Geometry of the Siegel modular threefold with paramodular level structure

In this paper we extend some results of Norman and Oort and of de Jong, and give an explicit description of the geometry of the Siegel modular threefold with paramodular level structure. We also discuss advantages and restrictions of three standard methods for studying moduli spaces of abelian varieties.


Introduction
In this paper we study the arithmetic model of the Siegel modular threefold with paramodular level structure at a prime p. This is a very special case of geometric objects considered in the works of Norman and Oort [9], and of de Jong [1]. We extend their work and determine the singularities in this case. Our result can be used for computing the cohomology groups of the Siegel 3-fold by the Picard-Leschetz formula (see [3], Expose XV).
Let p be a rational prime number, N ≥ 3 a prime-to-p positive integer. We choose a primitive N th root of unity ζ N in Q ⊂ C and an embedding Q ֒→ Q p . Let A 2,p,N denote the moduli scheme over Z (p) [ζ N ] of polarized abelian surfaces (A, λ, η) with polarization degree deg λ = p 2 and with a symplectic level N -structure with respect to ζ N . We denote by A 2,p,N := A 2,p,N ⊗ F p the reduction modulo p of the moduli scheme A 2,p,N .
In [9] Norman and Oort studies the p-rank stratification on Siegel moduli spaces (the case for principally polarized abelian varieties was studied earlier in Koblitz [6]) which we recall as follows. An abelian variety a field K of characteristic p is said to have p-rank f if dim Fp A[p](K) = f , which is an integer between zero and dim A. Conversely, given any integer 0 ≤ f ≤ g, there is a g-dimensional abelian variety with p-rank f . The discrete invariant p-rank defines locally closed subsets in the moduli space; it is showed in [9] that these subsets form a stratification. We now focus on the Siegel 3-fold. For each integer 0 ≤ f ≤ 2, let V f ⊂ A 2,p,N (resp. V ≤f ⊂ A 2,p,N ) be the reduced subscheme consisting of points with p-rank equal to f (resp. equal to or less than f ). The ordinary locus, which parametrizes the ordinary abelian varieties, is the p-rank 2 stratum; the non-ordinary locus is V ≤1 . We have the following fundamental results for A 2,p,N (see [10], Theorem 2.3.3, [9], Theorems 3.1 and 4.1, [1], Theorem 1.12 and Proposition 3.3): (1) The structure morphism A 2,p,N → SpecZ (p) [ζ N ] is flat and a complete intersection morphism of relative dimension 3. The fibers are geometrically irreducible. ( The non-ordinary locus V ≤1 has two irreducible components. Let S 2,p,N ⊂ A 2,p,N denote the supersingular locus, which is the reduced closed subscheme over F p consisting of supersingular abelian varieties. Recall that an abelian variety A in characteristic p is called supersingular if it is isogenous to a product of supersingular elliptic curves over an algebraically closed field k; it is called superspecial if it is isomorphic to a product of supersingular elliptic curves over k. Let Λ 2,1,N ⊂ A 2,1,N (F p ) be the set of superspecial points in the moduli space A 2,1,N of principally polarized abelian surfaces with level N -structure. Let Λ ⊂ S 2,p,N (F p ) be the subset consisting of superspecial points (A, λ, η) such that ker λ ≃ α p × α p . The following is a description of the supersingular locus (see [14], Theorem 4.7): (1) The scheme S 2,p,N is equi-dimensional and each irreducible component is isomorphic to P 1 over F p .
(3) The singular locus of S 2,p,N is the subset Λ. Moreover, at each singular point there are p 2 + 1 irreducible components passing through and any two of them intersect transversely.
In this paper we prove Theorem 1.3. The moduli scheme A 2,p,N is regular and smooth over SpecZ (p) [ζ N ] exactly away from the subset Λ. At each singular point x (in Λ) in the special fiber A 2,1,N , the formal completion of the local ring at x is isomorphic to where W (F p ) is the ring of Witt vectors over F p . Consequently, the moduli space A 2,p,N is normal. Theorem 1.3 improves Theorem 1.1 (1). Theorems 1.1-1.3 provide a good understanding of the geometry of the moduli scheme A 2,p.N . As the singularities are non-degenerate ordinary double points, one can use the Picard-Leschetz formula to compute the cohomology groups of the moduli space A 2,p,N . We intend to continue the work along this direction.
The paper is organized as follows. In Section 2 we study the singularity of the moduli scheme using the crystalline theory when p > 2. We also discuss the classification of deformations of non-degenerate double points. In Section 3 we determine the singularity of the moduli scheme using the local model for arbitrary residue characteristic. In Section 4 we explain that some higher terms of the defining equation in Norman's first example in [8] is required in order to describe the non-ordinary locus.

