Abstract
The moduli space of principally polarized Abelian varieties with real structure and with level N = 4m structure (with m≥1) is shown to coincide with the set of real points of a quasi-projective algebraic variety defined over ℚ, and to consist of finitely many copies of the quotient of the space GL(n, ℝ)/O(N) (of positive definite symmetric matrices) by the principal congruence subgroup of level N in GL(n, ℤ).
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