VARIATIONS ON A THEME OF MINKOWSKI AND SERRE

If a is a root of unity in an integral domain O of characteristic zero , (alpha - 1)(k) is an element of nO, and no prime divisor of n is a u nit in O, then alpha = 1 if n is a positive integer outside a finite s et determined by k. We prove this result and generalizations of it, an d give results when n is an element of the finite exceptional set. We give applications to endomorphisms of semi-abelian varieties, compatib le systems of l-adic representations, and the cohomology of projective varieties.