Central European Journal of Mathematics

CEJM 2(2) 2004 177–190 Hochschild Cohomology of skew group rings and invariants ∗ E.N. Marcos1† , R. Mart´ınez-Villa2‡ ...

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CEJM 2(2) 2004 177–190

Hochschild Cohomology of skew group rings and invariants ∗ E.N. Marcos1† , R. Mart´ınez-Villa2‡ , Ma.I.R. Martins1§ 1

Departamento de Matem´ atica - IME, Universidade de S˜ ao Paulo, C. Postal 66281, CEP 05315- 970, S˜ao Paulo, SP, Brasil 2 Instituto de Matematicas, UNAM- Campus Morelia, Apartado Postal 61-3, CP 58089, Morelia, Michoac´an, Mexico

Received 26 December 2003; accepted 18 March 2004 Abstract: Let A be a k-algebra and G be a group acting on A. We show that G also acts on the Hochschild cohomology algebra HH • (A) and that there is a monomorphism of rings HH • (A)G → HH • (A[G]). That allows us to show the existence of a monomorphism from G into HH • (A), where A is a Galois covering with group G. HH • (A) c Central European Science Journals. All rights reserved.

Keywords: Hochschild cohomology, skew group ring, Galois covering MSC (2000): 16E40, 16D20, 16S37

1

Introduction

The Hochschild cohomology groups were introduced by Hochschild ﬁfty years ago, but they have been investigated lately under diﬀerent aspects by many authors. In this work we are interested in studying the Hochschild cohomology of skew group rings and of certain Koszul algebras. For this let k be a ﬁeld and A be a ﬁnite-dimensional algebra over k, and we refer to it simply as a k-algebra. Our main purpose is comparing ∗

This work was done during the visit of the second author at the Universidade de S˜ ao Paulo on December 2000 and it is part of the work supported by a interchange grant from CNPq (Brasil) and CONACyT(Mexico). The first and the second authors thank respectively CNPq and CONACyT. † E-mail: [email protected] ‡ E-mail: [email protected] § E-mail: [email protected]

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the Hochschild cohomology algebra HH • (A) = ⊕i≥0 ExtiAe (A, A) of a k-algebra A with the Hochschild cohomology algebra HH • (A[G]) of the skew group algebra A[G], with G being a ﬁnite group acting on A. We obtain the following relation between these cohomology algebras. Theorem 2.8 Let A be a k-algebra and G be a finite group acting on A. Then G acts on the Hochschild cohomology k-algebra HH • (A), and there is a ring monomorphism: HH • (A)G → HH • (A[G]). For ﬁnite groups, it is known that there is a strong connection between skew group rings and Galois coverings, and between skew group rings and smash products of graded algebras (see [5]). These facts and the existence of the monomorphism in the Theorem above lead us to investigate a possible relation between the Hochschild cohomology algebras of G-graded algebras and their covering algebras deﬁned by G. In this direction, let A be a G-graded algebra, with G being a ﬁnite group. We consider the covering algebra of A deﬁned by G and we relate it with the smash product A#k[G]∗ . Then, as a consequence of Theorem 2.8 and using a theorem of duality coactions, we obtain that G is a and the existence of a ring monomorphism from HH • (A) G group of automorphisms of A into HH • (A). In this work we also study the Hochschild cohomology groups of Koszul algebras of ﬁnite global dimension. With this we reach the Hochschild cohomology groups of Cpreprojective algebras associated to Euclidean diagrams and the Auslander algebra of standard algebras, since these algebras are examples of Koszul algebras of global dimension two. We obtain a lower bound for the dimension of HH n (A), for A be a Koszul algebra of global dimension n. So as a consequence we get that the second Hochschild cohomology group of a C-preprojective algebra of Euclidean type does not vanish; and it is also true for the Auslander algebra of standard algebras having non projective indecomposable modules isomorphic to their own Auslander-Reiten translate. So, in both these cases the algebras are not rigid (see [9]). We now describe the contents of each section in the paper. In section 2, after recalling some notions and known facts needed for the work, we state and prove the main result of the section - Theorem 2.8. associated to a G-graded algebra A, In section 3 we deﬁne the covering algebra A where G is ﬁnite group. We also recall the notion of smash product, and we show that this product is isomorphic to the covering algebra. This isomorphism together with Theorem 2.8 and duality coactions gives us a similar relationship between the invariants and the Hochschild cohomology ring of A. of Hochschild cohomology ring HH • (A) In section 4 we deal with quadratic algebras. We construct the Koszul complex for quotients of path algebras by quadratic ideals through a similar procedure used by Berger in [2]. The Koszul complex (named bimodule Koszul complex by him) was constructed in [2] for quotients of free associative algebras A and it is a minimal graded resolution of A as an A − A bimodule, in case A is a generalized Koszul algebra (also called d-Koszul algebras). But it can be also constructed for algebras which are quotients of quiver algebras by ideals generated by elements of degree d ≥ 2 (see [11]). Our construction

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here follows closely the one in [11] for Koszul algebras (that is, 2-Koszul). It enables us to obtain a lower bound for the dimension of the n-Hochschild cohomology group of Koszul algebras of global dimension n, and in consequence the property mentioned above for preprojective algebras of Euclidean-type and the Auslander algebra of a standard algebra. While we were reviewing the ﬁnal version of this article we received from C. Cibils and M. J. Redondo a pre-print entitled Cartan-Leray spectral sequence for Galois coverings of categories, [4]. In this pre-print they gave a spectral sequence involving the Hochschild cohomology of an algebra and of a Galois covering and they show that the monomorphism obtained in our Proposition 3.5 is an isomorphism when the characteristic of the ﬁeld is zero.

2

Hochschild cohomology rings and invariants

Given a ring A we denote by Aop its opposite ring. For a ∈ A we denote by ao ∈ Aop the corresponding element in Aop . In case that A is an algebra over a ﬁeld k we will denote by Ae its enveloping algebra A ⊗k Aop . Moreover, if A and B are algebras over k, the algebra A ⊗k B will be denoted simply by A ⊗ B. Sometimes, for simplicity, we will not make explicit the ground ring of tensor product when it is clear in the context. We also remark that the category of left modules over the algebra Ae is canonically isomorphic to the category of A − A bimodules. So we use this isomorphism as identiﬁcation. Now we recall some deﬁnitions and basic facts. Definition 2.1. Let A be a ring and G a group. We say that G acts on A if there is a group homomorphism between G and the group Aut(A) of ring automorphisms of A. If this group homomorphism is injective, we say that G acts faithfully on A or that G is a group of automorphisms of A. We remark that if G acts on A, then G naturally also acts on the opposite ring Aop . In case that A is a k-algebra, we will assume that the group Aut(A) is the group of automorphisms of k-algebras. Moreover, if G and H are groups acting on the k-algebras A and B, respectively, then the group G × H acts on A ⊗k B, and consequently G × G acts on A ⊗ Aop . Now we are going to recall the deﬁnition of skew group algebra. Definition 2.2. Let A be a ring and G a ﬁnite group acting on A. The elements of the skew group ring A[G] are the same as those of the corresponding group ring. Addition is as usual coordinate-wise, and multiplication is extended by bilinearity from the formula (ag)(bh) = ag(b)gh, for a and b in A and g and h in G. The following statements will be useful later and their veriﬁcation are routine.

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Proposition 2.3. Let A be a k-algebra and G be a ﬁnite group acting on A. Then, i) The rings (A[G])op and Aop [G] are isomorphic, via the map o −1 o −1 θ((ag) ) = (g (a)) g . ii) A[G] ⊗k (A[G])op is isomorphic to (A ⊗k Aop )[G × G], via the map ψ(ag ⊗ (bh)o ) = a ⊗ (h−1 (b))o (g, h−1 ). Now we are going to describe certain approach on the category of the left A-modules and the category of the left A[G]-modules, where G is a ﬁnite group acting on the algebra A. This approach will be very useful for the next sections. Denoting by Mod-A the category of the left A-modules, for each g ∈ G, we can associate a functor, denoted by g (), on Mod-A. This functor associates to each M in Mod-A the module g M deﬁned as follows: g M = M as an abelian group (or k-vector space, in case A is a k-algebra) and for a ∈ A and m ∈ M , a ·g m := g(a)m. On morphisms, g () is deﬁned as the identity. Observe that the functor g () is clearly an exact functor and is an automorphism of Mod-A. We also observe that it is possible to deﬁne in a similar way an automorphism ()g on the category A-Mod of the right Amodules. Furthermore, analogously we could consider a functor g () on the category of A − A bimodules, by considering g M = M as an A − A bimodule where it has on the left the structure as above and on the right the original structure of MA (also in a similar way we could have M g as a A − A bimodule). The following facts can be veriﬁed easily. Proposition 2.4. Let A be a ring and G a group acting on A. If g and h are in G and M is a left A-module, then: i) g (h M ) = hgM ; ii) (M g )h = M gh ; −1 iii) Ag ∼ = Ag ∼ = g A (as A − A bimodules). Now we recall some basic facts related to A[G]-modules (see [16]) Proposition 2.5. Let A be a ring and G a ﬁnite group acting on A. If M is in ModA[G], then the map g Ψ : M →gM given by g Ψ(m) = g(m) deﬁnes an isomorphism of A-modules. Proof. Clearly g Ψ preserves the sum. Let a ∈ A and m ∈ M . So, we have g Ψ(am) = g(am) = g(a)g(m) = a ·g g(m) = a ·g g Ψ(m), which shows that g Ψ is a homomorphism of −1 A-modules. A similar veriﬁcation shows that the map m → g (m) is the A-morphism inverse of g Ψ. Proposition 2.6. (Lemma 4 in [16]) Let A be a k-algebra and G a ﬁnite group acting on A. Let M and N be A[G]-modules. Then the following statements hold: i) The abelian group HomA (M, N ) is a kG – module, with the action (g ∗ f )(m) = g(f (g −1 (m)), for any g ∈ G and f ∈ HomA (M, N ). Denote by HomA (M, N )G the set

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of ﬁxed points, then HomA (M, N )G ∼ = HomA[G] (M, N ); ii) For all i ≥ 1, there is a natural action of G on ExtiA (M, N ) and it veriﬁes that j ExtiA[G] (M, N ) ∼ = ExtiA (M, N )G . If g ∈ G, ξ ∈ ExtiA (M, N ) and η ∈ ExtA (M, N ), then g(ξη) = g(ξ)g(η). Proof. i) It is easy and well known (see for instance [16]). −1 ii) Let g ∈ G. Since g () is an exact functor, then a given element ξ ∈ ExtiA (M, N ) is −1 −1 −1 taken to an element g ξ ∈ ExtiA (g M, g N ). But, since M and N are A[G]-modules, by −1 using the isomorphism g Ψ and its inverse (see Proposition 2.5) we get an exact sequence, denoted by g(ξ), which is an element in ExtiA (M, N ). We note that if two exact sequences are representives of the same element in ExtiA (M, N ), then their correspondents under g(−) have the same property. Hence it indicates how to deﬁne the action. Proposition 2.4 guarantees that it really deﬁnes an action of G on ExtiA (M, N ). −1 The rest of the proof follows from the fact that the functor g () also preserves the A-projective modules and we leave the details to the reader. Note: We note that statement ii) in the last proposition could also be proved by remarking that A[G] is a projective A-module, and applying the functor HomA (−, N ) to the projective resolution of M as an A[G]-module, and observing that the homology obtained at each step is a kG-module. Now we are going to recall the deﬁnition of Hochschild cohomology groups. Let A be a k-algebra and A XA be an A-bimodule. We consider the enveloping algebra Ae = A⊗k Aop . The ith - Hochschild cohomology group of A with coeﬃcients in X, denoted by HH i (A, X), is the group ExtiAe (A, X), for each i ≥ 0. But our particular interest is the example X = A, whose HH i (A, A) = ExtiAe (A, A) is simply denoted by HH i (A), for i ≥ 0. These groups are used to deﬁne the Hochschild cohomology algebra HH • (A) = ⊕i≥0 HH i (A) = ⊕i≥0 ExtiAe (A, A) with the multiplication induced by the Yoneda product. In this way HH • (A) is a Z-graded algebra (see for instance [1, 13, 20]). The facts which we state next point up how important they are to establish the main result of this section: to relate the Hochschild cohomology algebras HH • (A) and HH • (A[G]). Let A be a k-algebra and G be a ﬁnite group acting on A. We have seen, in Proposition 2.3, that the enveloping algebra A[G]e of the skew group algebra A[G] is isomorphic to the k-algebra Ae [G × G]. We shall use this isomorphism together with Proposition 2.6 to describe an action of G × G on the Hochschild cohomology algebra HH • (A[G]), which is compatible with its Z-grading. With this in mind, we recall that A[G] is a Ae [G × G]-module, with the follow ing “ multiplications ”: (x ⊗ y o )( g∈G ag g) = g∈G xag g(y)g, and (σ, τ )( g∈G ag g) = −1 , for x, y, ag in A, and σ, τ in G. Then from Proposition 2.6, for g∈G σ(ag )σgτ each i ≥ 0, it follows that G × G acts on ExtiAe (A[G], A[G]) and ExtiA[G]e (A[G], A[G]) ∼ = G×G i (ExtAe (A[G], A[G]) . So, from the action of G × G on the grading we get the one

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wanted on HH • (A[G]). For obtaining the relationship between the Hochschild cohomology algebras of A and of A[G], we describe now a bit more in detail the action considered above (the one considered in Proposition 2.6) just for the cases i = 0 and i = 1, since for i ≥ 2 the procedure is analogous to the one for i = 1. First we write A[G] = Ag, as an Ae - module (or A − A bimodule), and so we get g∈G i i HH (A[G]) = ExtA[G]e (A[G], A[G]) ∼ ExtiAe (Ag, Ah), for each i ≥ 0. = (g,h)∈G×G

1. The action in case i = 0 For each g, h in G, let f(g,h) ∈ HomAe (Ag, Ah) and (σ, τ ) ∈ G × G. Then we consider the element (σ, τ )(f(g,h) ) ∈ HomAe (Aσgτ −1 , Aσhτ −1 ) deﬁned by (σ, τ )(f(g,h) )(aσgτ −1 ) = aσf (g)τ −1 , with a ∈ A. Clearly it deﬁnes an action on HomAe (A[G], A[G]) = HomAe (Ag, Ah) and it (g,h)∈G×G

is the action considered in Proposition 2.6.i). 2. The action in case i = 1. Let ξ(g,h) : 0 → Ah → L → Ag → 0 be a representative of an element in Ext1Ae (Ag, Ah) (σ−1 , τ−1 )

and (σ, τ ) ∈ G × G. By applying the exact functor () to this exact sequence (σ−1 , τ−1 ) Ah → Aσhτ −1 we obtain the element, denoted by and using the isomorphism 1 (σ, τ )(ξ(g,h) ) ∈ ExtAe (Aσgτ −1 , Aσhτ −1 ). It is easy to verify that it really deﬁnes an action on Ext1Ae (Ag, Ah), and it is the action mentioned in Proposition 2.6.ii). (g,h)∈G×G

Remark 2.7. We observe, in particular, that the subspace

ExtiAe (Ag, Ag) of

g∈G

under the action of G × G is taken to itself. On the other hand we also note that, for each i ≥ 0 and g ∈ G, ExtiAe (A, A) is canonically isomorphic to ExtiAe (Ag, Ag) (see 2.4.iii). So, with this identiﬁcation, we can consider an action of G on HH i (A) = ExtiAe (A, A) given by the following: for each g ∈ G and ξ ∈ HH i (A), g · ξ := (g, g)(ξ). Consequently, we obtain that G acts on HH • (A). ExtiAe (A[G], A[G])

Now we can show the main result of this section which gives a relation between the Hochschild cohomology algebras of A and A[G]. Theorem 2.8. Let A be a k-algebra and G be a ﬁnite group acting on A. Then G acts on the Hochschild cohomology k-algebra HH • (A), and there is a ring monomorphism: HH • (A)G → HH • (A[G]). Proof. First we write A[G] = Ag.

Then we remark again that the action of G ×

g∈G

G on HH i (A[G]) (i ≥ 0) enables having HH i (A[G]) ∼ = = (ExtiAe (A[G], A[G]))G×G ∼ G×G i i ( ExtAe (Ag, Ah)) . So, it suggests identifying an element ξ ∈ ExtAe (A[G], A[G]) (g,h)∈G×G

with a matrix ξ = (ξ(g,h) )g,h , with ξ(g,h) ∈ ExtiAe (Ag, Ah).

