  We discuss quantum Hall effects in a gapped insulator on a periodic
two-dimensional lattice. We derive a universal relation among the the quantized
Hall conductivity, and charge and flux densities per physical unit cell. This
follows from the magnetic translation symmetry and the large gauge invariance,
and holds for a very general class of interacting many-body systems. It can be
understood as a combination of Laughlin's gauge invariance argument and
Lieb-Schultz-Mattis-type theorem. A variety of complementary arguments, based
on a cut-and-glue procedure, the many-body electric polarization, and a
fractionalization algebra of magnetic translation symmetry, are given. Our
universal relation is applied to several examples to show nontrivial
constraints. In particular, a gapped ground state at a fractional charge
filling per physical unit cell must have either a nonvanishing Hall
conductivity or anyon excitations, excluding a trivial Mott insulator.
