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Primitive elements for pdivisible groups
 Robert Kottwitz^{1} and
 Preston Wake^{2}Email author
Received: 23 January 2017
Accepted: 5 June 2017
Published: 8 September 2017
Abstract
We introduce the notion of primitive elements in arbitrary truncated pdivisible groups. By design, the scheme of primitive elements is finite and locally free over the base. Primitive elements generalize the “points of exact order N,” developed by Drinfeld and Katz–Mazur for elliptic curves.
1 Background
In this paper, we observe that Raynaud’s theory of Haar measures on finite flat group schemes [10] may be used to define a “nontriviality” condition on sections, which we call nonnullity. For groups of order p, we show that nonnull sections are “generators” in the sense of Oort–Tate theory [11]. For truncated pdivisible groups, we use a nonnullity condition to define the notion of primitivity, generalizing the “points of exact order N” of Drinfeld [3] and Katz–Mazur [6].
In the case of elliptic curves, Drinfeld and Katz–Mazur go further and define full level structures. This allows them to construct and prove nice properties of integral models of modular curves at arbitrary levels in a very elegant fashion. We believe that our definition of primitive elements may be a first step toward defining full level structures in certain cases, as it was in previous work by one of us in the case \(\mu _p \times \mu _p\) [12]. However, for general pdivisible groups, we believe that new ideas are needed, and we hope that this work will lead to a better understanding of the issues involved in defining full level structures.
1.1 The problem of full level structures

\(\mathcal {F}_G\) is flat over S,

\(\mathcal {F}_G \times _S S[1/p] = \underline{\mathrm {Isom}}_{S[1/p]}\bigl ( (\mathbb {Z}/p^r\mathbb {Z})^g,G[1/p] \bigr )\).
1.2 Previous results
In the case where G embeds into a smooth curve C over S (for example if \(G=E[p^r]\) for an elliptic curve E), a satisfactory theory of full level structures has been built out of the ideas of Drinfeld [3]. However, Drinfeld’s definition crucially uses the fact that G is a Cartier divisor in C. Katz and Mazur developed a notion of “full set of sections,” which they show is equivalent to the Drinfeld level structure in the case that \(G \subset C\) [6, §1.10]. However, as Chai and Norman pointed out [1, Appendix], the Katz–Mazur definition does not give a flat space in general—it fails even for the relatively simple example of \(G=\mu _p \times \mu _p\).
More recently, one of the present authors developed a notion of full homomorphisms in the specific case \(G=\mu _p \times \mu _p\) [12].
1.3 Primitive elements
The first step in finding a basis for a free module is to find a primitive vector—that is, an element that can be extended to a basis. Analogously, a first step towards defining a notion of full level structure might be to define a notion of primitive element for group schemes. In addition, the notion of primitive element is needed to define the correct notion of “linear independence,” which is a key part of the method in [12] for \(G=\mu _p \times \mu _p\). In this paper we develop a formal theory of primitive elements, generalizing the ad hoc notion defined in [12].
1.4 Primitive elements and full homomorphisms
One may suggest defining a homomorphism \(\varphi :(\mathbb {Z}/p^r\mathbb {Z})^g \rightarrow G\) to be “full” if it sends primitive vectors to primitive vectors. Indeed, if G is constant, then this corresponds to the condition that the matrix of \(\varphi \) has linearly independent columns. However, the example of \(\mu _p \times \mu _p\) studied in [12] shows why this definition does not give a flat space of full homomorphisms. In that case, one may think of \(\varphi \) as a “\(2\times 2\)matrix with coefficients in \(\mu _p\).” If \(\varphi \) sends primitive vectors to primitive vectors, then the columns are “linearly independent,” but the rows may not be—hence the elements cutting out the condition that the rows be “linearly independent” are ptorsion elements in the coordinate ring of the space of full homomorphisms. On the other hand, the main theorem of [12] implies that column conditions together with the row conditions give a flat space.
For a general group G, there is no obvious analog of the row conditions, so it is not clear how to generalize from primitive vectors to full homomorphisms. A new idea is needed.
1.5 Summary
Let S be a scheme, and let G be a finite locally free (commutative) group scheme over S. Let G denote the rank of G. We define a closed subscheme \(G^\times \subset G\), which we call the nonnull subscheme. The ideal cutting out \(G^\times \) consists of invariant measures, as in Raynaud’s theory [10], on the Cartier dual of G. As a consequence of Raynaud’s results, \(G^\times \) is finite and locally free over S of rank \(G1\). We think of \(G^\times \) as the groupscheme version of the set of nonzero elements of G.
There does not seem to be any completely satisfactory word to use here. Since the identity element in G(S) can perfectly well lie in \(G^\times (S)\) (as happens in the second example below when S is a scheme over \(\mathbb F_p\)), it would be extremely confusing say that elements in \(G^\times (S)\) are nonzero. Instead we have chosen to say that they are “nonnull.”