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Therefore using remark 2.7, we deﬁne a morphism: for each i ≥ 0, given ξ ∈ HH i (A)G we take the element θi (ξ) ∈ ExtiAe (Ag, Ah) whose matrix representation θi = (g,h)∈G×G

i

(θ (ξ)(g,h) )g,h is such that:

i

θ (ξ)(g,h) =

0 if g = h ξ if g = h.

Since ξ ∈ (HH i (A))G , then, by construction, θi (ξ) is invariant under the action of G × G, and therefore the map θi : HH i (A)G → HH i (A[G]) is deﬁned; and it is not diﬃcult to verify that θ : HH • (A)G → HH • (A[G]), with θ = ⊕i θi is a monomorphism of rings.

3

Galois covering and Hochschild cohomology.

In this section we are going to apply the main theorem of the last section (Theorem 2.8) G into HH • (A), where A to show that there is also a ring monomorphism from HH • (A) is the covering algebra of a G-graded k-algebra A. In [5] it was proven that, for a k-algebra A graded by a ﬁnite group G, the smash product A#k[G]∗ plays the role for graded rings that the skew group algebra A[G] plays for group actions. So, in order to obtain the new monomorphism we shall use the notion of smash product A#k[G]∗ , for showing the existence of an isomorphism between this product and the covering algebra of A deﬁned by G and we apply the duality Theorem 3.5 in [5]. We recall here the deﬁnition of covering algebra associated to a graded algebra. This deﬁnition was introduced in a preliminary version of [12], and it can be found in [15]. The deﬁnition of covering algebra coincides with the one given by Green in [10] for quotients of path algebras of quivers. Definition 3.1. Let G be a ﬁnite group and A =

Ag be a G-graded k-algebra, with

g∈G

Ag indicating the k-subspace of the homogeneous elements of degree g. The covering is deﬁned as k-algebra associated to A, with respect to the given grading, denoted by A, follows. As k- vector space = ˜ h], where A[g, h] = Ag−1 h , and the multiplication is deﬁned in the A A[g, (g,h)∈G×G

, h ]. The product is in A[g, h ] and it is deﬁned h] and γ ∈ A[g following way: if γ ∈ A[g, by: 0 if g = h γγ = γγ if g = h. where the action of an element σ ∈ G Remark 3.2. We observe that G acts freely on A, h] to the same element, but now considered as an element in takes an element in A[g, → A which takes A[σg, σh]. Moreover the canonical vector space epimorphism F : A h] to Ag−1 h , is such that F σ = F , for all σ ∈ G, and the orbit space is A. So A is a A[g, Galois covering deﬁned by G in the sense of Gabriel and others,([3, 17]).

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Now we review the deﬁnition of smash product and some facts related to it (see [5]). Definition 3.3. Let G be a ﬁnite group and A =

Ag be a G-graded algebra. Let k[G]∗

g∈G

be the dual algebra of k[G], and its natural k-basis {pg |g ∈ G}; that is , for any g ∈ G and x = h∈G ah h ∈ k[G], pg (x) = ag ∈ k, and pg ph = δg,h ph , where δg,h is the Kronecker delta. The smash product, denoted by A#k[G]∗ , is the vector space A ⊗k k[G]∗ with the multiplication given by (a#pg )(b#ph ) = abgh−1 #ph (here a#pg denotes the element a ⊗ pg ). The next proposition was ﬁrst proved by Green, Marcos and Solberg in a preliminary version of [12]. Proposition 3.4. Let G a ﬁnite group and A = g∈G Ag be a G-graded algebra. Then of A are isomorphic algebras. the smash product A#k[G]∗ and the covering algebra A Proof. Any a ∈ A can be written uniquely as a = h∈G ah , where ah ∈ Ah . So we can by deﬁne the following map Ψ : A#k[G]∗ −→ A ˜ −1 h−1 , g −1 ]. ah )#pg ) = ah ∈ Ψ(( A[g h∈G

h∈G

h∈G

It is not hard to show that Ψ is a bijective homomorphism of algebras.

In Remark 3.2, we have seen the group of grading of A acts on the covering algebra A. Now we also note that using the isomorphism in Proposition 3.4 we get a corresponding action on the smash product A#k[G]∗ , which is given by g(a#ph ) = a#phg ; and it coincides with the one deﬁned in Lemma 3.3 in [5]. With these remarks, as a consequence of Theorem 2.8, of the isomorphism above and the duality coactions of Cohen-Montgomery (Th.3.5 in [5]) we obtain the following proposition. be the Proposition 3.5. Let G be a ﬁnite group and A be a G-graded k-algebra. Let A • covering algebra deﬁned by the grading. Then G acts on HH (A) and there is a ring G into HH • (A). monomorphism from (HH • (A)) as a group of automorphisms. Proof. As we have seen, in Remark 3.2, G acts on A and Then, on the one hand, from Theorem 2.8 it follows that G also acts on HH • (A) ˜ G to HH • ((A)[G]). there is a monomorphism from from HH • (A) But, on the other hand, ∗ ˜ by Proposition 3.4, A and A#k[G] are isomorphic, and according to our remark above this isomorphism leads G to act on the smash product A#k[G]∗ . So, by applying the duality theorem for coactions (Th. 3.5 in [5]), we get that (A#k[G]∗ )[G] is isomorphic to the matrix ring M|G| (A) where |G| denotes the order of the group G. Since Hochschild cohomology is an invariant by Morita equivalence (in reality is an invariant of derived equivalence, [19, 20]), then it follows that HH • (A[G]) is isomorphic to HH • (A), and the proposition is proved.

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4

185

The Hochschild cohomology of Koszul algebras

In this section we discuss some facts related to the Hochschild cohomology of Koszul algebras. In particular we consider preprojective algebras of Euclidean-type and Auslander algebras of standard algebras, which are Koszul algebras. In order to study the Hochschild cohomology of Koszul algebras of ﬁnite global dimension we introduce the construction of Koszul complex for quadratic algebras. We are using a similar procedure as was done in [2] and [11] for generalized Koszul algebras (or d- Koszul algebras). According to our comments in the introduction of this article, we use it for 2-Koszul algebra or simply Koszul algebras. So we review some deﬁnitions and ﬁx some notations. Let k be a ﬁeld and let Q be a ﬁnite quiver. We denote by kQ the path algebra of Q and we indicate by kQ0 the k-subalgebra whose underlying vector space is the subspace generated by the vertex set Q0 of Q. If Qi is the set of paths of length i, then we denote by kQi the subspace of kQ generated by Qi . It is worth noting that this subspace is a kQ0 bimodule. In this way, we will consider the path algebra kQ = ⊕i≥0 kQi as a graded algebra with the grading given by the length of the paths. Let A = kQ/I where I is a two side ideal of kQ generated by a set of quadratic relations (the k-algebra A is called a quadratic algebra). Denote by R the set of homogeneous elements of degree two in I, so R is viewed as a kQ0 sub-bimodule contained in kQ2 . For each n ≥ 2, we deﬁne Kn = r+s+2=n kQr .R.kQs . Now we consider the following Ae -modules: Qi = 0, for i < 0, Q1 = A ⊗kQ0 kQ1 ⊗kQ0 A,

and

Q0 = A ⊗kQ0 A, Qi = A ⊗kQ0 Ki ⊗kQ0 A for i ≥ 2.

It is clear that each Qi is a projective ﬁnitely generated Ae -module. Moreover, for each i ≥ 2, we have that Qi is a submodule of A ⊗kQ0 kQi ⊗kQ0 A, since Ki is contained in kQi . Now we construct, for each i ≥ 2, the following Ae -morphism fi : A ⊗kQ0 kQi ⊗kQ0 A → A ⊗kQ0 kQi−1 ⊗kQ0 A given by the formula: fi (a ⊗ α1 · · · αi ⊗ b) = aα1 ⊗ α2 · · · αi ⊗ b + (−1)i a ⊗ α1 · · · αi−1 ⊗ αi b. So in this way we get a family of morphisms (di )i , with di : Qi → Qi−1 deﬁned by di = 0, if i ≤ 0, d1 (1 ⊗kQ0 α ⊗kQ0 1) = α ⊗kQ0 1 − 1 ⊗kQ0 α, for each α ∈ Q1 , and di is the restriction of fi to Qi , if i ≥ 2. It is very easy to see that di ◦ di+1 = 0, for each i. Therefore we have a complex ((Qi )i , (di )i ) of ﬁnitely generated projective Ae -modules, and we have the following deﬁnition. Definition 4.1. Let A = kQ/I be a quadratic k-algebra. Let Qi be the Ae -modules and di ∈ HomAe (Qi , Qi−1 ) as above. The complex K ∗ (A) = ((Qi )i , (di )i ) is called the Koszul complex of A.

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With the deﬁnition of Koszul complex on hand, we shall utilize it for characterizing the Koszul algebras. In order to get it we take the augmented Koszul complex of A: dn−1

d

d

d

n · · · → Qn → Qn−1 → · · · → Q1 →1 Q0 →0 A → 0 where dn , n > 0, is as in K ∗ (A) and d0 (a ⊗kQ0 b) = ab

Theorem 4.2. [2, 11] Let A = kQ/I be a quadratic algebra. The augmented Koszul d

dn−1

d

d

n complex · · · → Qn → Qn−1 → · · · → Q1 →1 Q0 →0 A → 0 is exact if and only if A is a Koszul algebra.

In case A = kQ/I is a Koszul algebra, the augmented Koszul complex of A is a minimal graded projective resolution of A as Ae -module. Furthermore, if I is an admissible ideal, then this resolution is also a minimal projective resolution of A in Ae -mod. The augmented Koszul complex can be used to determine a lower bound for dimension of HH n (A), when A is a Koszul algebra of global dimension n. It is obtained in the corollary below. Corollary 4.3. Let A = kQ/I be a Koszul algebra of global dimension n. For each vertex v ∈ Q0 , ev denotes the associated idempotent of A. Then dimk (HH n (A)) ≥ dimk (

(ev Kn ev )).

v∈Q0

Proof. Since A is a Koszul algebra and has global dimension n, by Theorem 4.2 we have, using the notations ﬁxed above, that the long exact sequence d

d

d

d

n 0 → Qn → Qn−1 · · · → Q2 →2 Q1 →1 Q0 →o A → 0 is a graded minimal projective resolution of A in Ae -mod. Then the Hochschild cohomology of A can be computed as the cohomology of the complex:

d∗

0 → HomAe (A ⊗A0 A, A) →1 HomAe (A ⊗A0 kQ1 ⊗A0 A, A) → · · · → d∗

n HomAe (A ⊗A0 Kn−1 ⊗A0 A, A) → HomAe (A ⊗A0 Kn ⊗A0 A, A) → 0 · · ·

where A0 = kQ0 . On the other hand, it is easy to verify that HomAe (A ⊗A0 A, A) ∼ = HomAe0 (A0 , A) ∼ ∼ ev Aev , HomAe (A⊗A0 kQ1 ⊗A0 A, A) = HomAe0 (kQ1 , A), and, for j ≥ 2, HomAe (A⊗A0 = v∈Q0

Kj ⊗A0 A, A) ∼ = HomAe0 (Kj , A). Hence the last complex is isomorphic to the following one: d∗ d∗n 0 → v∈Q0 ev Aev →1 HomAe0 (kQ1 , A) · · · HomAe0 (Kn−1 , A) → HomAe0 (Kn , A) → 0, ∗ where we also are denoting by di the maps induced by the isomorphisms mentioned above. We observe that the vector space HomAe0 (Kj , A), with j ≥ 2, can be naturally graded by the induced grading of A; that is, we say that a map f is homogeneous of degree t if its image is contained in homogeneous component of degree t of A. So, it easy to see that the last complex is a complex of graded vector spaces and that the image of each d∗j is contained in the direct sum of the homogeneous subspaces of degree bigger than

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187

zero. Then, for j ≥ 2, the image of d∗j does not intersect the degree zero component (HomAe0 (Kj , A))0 ∼ ev ) of HomAe0 (Kj , A). In particular, the degree zero = v∈Q0 (ev Kj component (HomAe0 (Kn , A))0 ∼ = v∈Q0 (ev Kn ev ) does not intersect the image of d∗n and since HH n (A) = Coker d∗n , a simple computation of dimensions shows our statement. Among the Koszul algebras we are going to point out the C-preprojective algebras of Euclidean type and the Auslander algebra of a standard, representation ﬁnite-type kalgebra. We will see that these algebras are Koszul algebras and as consequence of it, via Corollary 4.3, we get interesting datum for HH 2 (A). First let us review the deﬁnition of preprojective algebras. ˆ whose Definition 4.4. Let Q be a ﬁnite quiver and k a ﬁeld. Consider the quiver Q op op ˆ 0 = Q0 and the arrows set Qˆ1 = Q1 ∪ Q1 , where Q denotes the opposite vertex set Q

quiver of Q. For each arrow α ∈ Q1 we write α ˆ for the corresponding arrow in the opposite quiver. The preprojective k-algebra associated to Q (or brieﬂy the preprojective ˆ k-algebra of Q), denoted by P(Q), is the k-algebra k Q/I, where I is the ideal generated by the relations αα ˆ and α ˆ α. α∈Q1

α∈Q1

We remark that it is well-known that the preprojective algebra constructed as above only depends on the underlying graph of the quiver Q; that is, quivers having the same underlying graph deﬁne isomorphic preprojective algebras. The Hochschild cohomology of preprojective algebras associated to Dynkin diagrams An were studied in [7, 8]. We mention the following result about preprojective C-algebras associated to Euclidean diagrams (see [6, 14, 18]). Theorem 4.5. The preprojective C-algebras associated to an Euclidean diagram are Morita equivalent to the skew group algebras C[x, y][G], with G a polyhedral group. So this theorem can be used in order to study some properties of the preprojective C-algebra of Euclidean-type through properties of certain skew group rings. For instance, from this theorem we obtain that a preprojective C-algebra associated to Euclidean diagrams has global dimension 2 (recall that gldim(C[x, y][G]) = gldim(C[x, y]) = 2). Moreover, since the preprojective algebras are always quadratic algebras, then we also get that preprojective C-algebras of Euclidean-type are Koszul algebras. In this point of view we obtain as a consequence of Corollary 4.3, the following fact about the second Hochschild cohomology group of preprojective C-algebras of Euclideantype. Corollary 4.6. Let A be a preprojective C-algebra associated to an Euclidean diagram. Then HH 2 (A) = 0 Proof.

