Hodge groups of abelian varieties with purely multiplicative reduction

Details Hodge groups of abelian varieties with purely multiplicative reduction Hodge groups of abelian varieties with purely multiplicative reduction

1995-05-31

arxiv.org/abs/alg-geom/9505037v2

Mathematics

Algebraic Geometry

This is an updated version of the paper. LaTeX2e or LaTeX2.09 or AMSLaTeX. Contact: silver@math.ohio-state.edu

Scientific paper

The main result of the paper is that if $A$ is an abelian variety over a subfield $F$ of ${\bold C}$, and $A$ has purely multiplicative reduction at a discrete valuation of $F$, then the Hodge group of $A$ is semisimple. Further, we give necessary and sufficient conditions for the Hodge group to be semisimple. We obtain bounds on certain torsion subgroups for abelian varieties which do not have purely multiplicative reduction at a given discrete valuation, and therefore obtain bounds on torsion for abelian varieties, defined over number fields, whose Hodge groups are not semisimple.

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