If \(G=\Gamma _S\), the constant groupscheme associated to a finite abelian group \(\Gamma \), then \(G^\times = (\Gamma \setminus \{0\})_S\), the scheme of nonidentity sections.

If \(G=\mu _p\), then \(G^\times =\mu _p^\times \), the scheme of primitive roots of unity.

If G is an Oort–Tate group [11] (i.e. \(G=p\)), then \(G^\times \) coincides with the scheme of generators defined by Haines and Rapoport [5].

If G is a Raynaud group [10] (i.e. G has an action of \(\mathbb {F}_q\) and \(G=q\) for some power q of p), then \(G^\times \) coincides with the scheme of \(\mathbb {F}_q\) generators defined by Katz–Mazur (c.f. [9])

If \(V=(\mathbb {Q}_p/\mathbb {Z}_p)^h\) and \(\mathcal {G}=V_S\) is a constant pdivisible group, then \(G^{{{\mathrm{prim}}}}\) is the scheme associated to the set of primitive vectors in the free \(\mathbb {Z}/p^r\mathbb {Z}\)module \(V[p^r]\).

If \(G=\mu _{p^r}\), then \(G^{{{\mathrm{prim}}}}\) is the subscheme of primitive roots of unity.

If \(G=E[p^r]\) for an elliptic curve E, then \(G^{{{\mathrm{prim}}}}\) is the scheme of sections “of exact order \(p^r\)” defined by Drinfeld–Katz–Mazur [3, 6].
1.6 Applications to Shimura varieties
Let X be a Shimura variety over \(\mathbb {Q}\) that has a universal abelian variety A over it, and suppose \(\mathfrak {X}\) and \(\mathcal {A}\) are models for X and A that are flat over \(\mathbb {Z}_{(p)}\). Then, for each \(r>1\), there is an interesting cover \(X_1(p^r)\) of X given by adding the additional data of a point of order \(p^r\) in A.
The scheme \(\mathfrak {X}_1(p^r):=\mathcal {A}[p^r]^{{{\mathrm{prim}}}}\) is an integral model for \(X_1(p^r)\) that is finite and flat over \(\mathfrak {X}\). Since \(\mathfrak {X}\) is flat over \(\mathbb {Z}_{(p)}\), this implies that \(\mathfrak {X}_1(p^r)\) is the Zariskiclosure of \(X_1(p)\) in \(\underline{\mathrm {Hom}}_\mathfrak {X}(\mathbb {Z}/p^r\mathbb {Z},\mathcal {A}[p^r])\). In particular, this “flatclosure” model, which is a priori only flat over \(\mathbb {Z}_{(p)}\), is actually flat over \(\mathfrak {X}\).
On the other hand, one can show that, except for modular curves (or the Drinfeld case), the scheme \(\mathfrak {X}_1(p^r)\) is not normal. In particular, \(\mathfrak {X}_1(p^r)\) is not the normalization of \(\mathfrak {X}\) in \(X_1(p^r)\), and this gives an example where the “normalization” and “flat closure” models differ.
This issue of nonnormality makes us doubtful that these models will have direct application to the Langlands program. Instead, we view the theory of primitive elements as an interesting tool to use in the future study of integral models. For example, it would be interesting to consider combining the notion of primitive element with parahoric models of Shimura varieties, in analogy with the work of Pappas on Hilbert modular varieties [9]. Using the theory of Raynaud group schemes, Pappas produces a model for \(\Gamma _1(p)\)type level that is normal (but not finite over the base).
2 Review of Raynaud’s Haar measures for finite flat group schemes
In this section we work over an affine base scheme \(S = {{\mathrm{Spec}}}(k)\), and G denotes a commutative group scheme over S that is finite, flat and finitely presented. So \(G = {{\mathrm{Spec}}}(A)\) with A locally free of finite rank as kmodule. This rank is a locally constant function, denoted G, on S.
We write \(G'={{\mathrm{Spec}}}(A')\) for the Cartier dual of G; it is another object of the same kind as G, and \(G' = G\). Recall that A and \(A'\) are the kduals of each other.