ˆ We have that A = CQ/I where Q is an Euclidean diagram. As we have

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commented above A is a Koszul algebra of global dimension two. Since ev K2 ev is not zero, for any vertex v, the statement follows from the corollary 4.3. We note that Theorem 4.5 can be used this to describe the centre of preprojective C-algebras of Euclidean-type, once their centres are the ones of the skew group rings C[x, y][G], for suitable groups G. Then it seems interesting to study the Hochschild cohomology of the skew group ring, in particular its centre. In order to study it, the following lemma will be useful. Lemma 4.7. Let R be a commutative ring and G be a group acting on R. Let g be an element in G and suppose that there is α ∈ R such that α − g(α) is not a zero divisor in R. Then HomRe (R, Rg) = 0. Proof. Let f ∈ HomRe (R, Rg). Then αf (1) = f (1).α = f (1)g(α) = g(α)f (1), and it implies that (α − g(α))f (1) = 0. Since α − g(α) is not a zero divisor in R, it follows that f (1) = 0, and consequently f = 0. Now we be able to describe the center of a skew group ring. Proposition 4.8. Let R be a commutative domain and G be a ﬁnite group of automorphisms of R. Then Center(R[G]) is isomorphic to RG . Proof. First we observe that HomRe (Rg, Rh) ∼ = HomRe (R, Rhg −1 ), for any g, h in G. So, since G acts faithfully on R and R is a domain, by Lemma 4.7 we obtain that HomRe (Rg, Rh) = 0, for any g = h in G. Now, if we write R[G] = Rg as Re -module, then we have that Center(R[G]) = g∈G

HH 0 (R[G]) = HomR[G]e (R[G], R[G]) = (HomRe (R[G], R[G]))G×G ( HomRe (Rg, Rh))G×G ∼ HomRe (Rg, Rg))G×G . =( (g,h)∈G×G

=

g∈G

Recalling the action deﬁned in section 2 and Remark 2.7 we have that ( HomRe (Rg, Rg))G×G ∼ = (HomRe (R, R))G ∼ = RG , and the statement is proved. g∈G

We remark that the proof of Proposition 4.8 could be obtained by a direct computation, but we have opted for the proof above to illustrate how to use our methods. For the next corollary we need some additional terminology. We are going to consider the Auslander algebra associated to a k-algebra of representation ﬁnite type. So, we recall the deﬁnition of Auslander algebras. Let A be a k-algebra of representation ﬁnite type (i.e. up to isomorphism there exist only ﬁnitely many indecomposable ﬁnitely generated A-modules). Let X1 , X2 , · · · Xm be a list of representatives from the isomorphism classes of indecomposable ﬁnirtely generated A-modules and let X = ⊕i Xi . The k-algebra Λ = EndA (X) is called Auslander algebra of A. Recall that A is said to be standard if Λ is isomorphic to the quotient of the path

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189

algebra kΓA of the Auslander-Reiten quiver ΓA of A by the ideal generated by the mesh relations. We denote by mod-A the category of ﬁnitely generated left A-modules, by ind A the subcategory of mod-A with one representative of each isoclass of indecomposable A-module and by τA the Auslander-Reiten translate DTr. Corollary 4.9. Let A be a standard representation-ﬁnite type k-algebra and Λ be its Auslander algebra. Then dim HH 2 (Λ) ≥ #{M ∈ ind A : τA M = M } Proof. It is known that the Auslander algebra Λ of a representation-ﬁnite algebra A has global dimension two. Moreover, since A is standard, we have that Λ ∼ = kΓA /I, where ΓA is the Auslander-Reiten quiver of A and I is the ideal generated by the mesh relations (so quadratic relations). Hence Γ is a Koszul algebra. By construction of the quiver ΓA and by the conditions on I, it is clear that the number of elements of the set {M ∈ ind A : τA M ) = M } is the dimension of the degree zero component of Homk(ΓA )e0 (K2 , Λ). But that component is isomorphic to v∈(ΓA )0 ev K2 ev , and therefore the result follows from corollary 4.3

References [1] I. Assem: Alg`ebres et modules. Enseignement des Math´ematiques, Les Presses de l’Universit´e, d’Ottawa, Masson, 1997. [2] R. Berger: Koszulity for nonquadratic algebras, J. Algebra, Vol. 239, (2001), pp. 705–734. [3] K. Bongartz and P. Gabriel: “Covering spaces in Representation-Theory“, Invent. Math, Vol. 65, (1982), pp. 331–378. [4] C. Cibils and M.J. Redondo: Cartan-Leray spectral sequence for Galois coverings of categories, preprint. [5] M. Cohen and S. Montgomery: “Group-graded rings, smash products, and group actions“, Trans. Amer. Math. Soc., Vol, 279, No. 1, (1984), pp. 237–258. [6] W. Crawley-Boevey and M.P. Holland: “Noncommutative deformations of Kleinian singularities“, Duke Math. J., Vol. 92, (1998), pp. 605–635. [7] K. Erdmann and N. Snashall: “On Hochschild cohomology of preprojective algebras II“, J. Algebra, Vol. 205, (1998), pp. 413–434. [8] K. Erdmann and N. Snashall: “Preprojective algebras of Dynkin type, periodicity and the second Hochschild cohomology“, In: CMS Conf. Proc.: Algebras and Modules, II. Geiranger, 1998, Vol. 24, AMS, 1998, pp. 183–193. [9] M. Gerstenhaber: “On the deformation of rings of algebras“, Ann. Math, Vol. 79, (1964), pp. 59–103. [10] E. Green: “Graphs with relations, coverings and group-graded algebras“, Trans. Amer. Math. Soc., Vol. 279, No. 1, (1983), pp. 297–310. [11] E. Green, R. Martinez-Villa, E.N. Marcos and P. Zhang: “D-Koszul algebras“, to appear in J. Pure and Appl. Algebra.

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[12] E. Green, E.N. Marcos and O . Solberg: “Representation and almost split sequences of Hopf algebras“,In: CMS Conf. Proceedings: Representation theory of algebras,Cocoyoc, Mexico, August 22-26, 1994, Vol. 18, AMS, Providence, 1996. [13] D. Happel: “Hochschild cohomology of ﬁnite-dimensional algebras“, In: S´eminaire d’Alg`ebre P. Dubreil et M.-P. Malliavin, 39`eme Ann´ee, Lecture Notes in Math, Vol. 1404, Springer-Verlag, 1989, pp. 108–126 [14] H. Lenzing: “Curve singularities arizing from the representation theory of tame hereditary algebras“, In: Rep. Theory I. Otawa, 1984, Lecture Notes in Math, Vol. 1177, Springer-Verlag, 1986, pp. 199–231. [15] E.N. Marcos: Singularidades, M´odulos sobre a´lgebras de Artin e a´lgebras de Hopf, Tese de Livre-Docencia, IME-USP, S˜ao Paulo, Brasil, 1996. [16] R. Martinez-Villa: “Skew group algebras and their Yoneda algebras“, Mathematical Journal of Okayama Univ., Vol. 43, (2001), pp. 1–16. [17] R. Mart´ınez-Villa and J.A. de la Pe˜ na: “The universal cover of a quiver with relations“, J.Pure Appl. Algebra, Vol. 30, (1983), pp. 277–292. [18] I. Reiten and M. Van Den Bergh: “Two-dimensional tame and maximal orders of ﬁnite representation type“, Mem. Amer. Math. Soc., Vol. 80, (1989). [19] J. Rickard: “Morita theory for derived category‘1‘, J. London Math. Soc., Vol. 39, (1989), pp. 436–456. [20] C.A. Weibel: An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, Vol. 38, Cambridge University Press, Cambridge, 1994.

CEJM 2(2) 2004 191–198

A numerical solution of a two-dimensional transport equation. Olga Martin

∗

Department of Mathematics, University “Politehnica” of Bucharest, Splaiul Independentei 313, Bucharest 16, Romania

Received 22 September 2003; revised 25 March 2004; accepted 5 April 2004 Abstract: In this paper we present a variational method for approximating solutions of the Dirichlet problem for the neutron transport equation in the stationary case. Error estimates from numerical examples are used to evaluate an approximation of the solution with respect to the steps of two grids. c Central European Science Journals. All rights reserved. Keywords: transport equation, variational calculus, diﬀerence scheme, Euler-Lagrange equation MSC (2000): 35J99, 65N99

1

Introduction

In a reactor, neutrons are produced by the ﬁssion of a nucleus and are called rapid neutrons if their average speeds are 2·107 m/s. Rapid neutrons are subjected to a slowness process, decreasing their energy until they are in a state of equilibrium with the other atoms in the environment. When the reactor is in a stationary state, the atoms have a tendency to move from a region with a high density to another with a low density, thus obtaining a uniform density. This process is called diﬀusion. The main problem in nuclear reactor theory is to ﬁnd the distribution of neutrons in a reactor, hence to ﬁnd the reactor’s density. The result is a scalar function that depends on the following variables: the position vector of the neutron in a datum coordinate system, the neutron speed and the time. The neutron density is the solution of an integral-diﬀerential equation called the neutron transport equation. Many authors treat this problem and its applications [1-3, 4-6]. ∗

E-mail: omartin [email protected]

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O. Martin / Central European Journal of Mathematics 2(2) 2004 191–198

In this paper we provide a solution for the two-dimensional stationary problem, where the solution u is the neutron density and where f from (1) is the source function. The neutron moves in a direction that makes an angle α with the Ox axis and an angle β with the Oy axis. We let µ=cosα and ν = cosβ. In order to ﬁnd the solution u of the Dirichlet problem for a transport equation, we use the variational calculus. Let ∆1 be a square grid with step k for D1 = [-1, 1]×[-1, 1]. For every value (µi , νj ) ∈ ∆1 , we deﬁne u by an interpolation polynomial that both satisﬁes the boundary conditions and minimizes a given functional J(u). The approximate solutions of numerical examples are compared with the accurate solution, thereby producing an estimation of the error.

2

Problem Formulation

Let us now consider the integral-diﬀerential equation of the transport theory: µ ∂u (x, y, µ, ν) + ν ∂u (x, y, µ, ν) + u(x, y, µ, ν) = ∂x ∂y u(x, y, µ , ν )dµ dν + f (x, y, µ, ν) =

(1)

D1

with (µ, ν) ∈ D1 = [ - 1, 1] × [ - 1, 1] , and u(x, y, µ, ν) |∂D2 = 0,

(x, y) ∈ D2 = [0, 1] × [0, 1]

∂D2 − the boundary of D2 (2)

(2)

where u(x, y, µ, ν) is the density of neutrons and where f is the source function. Let u be continuous on D1 (u ∈ C(D1 )) and have continuous second derivatives on D2 (u ∈ C 2 (D2 )). In order to solve the problem (1)–(2) using variational methods, we replace the problem with the equation obtained by addition of the partial derivatives of (1) with respect to the variables x and y, multiplied by µ and ν, respectively 2

2

2

∂ u µ2 ∂∂xu2 + 2νµ ∂x∂y + ν 2 ∂∂yu2 + µ ∂u + ν ∂u = ∂x ∂y ∂u ∂u dµ dν + ν dµ dν + µ ∂f + ν ∂f =µ ∂x ∂y ∂x ∂y D1

(3)

D1

or µ2

2 ∂2u ∂2u 2∂ u + 2νµ − u = F (x, y, µ, ν) + ν ∂x2 ∂x∂y ∂ y2

(4)

where F (x, y, µ, ν) = µ − D1

∂u (x, y, µ , ν )dµ dν + ν ∂x

D1

u(x, y, µ , ν ) dµ dν + µ

∂u (x, y, µ , ν )dµ dν − ∂y

D1

∂f ∂f (x, y, µ, ν) + ν (x, y, µ, ν) − f (x, y, µ, ν) ∂x ∂y

(5)

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193

in accordance with (1). Next, we present a variational method using the Ritz-Galerkin method. Let us deﬁne an approximate formulation of our problem (4)–(2). For this, we consider a square grid ∆1 for D1 with step k = 1/4, ∆1 = { (µi , νj ) ∈ D1 |{ i ∈ {0, 1, ..., 8}, j ∈ {0, 1, ..., 8}}, and a square grid with step h, ∆2 = { (xi , yj ) ∈ D2 | i ∈ {0, 1, ..., N + 1}, j ∈ {0, 1, ..., N + 1}}. Then, as shown in Fig. 1, each square of the h side from D2 is split into two isosceles triangles (N = 3). Consider the approximate solution of equation (4) given by u (x, y) =

N

αmn (µ, ν)ωmn (x, y)

(6)

m,n=1

where ηm , ξn − constants.

αmn (µ, ν) = µν + ηm µ + ξn ν,

(7)

Dklh

x2=y

2

3 1

4 Pk l 6

5

w 1

x2 Pkl

0

x1

0

x1=x Fig. 1

Fig. 2

From (5) and (7), we obtain: ∂ f (x, y, µ, ν) ∂ f (x, y, µ, ν) +ν · − f (x, y, µ, ν) (8) ∂x ∂y In order to obtain a Ritz-Galerkin approximation for the solution of equation (4), we minimize the functional: F (x, y, µ, ν) = µ ·

µ

J(u) =

2

∂u ∂x

2

∂u ∂u + 2µν · + ν2 ∂x ∂y

∂u ∂y

2

2

+ u (x, y, µ, ν) dxdy+2

D2

F u dxdy D2

(9) over the space of functions of the following form: uij (x, y) =

N

αmn (µi , νj )ωmn (x, y)

(10)

m,n=1

where the ωmn (x, y) are the pyramidal functions that correspond to the N hexagonal regions, Dmn , each of which is centered at the point Pmn of the network ∆2 . Each hexagonal sub-domain is the union of six triangles, {Dmn,,j }, j ∈{1,2,. . . ,6}, the numbering of

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O. Martin / Central European Journal of Mathematics 2(2) 2004 191–198

which is shown in Fig. 1. The geometric interpretation of the function ωmn is a pyramid with a height equal to unity and with projection P of its apex (Fig. 2). The function ωmn can be written in the following form:   h  1 − h1 (xm − x) − h1 (yn − y), if (x, y) ∈ Dnm,1      h  1 - h1 (xm − x) if (x, y) ∈ Dnm,2       h  1 + h1 (yn − y), if (x, y) ∈ Dnm,3    h ωmn (x, y) = 1 + h1 (xm − x) + h1 (yn − y), if (x, y) ∈ Dnm,4     h   1 + h1 (xm − x), if (x, y) ∈ Dnm,5      h  1 − h1 (yn − y), if (x, y) ∈ Dnm,6       0, in rest.

(11)

In order to minimize the functional J(u), we ﬁnd coeﬃcients αmn as solutions of the system of equations given by ∂J(uh ) = 0, ∂αij

i, j ∈ {1, 2, · · · , N }

(12)

or, in matrix form, given by A·α=g

(13)

where the elements of matrix A form a band along the main diagonal. The elements of A are N ∂ωkl ∂ωmn aN (k−1)+l,N (m−1)+n = Cst + ωkl ωmn dxdy , k, l, m, n = 1, 2, · · · , N ∂xs ∂xt D2

s,t=1

(14) where x1 = x, x2 = y and the coeﬃcients Cst are given by equation (11), expressed in the form 2 ∂2u Cst − u = F. (15) ∂ x ∂ x s t s,t=1 The vector g = (g1 , g2 , · · · , gN 2 ) with gN (k−1)+l = gkl , k, l, ∈ {1, 2, ..., N } has gkl = − F ωkl dxdy, k, l = 1, 2, · · · , N

(16)

h Dkl

but α = (α1 , α2 , · · · , αN 2 )t ( t – transpose matrix) is the unknown vector with αN (k−1)+l = αkl ,

k, l, ∈ {1, 2, ..., N } .

(17)

O. Martin / Central European Journal of Mathematics 2(2) 2004 191–198

3

195

Numerical results

Let us consider the transport equation µ

∂u ∂u (x, y, µ, ν) + ν (x, y, µ, ν) + u(x, y, µ, ν) = ∂x ∂y u(x, y, µ , ν )dµ dν + f (x, y, µ, ν) =

(18)

D1

with the conditions from (2) and f (x, y, µ, ν) = µν(µπ cos πx sin πy + νπ sin πx cos πy + sin πx sin πy).

(19)

For the domain D1 = [-1,1] × [-1,1] we use the network ∆1 with the step k = 1/4 and for D2 = [0,1] × [0,1] we have h = 1/3 (Fig. 3). y(x2)

(0,1)

(0,2/3)

(0,1/3)

0

12

22

11

21

(1/3,0)

(2/3,0)

(1,0)

x (x1)

Fig. 3

We begin the construction of an approximate solution by considering the sum (N = 2) u(x, y, µ, ν) = α11 (µ, ν)ω11 (x, y) + α12 (µ, ν)ω12 (x, y)+

(20)

+α21 (µ, ν)ω21 (x, y) + α22 (µ, ν)ω22 (x, y) with α11 (µ, ν) = η1 µ + ξ1 ν + µν,

α21 (µ, ν) = η2 µ + ξ1 ν + µν,

α12 (µ, ν) = η1 µ + ξ2 ν + µν,

α22 (µ, ν) = η2 µ + ξ2 ν + µν.

(21)

In accordance with (8), we obtain F (x, y, µi , νj ) = −µi νj (π 2 (µ2i + νj2 ) + 1) sin πx sin πy − 2π 2 µ2i νj2 cos πx cos πy. We ﬁnd A from (14). For example, when k = 1, l= 1, m= 1 and n =2, we have

(22)

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O. Martin / Central European Journal of Mathematics 2(2) 2004 191–198

a12 =

1/3

dx

0

+

2/3

2

11 ∂ω12 Cst ∂ω ∂xs ∂xt

s,t=1 2/3−x 2 2/3 1−x

dx

1/3

1/3

s,t=1

− ω11 ω12

11 ∂ω12 Cst ∂ω ∂xs ∂xt

dxdy+

− ω11 ω12

dxdy

corresponding to D11,3 , D11,4 , D12,1 , andD12,6 . Thus, we determine the matrices A and g of equation (13) in the form 

 2(µ2i + µi νi + νi2 ) + 0.054 −(µi νi + νi2 ) + 0.01

  −(µ ν + ν 2 ) + 0.01  ii i A=  −(µ2i + µi νi ) + 0.01 

2(µ2i + µi νi + νi2 ) + 0.054 µi νi + 0.01

0



−(µ2i + µi νi ) + 0.01

0 −(µ2i + µi νi ) + 0.01

µi νi + 0.01

2(µ2i + µi νi + νi2 ) + 0.54 −(µi νi + νi2 ) + 0.01

−(µ2i + µi νi ) + 0.01

−(µi νi + νi2 ) + 0.01

     

2(µ2i + µi νi + νi2 ) + 0.54



g  11     g12    g= ,    g21    g22

with

gij = −

F (x, y, µi , νj ) ωij dxdy

(23)

D2

h and, because ωij =0 only for (x, y) ∈ Dij , we obtain F (x, y, µi , νj )ωij dxdy. gij = −

(24)

Dij

In the following tables, we present the values of u(xk , yl , µi , νj ) where (xk , yl ) corresponds to the nodes of (11). From (10) and (11), it follows that u(xk , yl , µi , νj ) = αkl .

