Yu. G. Zarhin

Variations on a theme of Minkowski and Serre

Abstract If α is a root of unity in an integral domain O of characteristic zero, (α − 1)k ϵ n O , and no prime divisor of n is a unit in O , then α = 1 if n is a positive integer outside a finite set determined by k. We prove this result and generalizations of it, and give results when n is an element of the finite exceptional set. We give applications to endomorphisms of semi-abelian varieties, compatible systems of l-adic representations, and the cohomology of projective varieties.

CONNECTEDNESS EXTENSIONS FOR ABELIAN VARIETIES

Suppose

Semistable reduction of abelian varieties over extensions of small degree

A

Hodge groups of Abelian varieties with purely multiplicative reduction

is an abelian variety over a field

Reduction of Abelian Varieties

F

Isogenies of abelian varieties

, and

Polarizations on abelian varieties

\ell

Hodge groups of K3 surfaces.

is a prime not equal to the characteristic of

Hodge classes on abelian varieties of low dimension

F

Hodge classes and Tate classes on simple abelian fourfolds

. Let