the structure of [Formula: see text] as abelian group is: [Formula: see text] where [Formula: see text], [Formula: see text], and [Formula: see text] for [Formula: see text]. We are interested in finding an asymptotic formula for the number of prime ideals [Formula: see text] with [Formula: see text], [Formula: see text] has a good reduction at [Formula: see text], [Formula: see text]. We succeed in proving this under the assumption of the Generalized Riemann Hypothesis (GRH). Unconditionally, we achieve a short range asymptotic for abelian varieties of CM type, and the full cyclicity theorem for elliptic curves over a number field containing the CM field.

scholarly journals Semistable abelian varieties with good reduction outside 15

2011 ◽  
Vol 139 (1-2) ◽  
pp. 49-70 ◽  
Author(s):  
René Schoof
Keyword(s):  
Abelian Varieties ◽  
Good Reduction
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