(25)

The approximate solution u is then compared with the exact solution u¯(x, y, µ, ν) = µν sin πx sin πy.

O. Martin / Central European Journal of Mathematics 2(2) 2004 191–198

µ =-3/4

µ =-1/4

µ = 1 /2

ν = -3/4 ν = -1/2 ν = -1/4 ν = 1/4 ν = 1/2 ν = 3/3

ν = -3/4 ν = -1/2 ν = -1/4 ν = 1/4 ν = 1/2 ν = 3/3

ν = -3/4 ν= -1/2 ν = -1/4 ν = 1/4 ν = 1/2 ν= 3/3

U

u ¯

-0.458 -0.312 -0.164 0.194 0.41 0.623

-0.563 -0.375 -0.188 0.188 0.37 0.563

U

u ¯

-0.48 -0.329 -0.174 0.214 0.406 0.58

-0.563 -0.375 -0.188 0.188 0.375 0.563

U

u ¯

-0.614 -0.417 -0.203 0.164 0.313 0.47

-0.56 -0.375 -0.188 0.188 0.375 0.563

µ =-1/2

µ = 1/4

µ = 3/4

ν = -3/4 ν = -1/2 ν = -1/4 ν =1/4 ν = 1/2 ν = 3/3

ν = -3/4 ν = -1/2 ν = -1/4 ν = 1/4 ν = 1/2 ν = 3/3

ν = -3/4 ν= -1/2 ν = -1/4 ν = 1/4 1/2 3/3

197

U

u ¯

-0.47 -0.406 -0.164 0.203 0.417 0.614

-0.563 -0.375 -0.188 0.188 0.375 0.563

U

u ¯

-0.58 0.407 -0.214 0.171 0.329 0.48

-0.563 -0.375 -0.188 0.188 0.375 0.563

U

u ¯

-0.623 -0.41 -0.194 0.164 0.312 0.458

-0.563 -0.375 -0.188 0.188 0.375 0.563

Table 1

4

Conclusions

In solving equation (13) we ﬁnd angle α for the values (µi ,ν j ) ∈ ∆1 , hence we ﬁnd u(xk , yl , µi , νj ), where (xk , yl ) ∈ ∆2 . Note that all functions of the form (10) are not generalized solutions of (4), but are Ritz–Galerkin type approximations of the generalized solutions. As we have demonstrated in [4], an approximate solution u of the elliptic equation (4) with u ∈ C 2 (D2 ) veriﬁes the following inequality: u − u¯ < C1 h

(26)

where u¯ is the exact solution and h is the step of a square grid deﬁned over the domain D2 . In analyzing both the exact and numerical solutions, we ﬁnd that C1 = 5k where k is

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the step of a square grid deﬁned over the domain D1 and, hence, that the approximation of the solution with respect to steps h and k has the form u − u¯ ≤ 5hk.

(27)

References [1] K.M. Case and P.F. Zweifel: Massachusetts, 1967.

Linear Transport Theory,

Addison-Wesley,

[2] W.R. Davis: Classical Fields, Particles and the Theory of Relativity, Gordon and Breach, New York, 1970. [3] S. Glasstone and C. Kilton: The Elements of Nuclear Reactors Theory, Van Nostrand, Toronto – New York – London, 1982. ´ [4] G. Marchouk: M´ethodes de calcul num´erique, Edition MIR de Moscou, 1980. [5] G. Marchouk and V. Shaydourov: Raﬃnementdes solutions des sch´emas aux ´ diﬀ´erences, Edition MIR de Moscou, 1983. [6] G. Marciuk and V. Lebedev: Cislennie metodˆı v teorii perenosa neitronov, Atomizdat, Moscova, 1971. [7] O. Martin: “Une m´ethode de r´esolution de l’´equation du transfert des neutrons”, Rev.Roum.Sci.Tech.-M´ec.Appl., Vol. 37, (1992), pp. 623–646. [8] N. Mihailescu: “Oscillations in the power distribution in a reactor”, Rev. Nuclear Energy, Vol. 9, No. 1-4, (1998), pp. 37–41. [9] H. Pilkuhn: Relativistic Particle Physics, Springer Verlag, New York-HeidelbergBerlin, 1980.

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Pseudo-M V algebra of fractions and maximal pseudo-M V algebra of quotients Dana Piciu∗ Department of Mathematics, University of Craiova, 13, Al.I. Cuza st., 200585, Craiova, Romania

Received 5 March 2003; accepted 16 December 2003 Abstract: The aim of this paper is to define the notions of pseudo-M V algebra of fractions and maximal pseudo-M V algebra of quotients for a pseudo-M V algebra (taking as a guide-line the elegant construction of complete ring of quotients by partial morphisms introduced by G. Findlay and J. Lambek - see [14], p.36 ). For some informal explanations of the notion of fraction see [14], p. 37. In the last part of this paper the existence of the maximal pseudo-M V algebra of quotients for a pseudo-M V algebra (Theorem 4.5) is proven and I give explicit descriptions of this M V -algebra for some classes of pseudo M V -algebras. c Central European Science Journals. All rights reserved. Keywords: pseudo-M V algebra, multiplier, pseudo-M V algebra of fractions, maximal pseudoM V algebra of quotients MSC (2000): 06D35, 03G25

1

Introduction

The concept of maximal lattice of quotients for a distributive lattice was deﬁned by J.Schmid in [17]-[18] taking as a guide-line the construction of complete ring of quotients by partial morphisms introduced by G. Findlay and J. Lambek (see [14], p.36). For the case of Hilbert algebras, Heyting algebras and M V -algebras see [1], [10] respectively [5]. The central role in this construction is played by the concept of multiplier (deﬁned for a distributive lattice by W. H. Cornish in [8]-[9]). The paper is organized as follows. In Section 2 we recall the basic deﬁnitions and put in evidence many rules of calculus in pseudo-M V algebras which we need in the rest of paper. ∗

E-mail: [email protected]

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In Section 3 we deﬁne the notion of multiplier for a pseudo-M V algebra; also we put in evidence many results which we need in the rest of the paper (especially in section 3). In Section 4 we deﬁne the notions of pseudo-M V algebra of fractions and maximal pseudo-M V algebra of quotients for a pseudo-M V algebra. In the last part of this paper the existence of the maximal pseudo-M V algebra of quotients for a pseudo-M V algebra (Theorem 4.5) is proven and I give explicit descriptions of this M V -algebra for some classes of pseudo M V -algebras (local pseudo M V -algebras, pseudo M V -chains and Boolean algebras).

2

Definitions and preliminaries

We consider an algebra (A, +,− ,∼ , 0, 1) of type (2, 1, 1, 0, 0). We put by deﬁnition: y · x = (x− + y − )∼, and we consider that the operation · has priority to the operation +. Definition 2.1. ([12]) A pseudo-M V algebra is an algebra (2, 1, 1, 0, 0) satisfying the following equations: (a1 ) x + (y + z) = (x + y) + z, (a2 ) x + 0 = 0 + x = x, (a3 ) x + 1 = 1 + x = 1, (a4 ) 1∼ = 0, 1− = 0, (a5 ) (x− + y − )∼ = (x∼ + y ∼)− , (a6 ) x + x∼ · y = y + y ∼ · x = x · y − + y = y · x− + x, (a7 ) x · (x− + y) = (x + y ∼) · y, (a8 ) (x− )∼ = x.

(A, +,− ,∼ , 0, 1) of type

Pseudo-M V algebras were originally introduced by G. Georgescu and A. Iorgulescu in [12] and are a non-commutative generalization of M V − algebras and they can be taken as an algebraic semantics for a non-commutative generalization of a multiple valued reasoning. A non-commutative generalization of reasoning can be found, e.g., in psychological processes, see [11]: In clinical medicine on behalf of transplantation of human organs, an experiment was performed in which the same two questions have been posed to two groups of interviewed people: (1) Do you agree to dedicate your organs for medical transplantation after your death? (2) Do you agree to accept organs of a donor in the case of your need? In the order of questions was changed in the second group, the number of positive answers here was more higher than in the ﬁrst group. Following tradition, we denote a pseudo-M V algebra (A, +,− ,∼ , 0, 1) by its universe A. If A ⊆ A we write A ≤ A to indicate that A is a pseudo- M V subalgebra of A.

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Examples: E1 ) A singleton {0} is a trivial example of a pseudo-M V algebra; a pseudo-M V algebra is said to be nontrivial provided its universe has more that one element. E2 ) Let us consider an arbitrary l-group (G, ⊕, −, 0, ≤). For each u ∈ G, u > 0, let [0, u] = {x ∈ G : 0 ≤ x ≤ u} and for each x, y ∈ [0, u], let x+y = u ∧ (x ⊕ y), x− = u − x and x∼ = −x ⊕ u. Then ([0, u], +,− ,∼ , 0, u) is a pseudo-M V algebra. In [11] Dvureˇcenskij proved that every pseudo-M V algebra is isomorphic with an interval in an l -group. E3 ) Clearly, every M V − algebra is a pseudo-M V algebra, where the unary operations − ∼ , coincide. Every commutative pseudo-M V algebra (i.e. + is commutative) is an M V − algebra ([15], Proposition 1.29). For other classes of pseudo-M V algebras (local, archimedean) see [11] and [15]. Lemma 2.2. ([12])For x, y ∈ A, the following conditions are equivalent: (i) x− + y = 1, (ii) y ∼ · x = 0, (iii) y = x + x∼ · y, (iv) x = x · (x− + y), (v) There is an element z ∈ A such that x + z = y, (vi) x · y − = 0, (vii) y + x∼ = 1. For any two elements x, y ∈ A let us agree to write x ≤ y iﬀ x and y satisfy the equivalent conditions (i) − (vii) in the above lemma. So, ( [12], Proposition 1.10), ≤ is a partial order relation on A (which is called the natural order on A). Theorem 2.3. ([12]) If x, y, z ∈ A then the following hold: c1 ) y · x = (x∼ + y ∼)− , c2 ) x + y = (y − · x− )∼ = (y ∼ · x∼)− , c3 ) (x∼)− = x, c4 ) 0∼ = 0− = 1, c5 ) x · 1 = 1 · x = x, x · 0 = 0 · x = 0, c6 ) x + x∼ = 1, x− + x = 1, c7 ) x · x− = 0, x∼ · x = 0, c8 ) (x + y)− = y − · x− , (x + y)∼ = y ∼ · x∼, c9 ) (x · y)− = y − + x− , (x · y)∼ = y ∼ + x∼, c10 ) x∼ · y + y ∼ = y ∼ · x + x∼, c11 ) x · (x− + y) = y · (y − + x), c12 ) x · (y · z) = (x · y) · z, c13 ) x + x = x iﬀ x · x = x,

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c14 ) c15 ) c16 ) c17 ) c18 ) c19 ) c20 ) c21 ) c22 )

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x ≤ y iﬀ y − ≤ x− iﬀ y ∼ ≤ x∼, If x ≤ y, then x + z ≤ y + z and z + x ≤ z + y, If x ≤ y, then x · z ≤ y · z and z · x ≤ z · y, x · y ≤ z iﬀ y ≤ x− + z iﬀ x ≤ z + y ∼, x · y ≤ x, x · y ≤ y, x ≤ x + y, y ≤ x + y, If y + x = z + x and x · y = x · z then y = z, If x + y = x + z and y · x = z · x then y = z, x + y = y iﬀ x− + y − = x− , x + y = x iﬀ x∼ + y ∼ = y ∼.

Remark 2.4. If (A, +,− ,∼ , 0, 1) is a pseudo-M V algebra then (A, ·,∼ ,− , 1, 0) is also a pseudo-M V algebra. Remark 2.5. ([12]) On A, the natural order determines a bounded distributive lattice structure. Speciﬁcally, the join x ∨ y and the meet x ∧ y of the elements x and y are given by: x ∨ y = x + x∼ · y = y + y ∼ · x = x · y − + y = y · x− + x, x ∧ y = x · (x− + y) = y · (y − + x) = (x + y ∼) · y = (y + x∼) · x. Clearly, x · y ≤ x ∧ y ≤ x, y ≤ x ∨ y ≤ x + y. We shall denote this distributive lattice with 0 and 1 by L(A) (see [6]-[7]). For any pseudo-M V algebra A we shall write B(A) as an abbreviation of set of all complemented elements of L(A). Elements of B(A) are called the boolean elements of A. Theorem 2.6. ([12])For every element x in a pseudo-M V algebra A, the following conditions are equivalent: (i) x ∈ B(A), (ii) x ∨ x− = 1, (iii) x ∨ x∼ = 1, (iv) x ∧ x− = 0, (v) x ∧ x∼ = 0, (vi) x + x = x, (vii) x · x = x, (viii) x + y = x ∨ y = y + x, for all y ∈ A, (ix) x · y = x ∧ y = y · x, for all y ∈ A. Corollary 2.7. ([12]) (i) B(A) is subalgebra of the pseudo-M V algebra A. A subalgebra B of A is a boolean algebra iﬀ B ⊆ B(A), (ii) A pseudo-M V algebra A is a boolean algebra iﬀ the operation + is idempotent, i.e., the equation x + x = x is satisﬁed in A.

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Theorem 2.8. ([12]) If x, y, z, (xi )i∈I are elements of A, then if all suprema and inﬁma exist, the following hold: c23 ) x + xi = i∈I (x + xi ), i∈I c24 ) xi + x = i∈I (xi + x), i∈I xi = i∈I (x + xi ), c25 ) x + i∈I c26 ) xi + x = i∈I (xi + x), i∈I c27 ) x · xi = i∈I (x · xi ), i∈I c28 ) xi · x = i∈I (xi · x), i∈I c29 ) x · xi = i∈I (x · xi ), i∈I xi · x = i∈I (xi · x), c30 ) i∈I = c31 ) x ∧ x (x ∧ xi ), i i∈I i∈I c32 ) x ∨ i∈I xi = i∈I (x ∨ xi ). Theorem 2.9. ([12]) If x, y, z are elements of A, then the following hold: c33 ) (x ∧ y)− = x− ∨ y − , (x ∨ y)− = x− ∧ y − , c34 ) (x ∧ y)∼ = x∼ ∨ y ∼, (x ∨ y)∼ = x∼ ∧ y ∼, c35 ) x · y − ∧ y · x− = 0, x∼ · y ∧ y ∼ · x = 0, c36 ) (y + x∼) ∨ (x + y ∼) = 1, (y − + x) ∨ (x− + y) = 1, c37 ) x ∨ y = x · (x ∧ y)− + y, c38 ) x ∧ y = 0 ⇒ x + y = x ∨ y, c39 ) x ∧ y = 0 ⇒ x ∧ (y + z) = x ∧ z. Lemma 2.10. If a, b, x are elements of A, then: c40 ) [(a ∧ x) + (b ∧ x)] ∧ x = (a + b) ∧ x, c41 ) x ∧ a− ≥ x · (a ∧ x)− and a∼ ∧ x ≥ (a ∧ x)∼ · x . Proof. c40 ). By c23 and c24 we have [(a∧x)+(b∧x)]∧x = ((a∧x)+b)∧((a∧x)+x)∧x = ((a ∧ x) + b) ∧ x = (a + b) ∧ (x + b) ∧ x = (a + b) ∧ x. c29 c c41 ). We have x·(a∧x)− = x·(a− ∨x− ) = (x·a− )∨(x·x− ) =7 (x·a− )∨0 = x·a− ≤ x∧a− c30 c and (a ∧ x)∼ · x = (a∼ ∨ x∼) · x = (a∼ · x) ∨ (x∼ · x) =7 (a∼ · x) ∨ 0 = a∼ · x ≤ a∼ ∧ x. Corollary 2.11. If a ∈ B(A), then for all x, y ∈ A: c42 ) x ∧ a− = x · (a ∧ x)− and a∼ ∧ x = (a ∧ x)∼ · x, c43 ) a ∧ (x + y) = (a ∧ x) + (a ∧ y), c44 ) a ∨ (x + y) = (a ∨ x) + (a ∨ y). Proof. c42 ). See the proof of c41 . c24 c24 c43 ). We have: (a ∧ x) + (a ∧ y) = [(a ∧ x) + a] ∧ [(a ∧ x) + y] = [(a + a) ∧ (x + a)] ∧ [(a ∧ x) + y] = a ∧ (x + a) ∧ [(a + y) ∧ (x + y)] = a ∧ (a + y) ∧ (x + y) = a ∧ (x + y). c44 ). We have (a ∨ x) + (a ∨ y) = (a + x) + (a + y) = (a + a) + (x + y) = a + (x + y) = a ∨ (x + y). Definition 2.12. ([12]) Let A and B be pseudo-M V algebras. A function f : A → B

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is a morphism of pseudo-M V algebras if it satisﬁes the following conditions, for every x, y ∈ A : (a9 ) f (0) = 0, (a10 ) f (x + y) = f (x) + f (y), (a11 ) f (x− ) = (f (x))− , (a12 ) f (x∼) = (f (x))∼. If f is bijective then the morphism f is called an isomorphism of pseudo- M V algebras; in this case we write A ≈ B. Remark 2.13. It follows that: f (1) = 1, f (x · y) = f (x) · f (y), f (x ∨ y) = f (x) ∨ f (y), f (x ∧ y) = f (x) ∧ f (y), for every x, y ∈ A. Definition 2.14. ([12]) An ideal of a pseudo-M V algebra A is a subset I of A satisfying the following conditions: (a13 ) 0 ∈ I, (a14 ) If x ∈ I, y ∈ A and y ≤ x, then y ∈ I, (a15 ) If x, y ∈ I, then x+ y ∈ I. We denote by Id(A) the set of all ideals of A and by I(A) the set: I(A) = {I ⊆ A : if x, y ∈ A, x ≤ y and y ∈ I, then x ∈ I}. Remark 2.15. Clearly, Id(A) ⊆ I(A) and if I1 , I2 ∈ I(A), then I1 ∩ I2 ∈ I(A). Also, if I ∈ I(A) is a nonempty set, then 0 ∈ I. In general, a subset I of an ordered set (A, ≤) which verify only a14 is called ordered ideal (also known as down-set or decreasing set). Hence, if A is a pseudo M V − algebra, then I(A) is the set of all ordered ideals of A.

