The Moduli Space of Real Abelian Varieties with Level Structure

The moduli space of principally polarized Abelian varieties with real structure and with level N = 4m structure (with m1) is shown to coincide with the set of real points of a quasi-projective algebraic variety defined over , and to consist of finitely many copies of the quotient of the space GL(n, )/O(N) (of positive definite symmetric matrices) by the principal congruence subgroup of level N in GL(n, ).