Moduli space of real abelian varieties with level structure

Abstract

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