3

Multipliers on a pseudo-MV algebra

Definition 3.1. By a partial multiplier on A we mean a map f : I → A, where I ∈ I(A), which verify the following conditions: (a16 ) f (e · x) = e · f (x), for every e ∈ B(A) and x ∈ I, (a17 ) f (x) ≤ x, for every x ∈ I, (a18 ) If e ∈ I ∩ B(A), then f (e) ∈ B(A),

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(a19 ) x ∧ f (e) = e ∧ f (x), for every e ∈ I ∩ B(A) and x ∈ I (note that e · x ∈ I as e · x ≤ e ∧ x ≤ x). Remark 3.2. The condition a19 is not a consequence of a16 −a18 . As example, f : I → A, where I ∈ I(A), f (x) = x ∧ x− for every x ∈ I, verify a16 − a18 . Indeed, for x ∈ I and e ∈ B(A), we have f (e·x) = (e·x)∧(e·x)− = (e∧x)∧(e∧x)− = x∧[e∧(e∧x)− ] = x∧[e∧ (e− ∨x− )] = x∧[e·(e− ∨x− )] = x∧[e·(e− +x− )] = x∧(e∧x− ) = e∧(x∧x− ) = e∧f (x) = e · f (x). Clearly, f (x) ≤ x for every x ∈ I and for e ∈ I ∩ B(A), f (e) = e ∧ e− = 0 ∈ B(A). But if e ∈ I ∩ B(A) and x ∈ I, then x ∧ f (e) = x ∧ 0 = 0 = e ∧ (x ∧ x− ). By dom(f ) ∈ I(A) we denote the domain of f . If dom(f ) = A, we called f total. To simplify language, we will use multiplier instead of partial multiplier, using total to indicate that the domain of a certain multiplier is A. Examples 1. The map 0 : A → A deﬁned by 0(x) = 0, for every x ∈ A is a total multiplier on A; indeed if x ∈ A and e ∈ B(A), then 0(e · x) = 0 = e · 0 = e · 0(x) and 0(x) ≤ x. Clearly, if e ∈ A ∩ B(A) = B(A), then 0(e) = 0 ∈ B(A) and for x ∈ A, x ∧ 0(e) = e ∧ 0(x) = 0. 2. The map 1 : A → A deﬁned by 1(x) = x, for every x ∈ A is also a total multiplier on A; indeed if x ∈ A and e ∈ B(A), then 1(e · x) = e · x = e · 1(x) and 1(x) = x ≤ x. The conditions a18 − a19 are obviously veriﬁed. 3. For a ∈ B(A) and I ∈ I(A), the map fa : I → A deﬁned by fa (x) = a ∧ x, for every x ∈ I is a multiplier on A (called principal ). Indeed, for x ∈ I and e ∈ B(A), we have fa (e · x) = a ∧ (e · x) = a ∧ (e ∧ x) = e ∧ (a ∧ x) = e · (a ∧ x) = e · fa (x) and clearly fa (x) ≤ x. Also, if e ∈ I ∩ B(A), fa (e) = e ∧ a ∈ B(A) and x ∧ (a ∧ e) = e ∧ (a ∧ x), for every x ∈ I. Remark 3.3. In general, if we consider any a ∈ A, then fa : I → A veriﬁes only a16 , a17 and a19 but does not verify a18 . If dom(fa ) = A, we denote fa by fa ; clearly, f0 = 0. For I ∈ I(A), we denote M (I, A) = {f : I → A | f is a multiplier on A} and M (A) = ∪I∈I(A) M (I, A). If I1 , I2 ∈ I(A) and fi ∈ M (Ii , A), i = 1, 2, we deﬁne f1 ⊕ f2 : I1 ∩ I2 → A by (f1 ⊕ f2 )(x) = (f1 (x) + f2 (x)) ∧ x, for every x ∈ I1 ∩ I2 .

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Lemma 3.4. f1 ⊕ f2 ∈ M (I1 ∩ I2 , A). Proof. If x ∈ I1 ∩ I2 and e ∈ B(A), then (f1 ⊕ f2 )(e · x) = [f1 (e · x) + f2 (e · x)] ∧ (e · x) = c43 [(e · f1 (x)) + (e · f2 (x))] ∧ (e ∧ x) = [(e ∧ f1 (x)) + (e ∧ f2 (x))] ∧ (e ∧ x) = [e ∧ (f1 (x) + f2 (x))] ∧ (e ∧ x) = e ∧ [(f1 (x) + f2 (x)) ∧ x] = e · (f1 ⊕ f2 )(x). Clearly, (f1 ⊕ f2 )(x) ≤ x for every x ∈ I1 ∩ I2 and if e ∈ I1 ∩ I2 ∩ B(A) then (f1 ⊕ f2 )(e) = [f1 (e) + f2 (e)] ∧ e ∈ B(A). For e ∈ I1 ∩ I2 ∩ B(A) and x ∈ I1 ∩I2 we have: x ∧ (f1 ⊕ f2 )(e) = x ∧ [(f1 (e) + f2 (e)) ∧ e] = (f1 (e) + f2 (e)) ∧ x ∧ e, and e ∧ (f1 ⊕ f2 )(x) = e ∧ [(f1 (x) + f2 (x)) ∧ x] = e · [(f1 (x) + f2 (x)) ∧ x] c27

c

43 = [e · (f1 (x) + f2 (x))] ∧ (e · x) = [(e · f1 (x)) + (e · f2 (x))] ∧ (e · x)

= [x · f1 (e) + x · f2 (e)] ∧ (e · x) = [(f1 (e) ∧ x) + (f2 (e) ∧ x)] ∧ (e ∧ x) c

40 ((f1 (e) + f2 (e)) ∧ x) ∧ e = [[(f1 (e) ∧ x) + (f2 (e) ∧ x)] ∧ x] ∧ e =

= (f1 (e) + f2 (e)) ∧ x ∧ e, hence x ∧ (f1 ⊕ f2 )(e) = e ∧ (f1 ⊕ f2 )(x), that is f1 ⊕ f2 ∈ M (I1 ∩ I2 , A). For I ∈ I(A) and f ∈ M (I, A) we deﬁne f − , f ∼ : I → A by

f − (x) = x · (f (x))− , and f ∼(x) = (f (x))∼ · x, for every x ∈ I. Lemma 3.5. f − , f ∼ ∈ M (I, A). Proof. If x ∈ I and e ∈ B(A), then f − (e · x) = (e · x) · (f (e · x))− = e · x · (e · f (x))− = e · x · (e− + (f (x))− ) = x · (e · (e− + (f (x))− )) = x · (e ∧ (f (x))− ) = x · (e · (f (x))− ) = e · (x · (f (x))− ) = e · f − (x) and f ∼(e · x) = (f (e · x))∼ · (e · x) = (e · f (x))∼ · (e · x) = ((f (x))∼+e∼)·e·x = ((f (x))∼∧e)·x = ((f (x))∼·e)·x = ((f (x))∼·x)·e = f ∼(x)·e = e·f ∼(x) Clearly, f − (x) ≤ x and f ∼(x) ≤ x for every x ∈ I. Clearly, if e ∈ I ∩ B(A), then f − (e) = e · [f (e)]− ∈ B(A)

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and f ∼(e) = [f (e)]∼ · e ∈ B(A) Since f ∈ M (I, A), for e ∈ I ∩ B(A) and x ∈ I we have: x ∧ f (e) = e ∧ f (x) ⇒ x− ∨ (f (e))− = e− ∨ (f (x))− ⇒ x− + (f (e))− = e− + (f (x))− ⇒ e · x · [x− + (f (e))− ] = x · e · [e− + (f (x))− ] ⇒ e · [x ∧ (f (e))− ] = x · [e ∧ (f (x))− ] ⇒ e · x · (f (e))− = x · e · (f (x))− ⇒ x · [e · (f (e))− ] = e · [x · (f (x))− ] ⇒ x ∧ [e · (f (e))− ] = e ∧ [x · (f (x))− ] ⇒ x ∧ f − (e) = e ∧ f − (x), and x ∧ f (e) = e ∧ f (x) ⇒ x∼ ∨ (f (e))∼ = e∼ ∨ (f (x))∼ ⇒ (f (e))∼ + x∼ = (f (x))∼ + e∼ ⇒ [(f (e))∼ + x∼] · x · e = [(f (x))∼ + e∼] · e · x ⇒ [(f (e))∼ ∧ x] · e = [(f (x))∼ ∧ e] · x ⇒ (f (e))∼ · x · e = (f (x))∼ · e · x ⇒ [(f (e))∼ · e] · x = [(f (x))∼ · x] · e ⇒ [(f (e))∼ · e] ∧ x = [(f (x))∼ · x] ∧ e ⇒ x ∧ f ∼(e) = e ∧ f ∼(x), hence f − and f ∼ verify a19 , that is f − , f ∼ ∈ M (I, A).

For f ∈ M (I1 , A) and g ∈ M (I2 , A) with I1 , I2 ∈ I(A) we deﬁne f g on I1 ∩ I2 by f g = (g − ⊕ f − )∼. Lemma 3.6. For every x ∈ I1 ∩ I2 : (f g)(x) = (f (x) + x∼) · g(x) = f (x) · (x− + g(x)). Proof. For x ∈ I1 ∩ I2 we denote a = f (x), b = g(x); clearly a, b ≤ x. So: (f g)(x) = [(g − (x) + f − (x)) ∧ x]∼ · x = [(x · (g(x))− + x · (f (x))− ) ∧ x]∼ · x = c30 [(x · b− + x · a− ) ∧ x]∼ · x = [(x · b− + x · a− )∼ ∨ x∼] · x = [(x · b− + x · a− )∼ · x] ∨ (x∼ · x) = [(x · b− + x · a− )∼ · x] ∨ 0 = (x · b− + x · a− )∼ · x = (x · a− )∼ · (x · b− )∼ · x = ((a− )∼ + x∼) · ((b− )∼ +x∼)·x = (a+x∼)·(b+x∼)·x = (a+x∼)·(b∧x) = (a+x∼)·b = (f (x)+x∼)·g(x). Now we shall prove that (f (x) + x∼) · g(x) = f (x) · (x− + g(x)). Indeed, (f (x) + x∼) · g(x) = (f (x) + x∼) · (g(x) ∧ x) = (f (x) + x∼) · [x · (x− + g(x))] = [(f (x) + x∼) · x] · (x− + g(x)) = (f (x) ∧ x) · (x− + g(x)) = f (x) · (x− + g(x)). Proposition 3.7. (M (A), ⊕,− ,∼ , 0, 1) is a pseudo-M V algebra. Proof. We verify the axioms of pseudo-M V algebras. a1 ). Let fi ∈ M (Ii , A) where Ii ∈ I(A), i = 1, 2, 3 and denote I = I1 ∩ I2 ∩ I3 ∈ I(A). Also, denote f = f1 ⊕ (f2 ⊕ f3 ), g = (f1 ⊕ f2 ) ⊕ f3 and for x ∈ I, a = f1 (x), b = f2 (x), c = f3 (x). Clearly a, b, c ≤ x. Thus, for x ∈ I : f (x) = (f1 (x) + (f2 ⊕ f3 )(x)) ∧ x = (f1 (x) + ((f2 (x) + f3 (x)) ∧ x)) ∧ x =

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40 = (a + ((b + c) ∧ x)) ∧ x = ((a ∧ x) + ((b + c) ∧ x)) ∧ x = (a + b + c) ∧ x. Analogously, g(x) = (a+b+c)∧x, hence f = g, that is, the operation ⊕ is associative. a2 ). Let f ∈ M (I, A) with I ∈ I(A). If x ∈ I, then

(f ⊕ 0)(x) = (f (x) + 0(x)) ∧ x = f (x) ∧ x = f (x), and (0 ⊕ f )(x) = (0(x) + f (x)) ∧ x = f (x) ∧ x = f (x), hence f ⊕ 0 = 0 ⊕ f = f. a3 ). For f ∈ M (I, A) (with I ∈ I(A)) and x ∈ I, we have: (f ⊕ 1)(x) = (f (x) + 1(x)) ∧ x = (f (x) + x) ∧ x = x = 1(x), and (1 ⊕ f )(x) = (1(x) + f (x)) ∧ x = (x + f (x)) ∧ x = x = 1(x), hence f ⊕ 1 = 1 ⊕ f = 1. a4 ). For x ∈ A, we have 1− (x) = x · (1(x))− = x · x− = 0 = 0(x), and 1∼(x) = (1(x))∼ · x = x∼ · x = 0 = 0(x). So, 1∼ = 0, and 1− = 0. a5 ). Let f ∈ M (I, A), g ∈ M (J, A) (with I, J ∈ I(A)) and x ∈ I ∩ J. If denote a = f (x), b = g(x), then a, b ≤ x and from Lemma 3.6, (g − ⊕ f − )∼ = (f (x) + x∼) · g(x) = f (x) · (x− + g(x)). We have: (g ∼ ⊕ f ∼)− (x) = x · [((g(x))∼ · x + (f (x))∼ · x) ∧ x]− = x · [(b∼ · x + a∼ · x) ∧ x]− c29 = x·[(b∼ ·x+a∼ ·x)− ∨x− ] = [x·(b∼ ·x+a∼ ·x)− ]∨(x·x− ) = [x·(b∼ ·x+a∼ ·x)− ]∨0 = = x · (a∼ · x)− · (b∼ · x)− = x · (x− + a) · (x− + b) = (x ∧ a) · (x− + b) = a · (x− + b) = f (x) · (x− + g(x)), and by Lemma 3.6, we deduce that (g − ⊕ f − )∼ = (g ∼ ⊕ f ∼)− . a6 ). Let f ∈ M (I, A), g ∈ M (J, A) (with I, J ∈ I(A)) and x ∈ I ∩ J. We have: (f ⊕ f ∼ g)(x) = [f (x) + ((f (x))∼ · x + x∼) · g(x)] ∧ x = [f (x) + ((f (x))∼ · (x∼)− + x∼) · g(x)] ∧ x = [f (x) + ((f (x))∼ ∨ x∼) · g(x)] ∧ x = [f (x) + (f (x) ∧ x)∼ · g(x)] ∧ x = [f (x) + (f (x))∼ · g(x)] ∧ x = [f (x) ∨ g(x)] ∧ x = f (x) ∨ g(x); Analogously, (g ⊕ g ∼ f )(x) = g(x) ∨ f (x); (f g − ⊕ g)(x) = [f (x) · (x− + x · (g(x))− ) + g(x)] ∧ x = [f (x) · (x− + (x− )∼ · (g(x))− ) + g(x)] ∧ x = [f (x) · (x− ∨ (g(x))− ) + g(x)] ∧ x = [f (x) · (x ∧ g(x))− + g(x)] ∧ x = [f (x) · (g(x))− + g(x)] ∧ x = [f (x) ∨ g(x)] ∧ x = f (x) ∨ g(x); Analogously, (g f − ⊕ f )(x) = g(x) ∨ f (x); So, f ⊕ f ∼ g = g ⊕ g ∼ f = f g − ⊕ g = g f − ⊕ f.