write \(\star \) for the multiplication law on \(A'\) (intuitively, convolution of measures),

write 1 for the unit element in the ring A (intuitively, the constant function with value 1),

write \(\delta \) for the unit element in the ring \(A'\), i.e. the counit \(A \rightarrow k\) for the coalgebra A (intuitively, evaluation of functions at the identity element in the group G),

denote the natural Amodule structure on \(A'\) by \(f \mu \) (intuitively, pointwise multiplication of a measure by a function), and

denote the natural \(A'\)module structure on A by \(\mu \star f\) (intuitively, the convolution of a function by a measure).
 (1)
The natural pairing \(A' \otimes A \rightarrow k\) restricts to a perfect pairing between \(D_G\) and \(J_G\). So the line bundles \(D_G\) and \(J_G\) on S are canonically dual to each other.
 (2)
The map \(f \otimes \mu \mapsto f\mu \) is an isomorphism \(A \otimes D_G \rightarrow A'\). This map is an isomorphism of Amodules and of \(A'\)modules, with \(g \in A\) (respectively, \(\nu \in A'\)) operating on \(A \otimes D_G\) by the rule \(g(f \otimes \mu ):= (gf) \otimes \mu \) (respectively, \(\nu \star (f \otimes \mu ) := (\nu \star f) \otimes \mu \)).
2.1 Integration in stages
We need one more fact about Haar measures, namely an analog of the “integration in stages formula” in the theory of Haar measures on locally compact groups. It seems plausible that this is wellknown, but since we do not know a reference, we will provide a proof.
Lemma 2.1
Let \(\mu _H \in D_H\) and \(\mu _K \in D_K\). Choose \(\tilde{\mu }_K \in A'\) mapping to \(\mu _K\) under \(A' \twoheadrightarrow C'\). Then the element \(\mu _G \in A'\) defined by \(\mu _G = \tilde{\mu }_K \star \mu _H\) lies in \(D_G\) and is independent of the choice of \(\tilde{\mu }_K\). Moreover the map \(\mu _K \otimes \mu _H \mapsto \mu _G\) is an isomorphism \(D_K \otimes D_H \rightarrow D_G\).
Proof
It is evident that \(\mu _G\) is independent of the choice of the lifting \(\tilde{\mu }_K\), because this lifting is welldefined modulo \((I_{B'}) A'\), and \(I_{B'}\) annihilates \(\mu _H\). The rest of the lemma is most easily understood in terms of integration in stages, as we will now see.
The map \(\mathcal I_H : A \rightarrow C\) is equivariant with respect to \(G \twoheadrightarrow K\) (and the natural actions of G on A and K on C), and the composition \(\mu :=\mu _K \circ \mathcal I_H : A \rightarrow k\) is Gequivariant, i.e. \(\mu \in D_G\). Unwinding the definitions, one sees that \(\mu = \mu _G\). The work we did shows that \(\mu _G\) is surjective when both \(\mu _H\), \(\mu _K\) are surjective, and hence that \(\mu _K \otimes \mu _H \mapsto \mu _G\) is an isomorphism from \(D_K \otimes D_H\) to \(D_G\). \(\square \)
3 Nonnull elements in G
In this section we continue with k and G as in the previous section.
3.1 Definition of nonnullity of elements in G
The explicit description (2.2) of \(J_G\) shows that it is an ideal in A. We will refer to \(J_G\) as the nonnullity ideal. We denote by \(G^{\times } \hookrightarrow G\) the closed subscheme of G cut out by the ideal \(J_G\). Observe that \(G^{\times }={{\mathrm{Spec}}}(A/J_G)\) is locally free of rank \(G1\) over S.
For every kalgebra R, \(G^{\times }(R)\) is a subset of G(R). We say that an element \(g \in G(R)\) is nonnull when it lies in the subset \(G^{\times }(R)\). In the next subsections we will investigate this notion.
3.2 Nonnullity in the constant case
Start with a finite abelian group \(\Gamma \) and use it to build a constant group scheme G / S. Then A is the algebra of kvalued functions \(f:\Gamma \rightarrow k\), the ring structure being pointwise multiplication of functions. Then \(I_G=\{ f \in A : f(e_\Gamma ) =0 \}\) and \(J_G=\{ f \in A : f(\gamma ) =0 \quad \forall \, \gamma \ne e_\Gamma \}\). So \(A = J_G \oplus I_G\). In other words the scheme G decomposes as the disjoint union of two open (and closed) subschemes: \(G^{\times }\) and the identity section \(e_G(S)\). This example explains why we have chosen to call \(G^{\times }\) the closed subscheme of nonnull elements in G.
3.3 Testing nonnullity using an overring \(R' \supset R\)
An Rvalued point of G is given by a kalgebra homomorphism \(g:A \rightarrow R\). The element \(g \in G(R)\) is nonnull if and only if the ring homomorphism \(g:A \rightarrow R\) is 0 on the ideal \(J_G\) in A. Consequently, if \(R \rightarrow R'\) is an injective kalgebra homomorphism, then \(g \in G(R)\) is nonnull if and only if its image in \(G(R')\) is nonnull.
If \(R'/R\) is faithfully flat, then \(R \rightarrow R'\) is injective. So the notion of nonnullity is fpqc local, and therefore continues to make sense for any base scheme (or even algebraic space) S.
3.4 Nonnullity in the étale case

