  We introduce generalized Frobenius-Schur indicators for pivotal categories.
In a spherical fusion category C, an equivariant indicator of an object in C is
defined as a functional on the Grothendieck algebra of the quantum double Z(C)
via generalized Frobenius-Schur indicators. The set of all equivariant
indicators admits a natural action of the modular group. Using the properties
of equivariant indicators, we prove a congruence subgroup theorem for modular
categories. As a consequence, all modular representations of a modular category
have finite images, and they satisfy a conjecture of Eholzer. In addition, we
obtain two formulae for the generalized indicators, one of them a
generalization of Bantay's second indicator formula for a rational conformal
field theory. This formula implies a conjecture of Pradisi-Sagnotti-Stanev, as
well as a conjecture of Borisov-Halpern-Schweigert.