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a7 ). Let f ∈ M (I, A), g ∈ M (J, A) where I, J ∈ I(A). Thus, for x ∈ I ∩ J : (f (f − ⊕ g))(x) = f (x) · [x− + (x · (f (x))− + g(x)) ∧ x] = f (x) · ([x− + (x · (f (x))− + g(x))]∧[x− +x]) = f (x)·([x− +(x·(f (x))− +g(x))]∧1) = f (x)·[(x− +x·(f (x))− )+g(x)] = f (x) · [(x− + (x− )∼ · (f (x))− ) + g(x)] = f (x) · [(x− ∨ (f (x))− ) + g(x)] = f (x) · [(x ∧ f (x))− + g(x)] = f (x) · [(f (x))− + g(x)] = f (x) ∧ g(x). ((f ⊕ g ∼) g)(x) = [(f (x) + (g(x))∼ · x) ∧ x + x∼] · g(x) = [((f (x) + (g(x))∼ · x) + x∼) ∧ (x + x∼)] · g(x) = [((f (x) + (g(x))∼ · x) + x∼) ∧ 1] · g(x) = [f (x) + ((g(x))∼ · x + x∼)] · g(x) = [f (x) + ((g(x))∼ · (x∼)− + x∼)] · g(x) = [f (x) + ((g(x))∼ ∨ x∼)] · g(x) = [f (x) + (g(x) ∧ x)∼] · g(x) = [f (x) + (g(x))∼] · g(x) = f (x) ∧ g(x). So, f (f − ⊕ g) = (f ⊕ g ∼) g. a8 ). For f ∈ M (I, A) (with I ∈ I(A)) and x ∈ I, we have: (f − )∼(x) = (x·(f (x))− )∼ ·x = [((f (x))− )∼ +x∼]·x = (f (x)+x∼)·x = f (x)∧x = f (x). So, (f − )∼ = f. Lemma 3.8. Let f, g ∈ M (A). Then for every x ∈ dom(f ) ∩ dom(g): (i) (f ∧ g)(x) = f (x) ∧ g(x), (ii) (f ∨ g)(x) = f (x) ∨ g(x). Proof. We recall that in pseudo-M V algebra M (A) we have: f ∧ g = f (f − ⊕ g) = g (g − ⊕ f ) = (f ⊕ g ∼) g = (g ⊕ f ∼) f, and f ∨ g = f ⊕ f ∼ g = g ⊕ g ∼ f = f g − ⊕ g = g f − ⊕ f. So: (i). Follow immediately from Proposition 3.7, a7 ). (ii). Follow immediately from Proposition 3.7, a6 ).

Lemma 3.9. Let the map vA : B(A) → M (A) deﬁned by vA (a) = fa for every a ∈ B(A). Then vA is an injective morphism of pseudo-MV algebras. Proof. Clearly, vA (0) = f0 = 0. Let a, b ∈ B(A) and x ∈ A. We have: (vA (a) ⊕ vA (b))(x) = (vA (a)(x) + vA (b)(x)) ∧ x = ((a ∧ x) + (b ∧ x)) ∧ x c40

= (a + b) ∧ x = (vA (a + b))(x),

hence vA (a + b) = vA (a) ⊕ vA (b). Also, (vA (a))− (x) = x · (vA (a)(x))− = x · (a ∧ x)− = x · (a− ∨ x− ) = x · (x− + a− ) = (since a− ∈ B(A))

= x ∧ a− = vA (a− )(x),

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hence vA (a− ) = (vA (a))− , and (vA (a))∼(x) = (vA (a)(x))∼ · x = (a ∧ x)∼ · x = (a∼ ∨ x∼) · x = (a∼ + x∼) · x = (since a∼ ∈ B(A))

= a∼ ∧ x = vA (a∼)(x),

hence vA (a∼) = (vA (a))∼, that is, vA is a morphism of pseudo-M V algebras. To prove the injectivity of vA , let a, b ∈ B(A) such that vA (a) = vA (b). Then a ∧ x = b ∧ x, for every x ∈ A, hence for x = 1 we obtain that a ∧ 1 = b ∧ 1 ⇒ a = b. Definition 3.10. A nonempty set I ⊆ A is called regular iﬀ x ∧ e = y ∧ e for all e ∈ I ∩ B(A), implies x = y, for all x, y ∈ A. For example, A is a regular subset of A (since if x, y ∈ A and x ∧ e = y ∧ e for every e ∈ A ∩ B(A) = B(A), then for e = 1 we obtain x ∧ 1 = y ∧ 1 ⇔ x = y). More generally, every subset of A which contains 1 is regular. We denote R(A) = {I ⊆ A : I is a regular subset of A}. Remark 3.11. The condition I ∈ R(A) is equivalent with the condition: for every x, y ∈ A, if fx|I∩B(A) = fy|I∩B(A) , then x = y. Lemma 3.12. If I1 , I2 ∈ I(A) ∩ R(A), then I1 ∩ I2 ∈ I(A) ∩ R(A). Proof. By Remark 2.15, I1 ∩ I2 ∈ I(A). To prove I1 ∩ I2 ∈ R(A) let x, y ∈ A such that x ∧ e = y ∧ e for every e ∈ (I1 ∩ I2 ) ∩ B(A). If ei ∈ Ii ∩ B(A), i = 1, 2 are arbitrary, then e1 ∧ e2 ∈ I1 ∩ I2 ∩ B(A) so, we have (e1 ∧ e2 ) ∧ x = (e1 ∧ e2 ) ∧ y ⇔ e1 ∧ (e2 ∧ x) = e1 ∧ (e2 ∧ y). Since e1 ∈ I1 ∩ B(A) are arbitrary and I1 ∈ I(A) ∩ R(A), then we obtain e2 ∧ x = e2 ∧ y. Since e2 ∈ I2 ∩ B(A) are arbitrary and I2 ∈ I(A) ∩ R(A), we obtain x = y, hence I1 ∩ I2 ∈ I(A) ∩ R(A). Remark 3.13. By Lemma 3.12, we deduce that Mr (A) = {f ∈ M (A) : dom(f ) ∈ I(A) ∩ R(A)} is a pseudo-M V subalgebra of M (A).

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Definition 3.14. Given two multipliers f1 , f2 on A, we say that f2 extends f1 if dom(f1 ) ⊆ dom(f2 ) and f2|dom(f1 ) = f1 ; we write f1 ≤ f2 if f2 extends f1 . A multiplier f is called maximal if f cannot be extended to a strictly larger domain. Lemma 3.15. (i) If f1 , f2 ∈ M (A), f ∈ Mr (A) and f ≤ f1 , f ≤ f2 , then f1 and f2 agree on dom(f1 ) ∩ dom(f2 ), (ii) Every multiplier f ∈ Mr (A) can be extend to a maximal multiplier. Moreover, each principal multiplier fa with a ∈ B(A) and dom(fa ) ∈ I(A) ∩ R(A) can be uniquely extended to a total multiplier fa and each non-principal multiplier can be extended to a maximal non-principal one. Proof. (i). Assume, to the contrary that there exists x ∈ dom(f1 ) ∩ dom(f2 ) such that f1 (x) = f2 (x). Since dom(f ) ∈ R(A), there is e ∈ dom(f ) ∩ B(A) such that e ∧ f1 (x) = e ∧ f2 (x). But e ∧ fi (x) = fi (e · x) for i = 1, 2, thus f1 (e · x) = f2 (e · x). Since e · x ≤ e,we have e · x ∈ dom(f ), contradicting f ≤ f1 , f ≤ f2 . (ii).We ﬁrst prove that fa with a ∈ B(A) can not be extended to a non-principal multiplier. Let I = dom(fa ) ∈ I(A) ∩ R(A), fa : I → A and suppose by contrary that there exists I ∈ I(A), I ⊆ I (hence I ∈ I(A) ∩ R(A)) and a non-principal multiplier f ∈ M (I , A) which extends fa . Since f is non-principal, there exists x0 ∈ I , x0 ∈ / I such that f (x0 ) = x0 ∧ a (see Remark 3.11). Since I ∈ R(A), there exists e ∈ I ∩ B(A) such that e ∧ f (x0 ) = e ∧ (a ∧ x0 ) ⇔ f (e · x0 ) = e ∧ (a ∧ x0 ) ⇔ f (e · x0 ) = a ∧ (e · x0 ). Denoting x1 = e · x0 ∈ I (since x1 ≤ e), we obtain that f (x1 ) = a ∧ x1 , which is contradictory (since fa ≤ f ). Hence fa is uniquely extended by fa . Now, let f ∈ Mr (A) be non-principal and Mf = {(I, g) : I ∈ I(A), g ∈ M (I, A), dom(f ) ⊆ I and g|dom(f ) = f } (clearly, if (I, g) ∈ Mf , then I ∈ I(A) ∩ R(A)). The set Mf is ordered by (I1 , g1 ) ≤ (I2 , g2 ) iﬀ I1 ⊆ I2 and g2|I1 = g1 . Let {(Ik , gk ) : k ∈ K} be an chain in Mf . Then I = ∪k∈K Ik ∈ I(A) and dom(f ) ⊆ I . So, g : I → A deﬁned by g (x) = gk (x) if x ∈ Ik is correctly deﬁned (since if x ∈ Ik ∩ It with k, t ∈ K, then by (i), gk (x) = gt (x)). Clearly, g ∈ M (I , A) and g|dom(f ) = f (since if x ∈ dom(f ) ⊆ I , then x ∈ I and so there exists k ∈ K such that x ∈ Ik , hence g (x) = gk (x) = f (x)). So, (I , g ) is an upper bound for the family {(Ik , gk ) : k ∈ K}, hence by Zorn’s lemma, Mf contains at least one maximal multiplier h which extends f. Since f is non-principal and h extends f, h is also non-principal. On the pseudo-M V algebra Mr (A) we consider the relation ρA deﬁned by (f1 , f2 ) ∈ ρA iﬀ f1 and f2 agree on the intersection of their domains.

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Lemma 3.16. ρA is a congruence on Mr (A). Proof. The reﬂexivity and the symmetry of ρA are immediate. To prove the transitivity of ρA let (f1 , f2 ), (f2 , f3 ) ∈ ρA . Therefore f1 , f2 and respectively f2 , f3 agree on the intersection of their domains. If by contrary, there exists x0 ∈ dom(f1 ) ∩ dom(f3 ) such that f1 (x0 ) = f3 (x0 ), since dom(f2 ) ∈ R(A), there exists e ∈ dom(f2 ) ∩ B(A) such that e ∧ f1 (x0 ) = e ∧ f3 (x0 ) ⇔ e · f1 (x0 ) = e · f3 (x0 ) ⇔ f1 (e · x0 ) = f3 (e · x0 ) which is contradictory, since e · x0 ∈ dom(f1 ) ∩ dom(f2 ) ∩ dom(f3 ). To prove the compatibility of ρA with the operations ⊕,− and ∼ on Mr (A), let (f1 , f2 ), (g1 , g2 ) ∈ ρA . So, we have that f1 , f2 resp. g1 , g2 agree on the intersection of their domains. To prove (f1 ⊕ g1 , f2 ⊕ g2 ) ∈ ρA let x ∈ dom(f1 ) ∩ dom(f2 ) ∩ dom(g1 ) ∩ dom(g2 ). Then f1 (x) = f2 (x) and g1 (x) = g2 (x), hence (f1 ⊕ g1 )(x) = [f1 (x) + g1 (x)] ∧ x = [f2 (x) + g2 (x)] ∧ x = (f2 ⊕ g2 )(x), that is f1 ⊕ g1 , f2 ⊕ g2 agree on the intersection of their domains, hence ρA is compatible with the operation ⊕. If x ∈ dom(f1 ) ∩ dom(f2 ) then f1 (x) = f2 (x) and f1− (x) = x · (f1 (x))− = x · (f2 (x))− = f2− (x), and f1∼(x) = (f1 (x))∼ · x = (f2 (x))∼ · x = f2∼(x), hence f1− , f2− and f1∼, f2∼ agree on the intersection of their domains, hence ρA is compatible with the operations − and ∼ . Remark 3.17. We denote by A the quotient pseudo-M V algebra Mr (A)/ρA ; this pseudoM V algebra will have a very important role for this paper (see Theorem 4.5). For f ∈ Mr (A) with I = dom(f ) ∈ I(A) ∩ R(A), we denote by [f, I] the congruence class of f modulo ρA . Lemma 3.18. Let the map vA : B(A) → A deﬁned by vA (a) = [fa , A] for every a ∈ B(A). Then (i) vA is an injective morphism of pseudo-M V algebras, (ii) vA (B(A)) ⊆ B(A ), (iii) vA (B(A)) ∈ R(A ). Proof. (i). Follow from Lemma 3.9. c40 (ii). For a ∈ B(A) and x ∈ A we have (fa ⊕ fa )(x) = ((a ∧ x) + (a ∧ x)) ∧ x = (a + a) ∧ x = a ∧ x = fa (x), hence fa ⊕ fa = fa , that is [fa , A] ∈ B(A ). (iii). To prove vA (B(A)) ∈ R(A ), if by contrary there exist f1 , f2 ∈ Mr (A) such that [f1 , dom(f1 )] = [f2 , dom(f2 )] (that is there exists x0 ∈ dom(f1 ) ∩ dom(f2 ) such that f1 (x0 ) = f2 (x0 )) and [f1 , dom(f1 )] ∧ [fa , A] = [f2 , dom(f2 )] ∧ [fa , A] for every [fa , A] ∈ vA (B(A)) ∩ B(A ) (that is by (ii) for every [fa , A] ∈ vA (B(A)) with a ∈ B(A)), then (f1 ∧ fa )(x) = (f2 ∧ fa )(x) for every x ∈ dom(f1 ) ∩ dom(f2 ) and every a ∈ B(A) ⇔ f1 (x) ∧ a ∧ x = f2 (x) ∧ a ∧ x for every x ∈ dom(f1 ) ∩ dom(f2 ) and every a ∈ B(A). For a = 1 and x = x0 we obtain that f1 (x0 ) ∧ x0 = f2 (x0 ) ∧ x0 ⇔ f1 (x0 ) = f2 (x0 ), which is contradictory.

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Remark 3.19. Since for every a ∈ B(A), fa is the unique maximal multiplier on [fa , A] (by Lemma 3.15) we can identify [fa , A] with fa . So, since vA is an injective map, the elements of B(A) can be identiﬁed with the elements of the set { fa : a ∈ B(A)}. Lemma 3.20. In view of the identiﬁcations made above, if [f, dom(f )] ∈ A (with f ∈ Mr (A) and I = dom(f ) ∈ I(A) ∩ R(A)), then I ∩ B(A) ⊆ {a ∈ B(A) : fa ∧ [f, dom(f )] ∈ B(A)}. Proof. Let a ∈ I ∩ B(A). If to the contrary, fa ∧ [f, dom(f )] ∈ / B(A) (that is [fa ∧ f, dom(f )] ∈ / vA (B(A))), then fa ∧ f is a non-principal multiplier. Then by Lemma 3.15, (ii), fa ∧ f can be extended to a non-principal maximal multiplier f : I → A with I ∈ I(A) and I ⊆ I : ⊆

I fa ∧f

I

f

A Since f |I = fa ∧ f, for every x ∈ I, f (x) = (fa ∧ f )(x) = fa (x) ∧ f (x) = a ∧ x ∧ f (x) = a ∧ f (x) = a · f (x) = f (a · x) = x · f (a) (by a19 ) = x ∧ f (a), that is f |I is principal, which contradicts the assumption that f is non-principal.