the closed subscheme \(G^{\times }\) is also an open subscheme of G.

the identity section \(e_G : S \hookrightarrow G\) is an open and closed immersion, and

G decomposes as the disjoint unionof open subschemes.$$\begin{aligned} G = G^{\times } \, \coprod \, e_G(S) \end{aligned}$$
3.5 Behavior under base change
3.6 Nonnullity for Oort–Tate groups
Let k be a \(\Lambda \)algebra (e.g., a \(\mathbb Z_p\)algebra). Then, given suitable \(a,b \in k\), Oort–Tate construct a group \(G_{a,b}\) of order p over k, but we will fix a, b and just call the group G. The corresponding kalgebra is \(A = k[x]/(x^pax)\), and its augmentation ideal is generated by x. So the ideal \(J_G\) consists of all elements in A that are annihilated by x, and a short computation reveals that \(J_G\) is the ksubmodule of A generated by \(x^{p1}a\). This shows that an element \(g \in G(k)\) is nonnull in our sense if and only if g is a generator of G in the sense of Haines–Rapoport [5] (this notion of generator was first used by Deligne–Rapoport in [2, section V.2.6, p. 106]). Moreover, as Haines–Rapoport show [5, Remark 3.3.2], this is also equivalent to g having “exact order p” in the sense of Drinfeld–Katz–Mazur. This agreement suggests that the notion of nonnullity is a natural one.
Example 3.1
The above discussion applies to the group \(\mu _p\) of pth roots of unity. The result is that a section \(\zeta \in \mu _p(k)\) lies in \(\mu _p^{\times }\) if and only if \(\Phi _p(\zeta )=0\) (where \(\Phi _p(T) = 1 + T + \cdots + T^{p1}\) is the cyclotomic polynomial). In other words, \(\mu _p^\times \) is the subscheme of primitive pth roots of unity.
3.7 Nonnullity for Raynaud groups
Raynaud groups are a natural generalization of Oort–Tate groups, and in this case, again, the notion of nonnullity agrees with a wellstudied notion. We thank G. Pappas for communicating this generalization to us.
Let \(q=p^n\) be a power of p and let D be the ring defined analogously to \(\Lambda \), but with q in place of p (see [10, section 1.1]), and let k be a Dalgebra. Given a suitable 2ntuple \((\delta _1,\ldots ,\delta _n,\gamma _1,\ldots ,\gamma _n) \in k^{2n}\), Raynaud, in [10, Collolaire 1.5.1], defines a group scheme G over k with \(G=q\) together with an action of \(\mathbb {F}_q\) on G—that is, G is an \(\mathbb {F}_q\)vector space scheme of dimension 1. The corresponding kalgebra is \(A=k[x_i]/(x_i^p\delta _{i}x_{i+1})\) where i ranges over \(\{1,\ldots ,n\}\) and \(x_{n+1}:=x_1\). Then the augmentation ideal is generated by \((x_1,\ldots ,x_n)\), and using [4, Proposition 2.1], for example, one can see that \(J_{G}\) is the ksubmodule of A generated by \((x_1\cdots x_n)^{p1}  \delta _1 \cdots \delta _n\). By [9, Proposition 5.1.5], \(G^\times \) is the scheme of “\(\mathbb {F}_q\)generators of G”, in the sense of Katz–Mazur.
3.8 Products
Consider groups \(G_1\), \(G_2\) over k. The corresponding kalgebras, augmentation ideals, and nonnullity ideals will be denoted \(A_i\), \(I_i\), \(J_i\) (for \(i = 1,2\)). The ring of regular functions for the group \(G=G_1 \times G_2\) is \(A = A_1 \otimes A_2\), and its augmentation ideal \(I_G\) is \((I_1 \otimes A_2) + (A_1 \otimes I_2)\). Therefore the ideal \(J_G\) in A annihilated by I is the intersection of the ideal annihilated by \(I_1\), namely \(J_1 \otimes A_2\), and the one annihilated by \(I_2\), namely \(A_1 \otimes J_2\). (It follows that \( J_G= J_1 \otimes J_2 , \) and so \(J_G\) is also the product of the ideals \(J_1 \otimes A_2\) and \(A_1 \otimes J_2\).)
In more geometrical language, we just verified that \(G^\times \) is the “union” of the closed subschemes \(G_1 \times G_2^\times \) and \(G_1^\times \times G_2\) of G (i.e. it is the smallest closed subscheme containing the two given closed subschemes).
Some care is required in this situation. Consider an Rvalued point \(g=(g_1,g_2)\) of G. If \(g_1\) is nonnull or \(g_2\) is nonnull, then g is nonnull. However, the converse is false, as is illustrated by the next example (when considering points with values in a ring that is not an integral domain).
Example 3.2
Let \((x,y) \in \mu _p \times \mu _p\). Then (x, y) is nonnull if and only if \(\Phi _p(x)\Phi _p(y)\) vanishes. So, for this group, nonnullity coincides with the notion of primitivity introduced in [12].
3.9 Extensions
Lemma 3.3