Singularities using the crystalline theory
In this section we use the crystalline theory to determine the singularities of the moduli scheme A 2,p.N for p > 2. Standard references for the Grothendieck-Messing deformation theory are [5] and [7]. For convenience of discussion, we introduce the following definition.
and the other pairings are zero.
2. An equivalent definition (for general Siegel moduli spaces) is that a quasi-polarized Dieudonné module M is called Lagrangian if there is a maximally isotropic W -sublattice L ⊂ M such that L is co-torsion free and contained in V M . This is also equivalent to the condition that the corresponding polarized abelian variety can be lifted over W (cf. [13]).
The first order universal deformation of M is given by and R is the first order universal deformation ring. One computes Y 1 , Y 2 = −t 21 + pt 12 . This shows  (ii) a(A) = 2. In this case A is superspecial. It is not hard to classify the Dieudonné module M . There are two possibilities (notice that k is algebraically closed): (iia) There is a W -basis X 1 , X 2 , Y 1 , Y 2 for M with the following properties: and the remaining pairings are zero.
(iib) There is a W -basis X 1 , X 2 , Y 1 , Y 2 for M with the following properties: We have F M t ⊂ M and V M t ⊂ M . Therefore, ker λ ≃ α p × α p .
In the case (iib), one has We have F M t ⊂ M and V M t ⊂ M . Therefore, ker λ ≃ α p × α p . This proves the lemma.
Note that the point x in the case (iia) is Lagrangian. We will show (Lemma 2.10) that the moduli space has singularities at points in the case (iib). It follows from Lemma 2.2 that the point x in the case (iib) is not Lagrangian.
From the discussions above, we have proven where Q ′ (x) is any non-degenerate quadratic form over A, a ′ is an element in m A such that a ′ ≡ a mod m 3 A , and u is any unit in A. Since A is strictly Henselian, any two non-degenerate quadratic forms are equivalent. We may take Q ′ (x) to be the standard split form. By rescaling half of variables by suitable units in A, one shows that where Q(x) is a non-degenerate quadratic form and a ∈ m A , then there is an isomorphism R/(f (x)) ≃ R/(a + Q(x)).

Lemma 2.8. Let
A be a complete Noetherian local ring and m A be its maximal ideal. Let R, m R , I, and f (x) be as before. Suppose that where a, a i are elements in m A , and Q(x) is a non-degenerate quadratic form over A. Then there is an isomorphism for some element a ′ ∈ m A and some non-degenerate quadratic form Q ′ (x).
Proof. This is [4], Chapter III, Proposition 2.4. We sketch the proof as it will be used to prove Proposition 2.6. We will find an element b = (b 1 , . . . , b n ) with b i ∈ m A such that the linear term of f (x + b) vanishes, and apply Lemma 2.7. We have Proof of Proposition 2.6. Condition (2.1) implies that such that a ′ ≡ a mod m 3 A , each a i ∈ m 2 A and Q ′ (x) ≡ Q(x) mod m A . By Lemma 2.8, we have, for an appropriate element b, where a ′′ is an element in m A such that a ′′ ≡ a ′ mod m 4 A (since a i ∈ m 2 A , see Remark 2.9) and Q ′′ (x) ≡ Q ′ (x) mod m A . Therefore, by Lemma 2.7, where Q ′′ (x) is a non-degenerate quadratic form and a ′′ is an element in m A such that a ′′ ≡ a mod m 3 A .

2.3.
Singularities of A 2,p,N . Let x = (A, λ, η) be a point in Λ ⊂ A 2,p,N (F p ). Let M be the associated quasi-polarized Dieudonné module. By Lemma 2.4, we choose a W -basis X 1 , X 2 , Y 1 , Y 2 for M as in the case (iib). It is easy to compute that the Hodge filtration Fil ⊂ H DR 1 (A) is equal to < Y 1 , Y 2 > k , where k = F p . Let R u be the universal deformation ring of the quasipolarized Dieudonné module M . It follows from Theorem 1.1 (1) that for some power series f . By the Grothendieck-Messing deformation theory, the universal deformation of M over R u /m p R u is given by One computes the relation Y 1 , Y 2 = t 11 t 22 − t 12 t 21 + p = 0 in R u . This implies that

This shows
Lemma 2.10. If x ∈ Λ, then the moduli space A 2,p,N is not smooth at x. Proof. This follows immediately from Proposition 2.6 and (2.7).

