Endomorphism rings of abelian surfaces and projective models of their moduli spaces

Abstract

We construct projective models for Humbert surfaces and QCM-curves, i.e., Shimura curves together with their natural embedding into the coarse moduli space for principally polarized abelian surfaces. The points of a QCM-curve correspond to an abelian surface, such that its algebra of complex multiplications is an order in an indefinite rational quaternian algebra. Moreover, we determine the structure of such orders. Introduction.In this paper we are mainly concerned with constructing, in a concrete way, projective models for arbitrary QCM-curves. For maximal orders of the indefinite quaternion division algebras with discriminant 6 and 10, Hashimoto and Murabayashi constructed in a recent paper ([HM]) projective models of the corresponding QCM-curves. These examples inspired the author to study this question. In [HM] the authors furthermore construct a preimage under the Torelli map (which is an immersion).Let A be a simple principally polarized complex abelian surface, End(A) its ring of endomorphisms and L = End(A) <g> Q the algebra of endomorphisms (= the algebra of complex multiplications). Then, as is well-known, the ring End(A) is of one of the following types: an order in a CM-field of degree four, an order in an indefinite rational quaternion algebra, an order in a real quadratic field, or Z. The dimensions of the corresponding moduli spacesnamed Shimura varieties of PEL-type-is 0, 1,2, 3, respectively. In the first three cases we refer to these Shimura varieties, together with their embeddings into the Satake compactification Λ2 = Proj(A(/~2)) of Γ2\H2, as CM-points, QCM-curves (quaternionic complex multiplication), and Humbert surfaces. As a projective variety the Satake compactification Λ2 is a quotient of P 3 by a finite group G of order 46080, see [Rl]. We are mainly concerned with QCM-curves, i.e., Shimura curves with their natural embedding into Λ2. CM-points are special points on Humbert surfaces, which are easy to compute.Let us define a QCM-order to be any order in an indefinite rational quaternion algebra which occurs as an endomorphism ring of an abelian surface. The first result is to determine the structure of QCM-orders. We prove that a QCM-order may be written as R = Z 0 Zot 0 Zβ 0 Zaβ, where a and β are Rosati invariant elements of (positive) discriminant Δ(α),), such that the discriminant matrix Δ(α) Δ(α,j8)\ Δ(α,j8) A(β) I

Dynamics of SL2(ℝ) over moduli space in genus two

“…The above results on real multiplication in genus two originate in the work of Humbert in the late 1890s [Hu]; see also [vG, Ch. IX] and [Ru,§4]. Additional material on real multiplication and Hilbert modular varieties can be found in [HG], [vG] and [BL].…”

Higher-dimensional 3-adic CM construction

“…The latter equality follows by the symmetry of L. Note that p 23 → (2x, x, 3x). The kernel of ξ is given by the restriction of τ to A [6].…”

Quaternionic loci in Siegel’s modular threefold

“…Remark 22. Note that Theorem 10 of [35] states that if a b b c is a positive definite matrix in M (2, Z) satisfying a, c ≡ 0, 1 mod 4, 4|(b 2 − ac), gcd(a, b, c) = 1, and (b 2 − ac)/4 is squarefree, then any pair of two primitive singular relations ℓ 1 , ℓ 2 satisfying ( ℓ i , ℓ j ∆ ) = a b b c defines the same locus in A 2 . In Lemma 23, we find that when D ≡ 1, 2 mod 4, every quadratic form ax 2 + 2bxy + cy 2 in Q D satisfies Runge's conditions and Lemma 20 would follows immediately from Runge's theorem.…”