The closed subscheme \(H^{\times } \hookrightarrow H \hookrightarrow G\),

The closed subscheme \(\pi ^{1}(K^\times )\) of G obtained from \(K^{\times } \hookrightarrow K\) by base change along \(\pi : G \twoheadrightarrow K\).
Proof
We begin with the first item. The first item is true if and only if there exists an arrow j making the square (3.2) commute. This is the condition that \(i^*(J_A) \subset J_B\). That this condition holds follows from (3.1), which shows that \(i^*(J_A)\) is the product of the ideals \(J_B\) and \((i^*)(J_C)\) in B.
The first item can be strengthened to the statement that the square (3.2) is cartesian if and only if the inclusion \(i^*(J_A) \subset J_B\) is an equality. This is certainly the case when \(i^*(J_C)\) is the unit ideal in B.
When K is étale over S, we have seen that \(C = J_C \oplus I_C\). Therefore there exists \(f \in J_C\) such that \(1f \in I_C\). The image of f under \(i^*\) is equal to 1, showing that \(i^*(J_C)\) is indeed the unit ideal in B.
Finally, the second item is true if and only if the ideal \(AJ_C\) contains the ideal \(J_A\). The truth of this is obvious from (3.1). \(\square \)
Remark 3.4
Let \(h \in H(R)\). The lemma implies that, if h is nonnull for H, then it is nonnull for G. It also implies that the converse is true provided that K is étale over S. In general the converse is false. For example, \((1,y) \in \mu _p(R) \times \mu _p(R)\) is nonnull if and only if \(p\Phi _p(y)=0\), while \(y \in \mu _p(R)\) is nonnull if and only if \(\Phi _p(y)=0\). These are equivalent conditions when p is invertible in R, but not in general.
4 Primitivity of points in truncated pdivisible groups
In this section we fix a prime number p.
4.1 Definition of primitivity
Let R be a kalgebra, let x be an Rvalued point of \(\mathcal {G}_i\), and write \(\bar{x}\) for the image of x under the canonical homomorphism \(\mathcal {G}_i \twoheadrightarrow \mathcal {G}_1\) (raising to the power \(p^{i1}\)). We say that x is primitive if \(\bar{x}\) is nonnull in \(\mathcal {G}_1(R)\).

