Hodge Groups of Abelian Varieties with Purely Multiplicative Reduction

摘要

The authors show that if A is an abelian variety over a subfield F of C, and Ahas purely multiplicative reduction at a discrete valuation of F, then the Hodge group of A is semisimple (Theorem 4.1). Since the non-semisimplicity of the Hodge group of an abelian variety can be translated into a condition on the endomorphism algebra and its action on the tangent space (see Theorem 3.1), this gives a useful criterion for determining when an abelian variety does not have purely multiplicative reduction. For abelian varieties over number fields, a result analogous to Theorem 4.1 holds where the Hodge group is replaced by a certain linear algebraic group H(l) over Q(l) arising from the image of the l-adic representation associated to A (see Theorem 3.2). The Mumford-Tate conjecture predicts that H(l) is the extension of scalars to Q(l) of the Hodge group. The authors' result generalizes a result of Mustafin which says that for a Hodge family of abelian varieties admitting a 'strong degeneration', generically the fibers have semisimple Hodge group. In section 5 the authors provide bounds on torsion for abelian varieties which do not have purely multiplicative reduction at a given discrete valuation. The authors apply this and Theorem 4,1 to obtain bounds on torsion for abelian varieties whose Hodge groups are not semisimple.