The Moduli Space of Real Abelian Varieties with Level Structure

Abstract

Abstract. The moduli space of principally polarized Abelian varieties with real structure and with level N ¼ 4m structure (with m 5 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 GLðn; RÞ=OðN Þ (of positive definite symmetric matrices) by the principal congruence subgroup of level N in GLðn; ZÞ.

Polarized Real Tori

“…However, neither the moduli space nor this compactification has an algebraic structure. On the other hand, by considering real abelian varieties with a suitable level structure Goresky and Tai [8] show that the moduli space of real principally polarized abelian varieties with level 4m structure (m ≥ 1) coincides with the set of real points of a quasi-projective algebraic variety defined over Q and consists of finitely many copies of the quotient G g (4m)\P g with a discrete subgroup G g (4m) of GL(g, Z), where G g (4m) = {γ ∈ GL(g, Z) | γ ≡ I g (mod 4m)}. This paper is organized as follows.…”

“…In Section 4, we discuss a moduli space for real abelian varieties and recall some basic properties of a moduli for real abelian varieties. In Section 5 we discuss compactifications of the moduli space for real abelian varieties and review some results on this moduli space obtained by Silhol [26], Goresky and Tai [8]. In Section 6 we introduce a notion of polarized real tori and investigate some properties of polarized real tori.…”

“…Finally I would like to mention that this work was motivated and initiated by the works of Silhol [26] and Goresky-Tai [8].…”

“…In this section we review some results on the first cohomology set H 1 ( τ , Γ) obtained by Goresky and Tai [8], where τ = {1, τ } is a group of order 2 and γ is a certain arithmetic subgroup. These results are often used in this article.…”

Polarized Real Tori

“…However, neither the moduli space nor this compactification has an algebraic structure. On the other hand, by considering real abelian varieties with a suitable level structure Goresky and Tai [9] shows that the moduli space of real principally polarized abelian varieties with level 4m structure (m ≥ 1) coincides with the set of real points of a quasi-projective algebraic variety defined over Q and consists of finitely many copies of the quotient G g (4m) P g with a discrete subgroup G g (4m) of GL(g, Z), where G g (4m) = {γ ∈ GL(g, Z) γ ≡ I g (mod 4m) }.…”