\(\mathcal {G}_i^{{{\mathrm{prim}}}} \rightarrow \mathcal {G}_i\) is a closed immersion, and

\(\mathcal {G}_i^{{{\mathrm{prim}}}} \rightarrow \mathcal {G}_1^{\times }\) is finite, locally free of rank \(p^{h(i1)}\).
4.2 Comparison with points of exact order N on elliptic curves
 Step 1 :

It is evident that (\(\star \)) holds when p is invertible on S, because \(E_N/S\) is then étale.
 Step 2 :

Next we check that (\(\star \)) holds for E / S whenever S is flat over \(\mathbb Z\). In this situation \(E_N\), \(E_N^{{{\mathrm{prim}}}}\), and \(E_N^\sharp \) are flat over \(\mathbb Z\). Here we used that \(E_N/S\) is flat (standard), that \(E_N^{{{\mathrm{prim}}}}/S\) is flat (see Sect. 4.1), and that \(E_N^\sharp /S\) is flat (see [6, Thm. 5.1.1]). By Step 1 the closed subschemes \(E_N^{{{\mathrm{prim}}}}\) and \(E_N^\sharp \) of \(E_N\) coincide over the locus in S where p is invertible, so the flatness of \(E_N\), \(E_N^{{{\mathrm{prim}}}}\) and \(E_N^\sharp \) over \(\mathbb Z\) forces \(E_N^{{{\mathrm{prim}}}}\) to coincide with \(E_N^{\sharp }\) over all of S.
 Step 3 :

Let \(\mathcal E\) denote the moduli stack (over \(\mathbb Z\)) of elliptic curves, and choose a presentation (see [7]) \(f : \mathcal M \twoheadrightarrow \mathcal E\) for it. Here f is étale and surjective, and \(\mathcal M\) is a smooth scheme of finite type over \(\mathbb Z\). Pulling back the universal elliptic curve on \(\mathcal E\), we obtain an elliptic curve \(\mathbf E\) on the scheme \(\mathcal M\). In the the terminology of [6], \(\mathbf E/\mathcal M\) is a “modular family.”
Now consider an elliptic curve E over an arbitrary base scheme S. We consider the product \(\mathcal M \times S\) and write \(p_1\), \(p_2\) for the two projections. We then have two elliptic curves over \(\mathcal M \times S\), namely \(p_1^*\mathbf E\) and \(p_2^*E\), and we form the \(\mathcal M \times S\)scheme T of isomorphisms between \(p_1^*\mathbf E\) and \(p_2^*E\). Over T the elliptic curves \(\mathbf E\) and E become tautologically isomorphic; the resulting elliptic curve on T will be denoted \(\tilde{E}\). At this point we have a commutative diagram in which both squares are cartesian. The two arrows in the bottom row exhibit T as the fiber product of \(\mathcal M\) and S over \(\mathcal E\), so \(T \rightarrow S\) is étale and surjective.Now \(\mathcal M\) is flat over \(\mathbb Z\), so (\(\star \)) holds for \(\mathbf E/\mathcal M\). It follows that (\(\star \)) holds for \(\tilde{E}/T\). Here we used that the operations of forming \(E_N^{{{\mathrm{prim}}}}\) and \(E_N^\sharp \) both commute with base change (use Sect. 3.5 and the first chapter of [6], especially their Corollary 1.3.7). So \(E_N^{{{\mathrm{prim}}}}\) and \(E_N^\sharp \) become equal after the étale surjective base change \(T \rightarrow S\). By descent theory \(E_N^{{{\mathrm{prim}}}}\) and \(E_N^\sharp \) are themselves equal.
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Acknowledgements
PW was supported by the National Science Foundation under the Mathematical Sciences Postdoctoral Research Fellowship No. 1606255. We thank G. Boxer, B. Levin and K. Madapusi Pera for interesting conversations about integral models. We are grateful to T. Haines, G. Pappas, and M. Rapoport for helpful comments on a preliminary version of this paper. We thank the referees for comments and suggestions.
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