Singularities using the local model method
In this section we use the method of local models to determine the singularities of the moduli scheme A 2,p.N , including the case where p = 2. Our references are de Jong [2] and Rapoport-Zink [11].
3.1. Local model diagrams. Let Λ 1 := Z 4 p and e 1 , . . . , e 4 be the standard basis. Let ψ be the alternating pairing on Λ 1 so that its representing matrix with respect to the standard basis is Let M loc be the local model associated to the lattice Λ 1 . This is the projective scheme over Z p representing the following functor. For any Z p -scheme S, the set M loc (S) of its S-valued points is the set of locally free O S -submodules F ⊂ Λ 1 ⊗ O S , locally on S direct summands of Λ 1 ⊗ O S , which are isotropic with respect to the pairing ψ.
Let A 2,p,N be the moduli space over Z p [ζ N ] parametrizing equivalence classes of objects (A, ξ), where A is an object in A 2,p,N and ξ : H DR 1 (A/S) ≃ Λ 1 ⊗ O S is an isomorphism which preserves the polarizations. Write A 2,p,N for the reduction A 2,p,N ⊗ F p modulo p.
Let G be the group scheme over Z p representing the functor S → Aut(Λ 1 ⊗ O S , ψ). This group acts on the schemes A 2,p,N and M loc from the left. We have the following (local model) diagram where • ϕ 2 is the morphism that sends each object (A • , ξ) to the image ξ(ω) of the Hodge submodule ω ⊂ H DR 1 (A/S), and • ϕ 1 is the morphism that forgets the trivialization ξ. We know that (i) the morphism ϕ 2 is G-equivalent, surjective and smooth, and has the same relative dimension as ϕ 1 does, and (ii) the morphism ϕ 1 : A 2,p,N → A 2,p,N is a G-torsor.

3.2.
Singularities of A 2,p,N . Let x = (A, λ, η) be a point in Λ ⊂ A 2,p,N (k), where k = F p . Let M be the associated quasi-polarized Dieudonné module. By Lemma 2.4, we choose a W -basis X 1 , X 2 , Y 1 , Y 2 for M as in the case (iib). We choose a trivialization ξ which sends Y 1 , X 1 , X 2 , Y 2 to e 1 , e 2 , e 3 , e 4 , respectively. This defines a point y ∈ A 2,p,N mapping to x. We put z = ϕ 2 (y) ∈ M loc (F p ), which is given by Around the point z we choose coordinates same as in Subsection 2.3 so that an affine open chart U is SpecW [t 11 , t 12 , t 21 , t 22 ]/(p + t 11 t 22 − t 12 t 21 ).
Let R x (resp. R y and R z ) be the completion of the local ring of A 2,p,N at x (resp. of A 2,p,N at y and of M loc at z). We have Remark 3.1. The local model is a very effective method for computing local moduli spaces. However, one can apply it in the special case when the kernel of the polarization is contained in the p-torsion subgroup.

Some fine information from the Cartier theory
As experts all know that there is another important tool for computing local moduli spaces -using the Cartier theory, we discuss our situation with this method. First of all, the method works exclusively only in characteristic p, namely the best information we can get is the local moduli spaces modulo p. In [8] Norman established the theoretic background for an algorithm for computing local moduli spaces. In principle the algorithm can only compute the coordinate ring of the universal deformation space modulo a specific power of the maximal ideal. However, since the singularity is determined by its sufficiently well approximation, one would be able to determine the singularity eventually. Norman gave two examples in [8] and the first example is exactly the case of A 2,p,N at points in Λ.
Let M be the Dieudonné module associated to a point x in Λ. We can choose basis {e i } for M so that (cf. (iib)):  and that the quasi-polarization P from M to its dual M t is given by P (e 1 ) = f 2 , P (e 2 ) = −f 1 , where {f i } is the dual basis for M t . Using the theory of displays, one constructs the equi-characteristic universal deformation M over the universal deformation ring R 0 = k[[t 11 , t 12 , t 21 , t 22 ]], which is generated by {e i } over the Cartier ring Cart p (R 0 ) (the ring A R 0 in [8]) with the following relations  where T ij = [t ij ] is the Teichmüller lifting of t ij . Let f ∈ R 0 be the defining equation (up to a unit) of the maximal closed formal subscheme over which the quasi-polarization P extends. Norman showed that (when p > 2): f = t 11 t 22 − t 12 t 21 + r(t), for some r(t) ∈ (t ij ) p .
Since the singularity of R 0 /(f ) is determined by its leading term, one computes the local moduli space in characteristic p. However, the following lemma shows that the calculation of the remaining term r(t) is required in order to determine the non-ordinary locus. Therefore, the defining equation of the non-ordinary locus is t 11 t 22 − t 12 t 21 .