4

Maximal pseudo-MV algebra of quotients

Definition 4.1. A pseudo-M V algebra F is called pseudo-M V algebra of fractions of A if: (a20 ) B(A) is a pseudo-M V subalgebra of F (that is B(A) ≤ F ), (a21 ) For every a , b , c ∈ F, a = b , there exists e ∈ B(A) such that e ∧ a = e ∧ b and e ∧ c ∈ B(A), (a22 ) For every a ∈ F and e ∈ B(A) such that e ∧ a ∈ B(A), if f ∈ B(A) and f ≤ e, then f ∧ a ∈ B(A). So, pseudo-M V algebra B(A) is a pseudo-M V algebra of fractions of itself (since 1 ∈ B(A)). As a notational convenience, we write A ≺ F to indicate that F is a pseudo-M V algebra of fractions for A. Definition 4.2. Q(A) is the maximal pseudo-M V algebra of quotients of A if A ≺ Q(A) and for every pseudo-M V algebra F with A ≺ F there exists an injective morphism of pseudo-M V algebras i : F → Q(A). As for nomenclature, the obvious ”pseudo-M V algebra of quotients” is excluded by the wide-spread use of ”quotient” as a synonym for ”interval” in lattice theory. So we

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settle for ”fraction” as introduced by Findlay and Lambek in [14] in their construction of rings of quotients. Lemma 4.3. Let A ≺ F ; then for every a , b ∈ F, a = b , and any ﬁnite sequence c1 , ..., cn ∈ F, there exists e ∈ B(A) such that e ∧ a = e ∧ b and e ∧ ci ∈ B(A) for i = 1, 2, ..., n (n ≥ 2). Proof. Assume lemma holds true for n − 1. So we may ﬁnd f ∈ B(A) such that f ∧ a = f ∧ b and f ∧ ci ∈ B(A) for i = 1, 2, ..., n − 1. Since A ≺ F , we ﬁnd g ∈ B(A) such that g ∧ (f ∧ a ) = g ∧ (f ∧ b ) and g ∧ cn ∈ B(A). The element e = f ∧ g ∈ B(A) has the required properties. Lemma 4.4. Let A ≺ F and a ∈ F. Then Ia = {e ∈ B(A) : e ∧ a ∈ B(A)} ∈ I(B(A)) ∩ R(A). Proof. By a22 we deduce that Ia ∈ I(B(A)). To prove Ia ∈ R(A), let x, y ∈ A such that e ∧ x = e ∧ y for every e ∈ Ia ∩ B(A). If by contrary, x = y, since A ≺ F , there exists e0 ∈ B(A) such that e0 ∧ a ∈ B(A) (that is e0 ∈ Ia ) and e0 ∧ x = e0 ∧ y, which is contradictory. Theorem 4.5. A (deﬁned in section 3) is the maximal pseudo-M V algebra Q(A) of quotients of A. Proof. The fact that B(A) is a pseudo-M V subalgebra of A follows from Lemma 3.18, (i). To prove a21 , let [f, dom(f )], [g, dom(g)], [h, dom(h)] ∈ A with f, g, h ∈ Mr (A) such that [g, dom(g)] = [h, dom(h)] (that is there exists x0 ∈ dom(g) ∩ dom(h) such that g(x0 ) = h(x0 )). Put I = dom(f ) ∈ I(A) ∩ R(A) and I[f,dom(f )] = {a ∈ B(A) : fa ∧ [f, dom(f )] ∈ B(A)} (by Lemma 3.18, fa ∈ B(M (A)) if a ∈ B(A)). Then by Lemma 3.20, I ∩ B(A) ⊆ I[f,dom(f )] . If we suppose that for every a ∈ I ∩ B(A), fa ∧ [g, dom(g)] = fa ∧ [h, dom(h)], then [fa ∧ g, dom(g)] = [fa ∧ h, dom(h)], hence for every x ∈ dom(g) ∩ dom(h) we have (fa ∧ g)(x) = (fa ∧ h)(x) i.e. a ∧ g(x) = a ∧ h(x). Since I ∈ R(A) we deduce that g(x) = h(x) for every x ∈ dom(g) ∩ dom(h) so [g, dom(g)] = [h, dom(h)], which is contradictory. Hence, if [g, dom(g)] = [h, dom(h)], then there exists a ∈ I ∩ B(A), such that fa ∧ [g, dom(g)] = fa ∧ [h, dom(h)]. But for this a ∈ I ∩ B(A) we have fa ∧ [f, dom(f )] ∈ B(A)

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(since by Lemma 3.20, I ∩ B(A) ⊆ I[f,dom(f )] ). To prove a22 let a ∈ B(A) such that fa ∧ [f, dom(f )] ∈ B(A), that is there exists b ∈ B(A) with fa ∧ f = fb . Thus (fa ∧ f )(x) = fb (x) so a ∧ f (x) = b ∧ x, for every x ∈ dom(f ). If we consider c ∈ B(A) such that fc ≤ fa then c ≤ a (hence c = c ∧ a) and for every x ∈ dom(f ) we have: (fc ∧ f )(x) = c ∧ f (x) = (c ∧ a) ∧ f (x) = c ∧ (a ∧ f (x)) = c ∧ (b ∧ x) = (c ∧ b) ∧ x = fc∧b (x), for every x ∈ dom(f ), hence fc ∧ f = fc∧b ∈ vA (B(A)) (since c ∧ b ∈ B(A)). To prove the maximality of A , let F be a pseudo-M V algebra such that A ≺ F ; thus B(A) ⊆ B(F ) A≺ F i A For a ∈ F, Ia = {e ∈ B(A) : e ∧ a ∈ B(A)} ∈ I(B(A)) ∩ R(A) (by Lemma 4.4 ). Thus fa : Ia → A deﬁned by fa (x) = x ∧ a is a multiplier. Indeed, if e ∈ B(A) and x ∈ Ia , then fa (e · x) = (e · x) ∧ a = (e ∧ x) ∧ a = e ∧ (x ∧ a ) = e · (x ∧ a ) = e · fa (x), and fa (x) ≤ x, hence a16 and a17 are veriﬁed. To verify a18 , let e ∈ Ia ∩ B(A) = Ia . Thus, fa (e) = e ∧ a ∈ B(A) (since e ∈ Ia ). The condition a19 is obviously veriﬁed, hence [fa , Ia ] ∈ A . We deﬁne i : F → A , by i(a ) = [fa , Ia ], for every a ∈ F. Clearly i(0) = 0. c40 For a , b ∈ F and x ∈ Ia ∩ Ib , we have (i(a ) ⊕ i(b ))(x) = [(a ∧ x) + (b ∧ x)] ∧ x = (a + b ) ∧ x = i(a + b )(x), hence i(a ) ⊕ i(b ) = i(a + b ). Also, for x ∈ Ia we have (i(a ))− (x) = x · [i(a )(x)]− = x · (a ∧ x)− = x · (a · x)− = x · [x− + (a )− ] = x ∧ (a )− = f(a )− (x) = i((a )− )(x), and (i(a ))∼(x) = [i(a )(x)]∼ · x = (a ∧ x)∼ · x = (x · a )∼ · x = [(a )∼ + x∼] · x = (a )∼ ∧ x = f(a )∼ (x) = i((a )∼)(x), hence i((a )− ) = (i(a ))− , and i((a )∼) = (i(a ))∼, that is, i is a morphism of pseudo-M V algebras. To prove the injectivity of i, let a , b ∈ F such that i(a ) = i(b ). It follows that [fa , Ia ] = [fb , Ib ] so fa (x) = fb (x) for every x ∈ Ia ∩ Ib . We get a ∧ x = b ∧ x for every x ∈ Ia ∩ Ib . If a = b , by Lemma 4.3 (since A ≺ F ), there exists e ∈ B(A) such that e ∧ a , e ∧ b ∈ B(A) and e ∧ a = e ∧ b which is contradictory (since e ∧ a , e ∧ b ∈ B(A) implies e ∈ Ia ∩ Ib ).

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Remark 4.6. 1. If A is a pseudo- M V algebra with B(A) = {0, 1} = L2 and A ≺ F then F = {0, 1}, hence Q(A) = A ≈ L2 . Indeed, if a, b, c ∈ F with a = b, then by a21 there exists e ∈ B(A) such e ∧ a = e ∧ b (hence e = 0) and e ∧ c ∈ B(A). Clearly, e = 1, hence c ∈ B(A), that is F = B(A). As examples of pseudo -M V algebras with this property we have local pseudo-M V algebras and pseudo-M V chains. 2. If A is an M V -algebra, then Q(A) is the maximal MV-algebra of quotients obtained in [5]. 3. If A is a Boolean algebra, then B(A) = A and Q(A) is the classical DedekindMacNeille completion of A (see [18], p.687). Open Problem: Are there nonboolean pseudo-M V algebra A such that Q(A) is the classical Dedekind-MacNeille completion of B(A)? Note that in [11] is proved that: (i) Any Archimedean pseudo-M V algebra is commutative, i.e., an M V algebra. (ii) A pseudo-M V algebra has the Dedekind-MacNeille completion as a pseudo-M V algebra iﬀ A is Archimedean.

Acknowledgements We would like to express our gratitude for the guidance given by the referee in the elaboration of this paper.

References [1] D. Bu¸sneag: “Hilbert algebra of fractions and maximal Hilbert algebras of quotients“, Kobe Journal of Mathematics, Vol. 5, (1988), pp. 161–172. [2] D. Bu¸sneag and D. Piciu: Meet-irreducible ideals in an MV-algebra, Analele Universit˘a¸tii din Craiova, Seria Matematica-Informatica, Vol. XXVIII, 2001, pp. 110– 119. [3] D. Bu¸sneag and D. Piciu: “On the lattice of ideals of an MV-algebra“, Scientiae Mathematicae Japonicae, Vol. 56, No. 2, (2002), pp. 367–372, e6, pp. 221–226. [4] D. Bu¸sneag and D. Piciu: MV-algebra of fractions relative to an ∧−closed system, Analele Universit˘a¸tii din Craiova, Seria Matematica-Informatica, Vol. XXX, 2003, pp. 1–6. [5] D. Bu¸sneag and D. Piciu: “MV - algebra of fractions and maximal MV-algebra of quotients“, to appear in Journal of Multiple-Valued Logic and Soft Computing. [6] C.C. Chang: “Algebraic analysis of many valued logics“, Trans. Amer. Math. Soc., Vol. 88, (1958), pp. 467–490. [7] R. Cignoli, I.M.L. D Ottaviano and D. Mundici: Algebraic foundation of many -valued Reasoning, Kluwer Academic Publishers, Dordrecht, 2000. [8] W.H. Cornish: “The multiplier extension of a distributive lattice“, Journal of Algebra, Vol. 32, (1974), pp. 339–355.

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[9] W.H. Cornish: “A multiplier approach to implicative BCK-algebras“, Mathematics Seminar Notes, Kobe University, Vol. 8, No. 1, (1980), pp. 157–169. [10] C. Dan: F-multipliers and the localisation of Heyting algebras, Analele Universit˘a¸tii din Craiova, Seria Matematica-Informatica, Vol. XXIV, 1997, pp. 98–109. [11] A. Dvureˇcenskij: “Pseudo-MV-algebras are intervals in l-groups“, Journal of Australian Mathematical Society, Vol. 72, (2002), pp. 427–445. [12] G. Georgescu and A. Iorgulescu: “Pseudo-MV algebras“, Multi. Val. Logic, Vol. 6, (2001), pp. 95–135. [13] P. H´ajek: Metamathematics of fuzzy logic, Kluwer Acad. Publ., Dordrecht, 1998. [14] J. Lambek: Lectures on Rings and Modules, Blaisdell Publishing Company, 1966. [15] I. Leu¸stean: “Local Pseudo-MV algebras“, Soft Computing, Vol. 5, (2001), pp. 386– 395. [16] D. Piciu: Pseudo-MV algebra of fractions relative to an ∧−closed system, Analele Universit˘a¸tii din Craiova, Seria Matematica-Informatica, Vol. XXX, 2003, pp. 7–13. [17] J. Schmid: “Multipliers on distributive lattices and rings of quotients“, Houston Journal of Mathematics, Vol. 6, No. 3, (1980), pp. 401–425. [18] J. Schmid: Distributive lattices and rings of quotients, Coll. Math. Societatis Janos Bolyai, Szeged, Hungary, 1980.

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Representation of finite groups and the first Betti number of branched coverings of a universal Borromean orbifold Masahito Todaa∗ Department of Mathematics, Ochanomizu University, 2-1-1, Ohtsuka, Bunkyo-ku, Tokyo 112-8610, Japan

Received 19 June 2003; accepted 5 April 2004 Abstract: The paper studies the first homology of finite regular branched coverings of a universal Borromean orbifold called B4,4,4 \H3 . We investigate the irreducible components of the first homology as a representation space of the finite covering transformation group G. This gives information on the first betti number of finite coverings of general 3-manifolds by the universality of B4,4,4 . The main result of the paper is a criterion in terms of the irreducible character whether a given irreducible representation of G is an irreducible component of the first homology when G admits certain symmetries. As a special case of the motivating argument the criterion is applied to principal congruence subgroups of B4,4,4 . The group theoretic computation shows that most of the, possibly nonprincipal, congruence subgroups are of positive first Betti number. c Central European Science Journals. All rights reserved. Keywords: hyperbolic geometry, 3-manifold, arithmetic lattice, finite groups of Lie type MSC (2000): 57M12, 57M50, 57M60, 57S17, 20C05, 20C33 a

1

This work is partially supported by the Sonderforschungsbereich 288.

Introduction

1.1 Motivation The object of interest in this paper is the ﬁrst homology of ﬁnite regular branched coverings of a hyperbolic 3-orbifold. We shall stick to a single, but universal, example of ∗

E-mail: [email protected]

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3-orbifolds, which is called B4,4,4 \H3 in [6]. The homology is given a structure of C[G]module by the action of the ﬁnite covering transformation group G. In this paper we shall study the irreducible components of the C[G]-module. The investigation is motivated by the following problem in 3-dimensional topology: Problem 1.1. Does any 3-manifold of inﬁnite fundamental group have a ﬁnite-sheeted cover of positive ﬁrst Betti number? This problem was raised by Thurston in his paper [14], which can be one of the crucial steps towards his hyperbolization conjecture of irreducible atoroidal 3-manifolds through his hyperbolization theorem for Haken atoroidal 3-manifolds. We shall sketch how irreducible components of the C[G]-module of the ﬁrst homology is related to the ﬁrst Betti numbers of unbranched coverings of a given 3-manifold. The following lemma shows that the C[G]-module structure of the ﬁrst homology of the regular branched covering gives the ﬁrst Betti number of all branched coverings covered by the regular branched covering. Lemma 1.2. Suppose that Γ is an orientation-preserving cocompact Kleinian group and Γ0 a normal subgroup of ﬁnite index in Γ. Then we have H∗ (Γ\H3 , C) H∗ (Γ0 \H3 , C)Γ/Γ0 where superscript Γ/Γ0 denotes the ﬁxed point set by the action of Γ/Γ0 . Proof 1.3. Since Γ is orientation preserving we can ﬁnd the cell decompositions of Γ\H3 such that covering transformation group G := Γ/Γ0 acts as a cellular map preserving the orientation of the cell. Hence the transfer map is the inverse of the G-isomorphism of H∗ (Γ0 \H3 , C)G to H∗ (Γ\H3 , C) induced by the covering map. Now let us recall the deﬁnition of universal groups. Definition 1.4. A Kleinian group Γ is universal if, for any given closed 3-manifold M , there is a subgroup ΓM of ﬁnite index in Γ such that ΓM \H3 is homeomorphic to M . See [7] for the universality of Kleinian group B4,4,4 . We denote by TΓ the subgroup of Γ generated by all elements of ﬁnite order in Γ. Proposition 1.5. For given closed 3-manifold M , any subgroup ΓM of universal group B4,4,4 associated to M in the deﬁnition and each normal subgroup Γ0 of ﬁnite index in ˜ Γ0 of M with B4,4,4 , we can ﬁnd a ﬁnite-sheeted (unbranched) covering M ˜ Γ0 ) ≥ dim(H1 (Γ0 \H3 , C)TΓM Γ0 /Γ0 ) b1 (M where b1 (·) denotes the ﬁrst Betti number.

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Proof 1.6. Since Γ0 is normal and of ﬁnite index, Γ1 : = TΓM Γ0 forms a subgroup of ˜ Γ0 is a ﬁnite˜ Γ0 := ΓM ∩ Γ1 \H3 . By deﬁnition of TΓ M ﬁnite index in B4,4,4 . Set M M 3 sheeted unbranched covering of ΓM \H = M . The argument of Lemma shows that the covering map ΓM ∩ Γ1 \H3 → Γ1 \H3 induces a surjection on the ﬁrst homology with complex coeﬃcient since we can take a normal subgroup N of ﬁnite index in Γ1 with ˜ Γ0 ) ≥ b1 (Γ1 \H3 ). Then, again by Lemma, we have b1 (MΓ0 ) ≥ N ⊂ Γ1 ∩ ΓM . Hence b1 (M b1 (Γ1 \H3 ) = dim(H1 (Γ0 \H3 , C)Γ1 /Γ0 ). This proves the proposition. The lower bound in the Proposition depends only on the image group of TΓ1 in G and the irreducible components in G-module H1 (Γ0 \H3 , C). In this formulation, our motivating problem may be reduced to ﬁnding a ﬁnite quotient G in which the image of TΓ1 is so ”small” that it has a nontrivial ﬁxed point in H1 (Γ0 \H3 , C). However it is not clear how ”small” the image should be. To make it clear we shall investigate an irreducible component of G-module H1 (Γ0 \H3 , C) for a given ﬁnite quotient.

1.2 General result To obtain a general criterion for a given irreducible representation of G to be a component of G-module H1 (Γ0 \H3 , C), we shall make use of the symmetry of B4,4,4 \H3 . B4,4,4 has three normalizers r1 , r2 , r3 in Isom(H3 ), all of which are reﬂections with respect to hyperbolic planes, hence are of order 2. These reﬂections generates ﬁnite group S = r1 , r2 , r3 Z2 × Z2 × Z2 in Isom(H3 ). If Γ0 is a normal subgroup of B4,4,4 normalized by subgroup S0 in S (we shall call such a normal subgroup S0 -normal), H∗ (Γ0 \H3 , C) naturally carries the action of semidirect product G S0 . For S0 = r1 or r1 r2 r3 we shall prove the following criterion in Section 3. Theorem 1.7. Let r = r1 or r1 r2 r3 Suppose Γ0 is an r-normal subgroup of ﬁnite index in B4,4,4 and ρ is a nontrivial r-invariant irreducible representation of G. Then ρ is an irreducible component of G-module H1 (Γ0 \H3 , C), if there holds αi χρ¯(gi r) = 0 i

where χρ¯ denotes an irreducible character of G r which restricts to that of ρ on G ,αi ∈ Q and gi ’s are elements of G which are the image in G of elements of ﬁnite order in B4,4,4 . (αi and gi are to be speciﬁed in Theorems in 3.3.) gi r’s in the formula are elements of order two or four in B4,4,4 . This apriori knowledge admits the explicit computation of the generalized character when we have enough information on the structure of ﬁnite group G. In any case the role played by the symmetry is essential in the proof (See 3.1). The usage of the symmetry is inspired by the work of Millson [10].

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1.3 Computation for congruence subgroups √ Since B4,4,4 is an arithmetic lattice of SO(3, 1) over Q( 5) (cf.[6]) we can consider the congruence subgroups. Since the principal congruence subgroup Γm associated to ideal m √ of Q( 5) is S-normal they provides an inﬁnite sequence of examples of Γ0 in Theorem 1.7. In section 5 and 6 we shall apply Theorem 1.7 to the case of congruence subgroup Γp associated to prime ideal p. For instance we shall prove the following theorem in 6.5. √ Theorem 1.8. There exists ﬁnite set PB4,4,4 of prime ideals of k = Q( 5) such that if the prime ideal p ∈ PB4,4,4 and its norm q veriﬁes q ≡ ±1 mod 8, any nontrivial r1 -invariant irreducible representation is an irreducible component of G-module H1 (Γp\H3 , C). √ To prove theorem 1.8 we should compute the character of ρ¯. If −(1 + 5)/2 is a square in the residue ﬁeld at p G is basically the direct product of two copies of SL2 (Fq ) and r permutes the component. This special structure of G makes the computation easy. Elementary group theoretic argument shows that this situation is the case under the following general assumption. Theorem 1.9. Let Γ be a maximal r1 r2 r3 -normal, but not maximal normal subgroup of ﬁnite index in B4,4,4 . Then any nontrivial r1 r2 r3 -invariant irreducible representation of G = B4,4,4 /Γ is an irreducible component of H1 (Γ\H3 , C). √ If −(1 + 5)/2 is not square we develop a general theory of the semidirect extension of irreducible characters in 6.2. As a conclusion to our motivation we shall sketch the proof of the following consequence in 7.3. Theorem 1.10. Let Γ be a subgroup of ﬁnite index in B4,4,4 . If there exists prime ideal √ p ∈ PB4,4,4 of Q( 5) such that TΓ projects to a subgroup of G = B4,4,4 /Γp which does not contain a non-central normal subgroup, Γ\H3 has a ﬁnite sheeted unbranched covering of positive ﬁrst Betti number. For classical algebraic theories, refer to [8] for general representation theory of ﬁnite groups and theory of quadratic forms, [2] and [4] for ﬁnite groups of Lie type, [13] for general theory of ﬁnite groups.

Acknowledgment The author would like to thank Professor Dr.Frank Duzaar for giving him the opportunity to stay at Humboldt Universitaet zu Berlin with the support of SFB 288. He is also grateful to the hospitality of Dr J.F. Grotowski during the stay.

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The chain complex with group action

2.1 Description of B4,4,4 We shall recall the construction of universal group B4,4,4 following [6] with some modiﬁ√ cation to work in S(3, 1; R) rather than in SL2 (C). In this paper always a = (1 + 5)/2, which is denoted by A2 in [6]. Let f = a−1 (x2 +y 2 +z 2 )−t2 be a quadratic form on RP 3 and L0 be the negative leaf of RP 3 \{f = 0}. L0 is a rescaled hyperbolic 3-space which is iden√ √ √ tiﬁed with the canonical one by (x : y : z : t) → (ζ : η : θ : τ ) = ( ax : ay : az : t). Let P be the polyhedron located in L0 as in Fig. 1 and r1 , r2 and r3 be hyperbolic reﬂections with respect to Y Z−, ZX− and XY − planes in L0 . respectively. The eight images of P by these reﬂections form a regular dodecahedron R in L0 with all dihedral angles being π/2. (See [6] Section 3) We label each sides of R as in Fig. 2.

Z (0,0,1) (1-1/a,0,1)

C (0,1,1-1/a)

A B (1,0,0)

X

(0,1,0) Y

(1,1-1/a,0)

Fig. 1 Polyhedron P.

We obtain the compact hyperbolic orbifold O, gluing sides X and X (X = A, B, .., F ) by the elliptic transformation θX of order 4 which takes X to X with its axis on the common ridge. B4,4,4 is deﬁned to be the corresponding cocompact Kleinian group. As topologically observed, O is homeomorphic to the 3-sphere and ramiﬁes over the Borromean ring with degree 4 on all components, which comes from the bold-faced ridges in Fig. 2. The following three lemmas can be observed from Fig. 1, Fig. 2 and Poincare’s theorem(see e.g. [11] Section 13).

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D' C'

C

E

B'

B

E'

D

A

F'

F

A'

Fig. 2 Arrangement of Sides of Regular Dodecahedron R.

Lemma 2.1. B4,4,4 is represented by generators θA , θB , .., θF and the following relations. θX 4 = ε,

(X = A, B, .., F ),

−1 −1 θB −1 θD θB = θA , θC −1 θE θC = θB , θA −1 θF θA = θC−1 , −1 , θF −1 θB θF = θE−1 , θD −1 θC θD = θF−1 , θE −1 θA θE = θD

Lemma 2.2. (1) The subgroup generated by r1 , r2 , and r3 in Isom(H3 ) is isomorphic to Z2 × Z2 × Z2 . (2) r1 , r2 , and r3 are normalizers of B4,4,4 . Lemma 2.3. The following relations hold r1

−1 θA = θ D ,

r1

−1 θB = θB ,

r1

θC = θC ,

r2

θA = θA ,

r2

θB = θE−1 ,

r2

θC = θC−1 ,

r3

−1 θA = θ A ,

r3

θB = θB ,

r3

θC = θF−1 ,

r1

θE = θE−1 ,

r2 r2

r1

r2

θD = θD ,

θD = θD ,

θF = θF ,

r1

θF = θF−1 ,

θE = θE .

2.2 Cell decomposition Let Fi be the set of the interiors of i-faces of Polyhedron R. Let F˜i := {γ · f : γ ∈ B4,4,4 , f ∈ Fi }. F˜i gives the regular cell decomposition of H3 . For a subgroup Γ of B4,4,4 ,

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the cell decomposition homeomorphically projects to that of Γ\H3 since any stabilizer of a cell ﬁxes the cell pointwise. Each i-cell in the decomposition of Γ\H3 is identiﬁed with ˜ i . If Γ is normal, G = B4,4,4 /Γ the Γ-orbit in F˜i . We denote by (Fi )Γ the set of Γ-orbits in F ˜ i )Γ in the usual manner. The following lemma is the concrete description of the acts (F G-action in terms of the labelling of the cells in Fig. 3.

D' C'

C

E

B'

B

E'

D

A

F'

F

A'

Fig. 3 Edges and Vertexes.

Lemma 2.4. Let Γ be a normal subgroup of B4,4,4 and G = B4,4,4 /Γ. ( denotes the isomorphism as G-set.) (0)

(F0 )Γ

= G(ΓQ) ∪ {G(ΓPx ); x = a, b, ..f } ,

G(ΓQ) G, (1)

(F1 )Γ

G(ΓPx ) G/ θX (x = a, b, .., f, X = A, B, .., F ).

= {G(Γxx ); x = a, b, ..f } ∪ {G(Γy); y = ab, bc, ca, de, ef, f d} ,

G(Γxx ) G/ θx (x = a, b, .., f ),

G(Γy) G (y = ab, bc, ca, de, ef, f d).

(2)

(F2 )Γ

= {G(ΓX); X = A, B, .., F } .G(ΓX) G (X = A, B, .., F )

(3)

(F3 )Γ

= G(ΓR) G.

Proof 2.5. The action of B4,4,4 decomposes F˜0 into the orbits B4,4,4 Q and B4,4,4 P x(x = a, b, .., f ). Decomposing these B4,4,4 -orbits into Γ-orbits, we have the ﬁrst half of the statement (0). Since the isotropy groups of Q and P x’s by B4,4,4 are {e} and θX , respectively, we have the second half of the statement (0). The others follows similarly.

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To describe the chain complex, we shall give an orientation of the cells as follows. In Fig. 3, each 1-cell is oriented by arrows, 2-cell X (X=A,B,..,F) is oriented by the outer ¯ 3-cell R ¯ is oriented by the canonical orientation of L0 . Since all stabilizers normal of R, of a cell is orientation preserving, the convention consistently orients cells in (F∗ )Γ for any subgroup Γ of B4,4,4 . We denote the G−chain complex by (C∗ , ∂∗ ) associated to the cell decomposition. If Γ is S0 -normal, the complex has the action of the semidirect product G S0 since the left action of S0 permutes the cells. By Lemma 2.4 and our orientation of cells, i-chain module Ci is described as G-module as follows. C0 C[G] · vQ ⊕ x C[G/ θX ] · vx := C0 ⊕ C0 C1 = x C[G/ θX ] · ex ⊕ y C[G] · ey := C1 ⊕ C1 C2 = x C[G] · sX , C3 = C[G] · cR . where the summation indexes varies as in Lemma 2.4 and v∗ , e∗ , s∗ , and c∗ are the oriented cells of the corresponding 0-,1-,2- and 3-cells. We also decompose C0 and C1 into two summands. In the following lemma we use the ”coordinate notions” according to this decomposition of the chain modules. Lemma 2.6. Suppose Γ is r1 -normal. Then the action of r1 on C∗ is described as follows. C0 α → r1 αθB ∈ C0 , C0 (αA , αB , .., αF ) −→ (r1 αA θB , r1 αB , r1 αF θA , r1 αD θE , r1 αE , r1 αC θD ) ∈ C0 , C1 (αa , αb , .., αf ) → (−r1 αd , r1 αb , −r1 αc , −r1 αa , r1 αe , −r1 αf ) ∈ C1 , C1 (αab , αbc , αca , αde , αef , αf d ) −→ (r1 αab θB , r1 αbc θB , r1 αf d θA θC , r1 αde θE , r1 αef θE , r1 αca θD θF ) ∈ C1 , C2 (αA , αB , .., αF ) −→ (r1 αD θA , r1 αB θB , −r1 αC , r1 αA θD , r1 αE θE , −r1 αF ) ∈ C2 , C3 α → −r1 α ∈ C3 . Moreover if ρ is a r1 -invariant irreducible representation of G these actions restrict to the homogeneous components of ρ. Proof 2.7. The statement follows from observing the group action in Fig. 2 with our convention of the cell orientation. Lemma 2.8. Suppose Γ is r1 r2 r3 -normal. The actions of r1 r2 r3 on six term modules C0 , C1 , C1 and C2 permute the components of pairs A ↔ D,B ↔ E and C ↔ F . The

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actions on C0 and C3 are given by C0 α → r1 r2 r3 αθE−1 θF−1 θA ∈ C0 ,

C3 α → −r1 r2 r3 α ∈ C3 .

If ρ is a r1 r2 r3 -invariant irreducible representation of G, these actions restrict to the homogeneous components of ρ. Proof 2.9. The statement for the six term modules can be observed from Fig. 2. The action on C0 is computed as r1 r2 r3 · αvQ = r1 r2 r3 αr2 r3 θB r3 θC θA . Hence the formula for C0 follows from Lemma 2.3. The formula for C3 and the last statement is immediate.

3

General principle and Character formulae

Notations For the irreducible representation ρ χρ denotes the corresponding irreducible character. By Irr(G) we denote the set of irreducible representations of G. For r ∈ Aut(G) (resp. subgroup S ⊂ Aut(G)) Irrr (G) (resp. IrrS (G)) denotes the set of r-invariant (resp. Sinvariant) irreducible representations. For G-module M and ρ ∈ Irr(G), Mρ denotes the homogeneous component of M associated to ρ. By calligraphic letter M we denote the corresponding character of G-module M . ρ∗ and χ∗ denote the cogridients of representation ρ and character χ, respectively. Res and Ind denote the restriction and the induction operator on the representations, respectively. ∗, ∗G denotes the usual inner product of class functions on ﬁnite group G.

3.1 General principle Let (C∗ , ∂∗ ) be the chain complex described in 2.2. Lemma 3.1. Suppose Γ is S0 -normal and of ﬁnite index. For any ρ¯ ∈ Irr(G S0 ) chain complex (C∗ , ∂∗ ) restricts to G S0 -subcomplex (C∗,¯ρ , ∂∗ |C∗,ρ¯ ) and H∗ (Γ\H3 , C)ρ¯ H∗ (C∗,¯ρ , ∂∗ |C∗,ρ¯ ). Proof 3.2. Since G S0 is ﬁnite, C[G S0 ]-modules are completely reducible. Then the lemma follows from the deﬁnition of homogeneous components. Lemma 3.3. Suppose Γ is normal and of ﬁnite index in B4,4,4 . For any ρ ∈ Irr(G), H1 (Γ\H3 , C)ρ H2 (Γ\H3 , C)ρ as G-module. Proof 3.4. Since Γ\H3 is a closed oriented 3-manifold and G acts as the orientation preserving isometries, the Poincare duality induces the isomorphism of G-module H1 (Γ\H3 , C)∗ H2 (Γ\H3 , C) as G-module. Hence (H1 (Γ\H3 , C)ρ )∗ H2 (Γ\H3 , C)ρ∗ .

(3.1)

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On the other hand, since the homology and the action of G is deﬁned over R, the homology admits the complex conjugate which commutes with the action of G. Hence the complex conjugate induces an isomorphism of R-vector spaces H2 (Γ\H3 , C)ρ∗ H2 (Γ\H3 , C)ρ .

(3.2)

Since the structure of homogeneous components as C[G]-module is determined by its dimension, the result follows from (3.1) and (3.2). If ρ ∈ IrrS0 G and M is a G S0 -module, then S0 stabilizes homogeneous component 0 Mρ of G-module ResGS M . Hence Mρ carries the action of G S0 and we denote by G ¯ Mρ the associated character of G S0 . Proposition 3.5. Suppose that Γ is S0 -normal and ρ ∈ IrrS0 (G) is nontrivial. Then ρ is an irreducible component of H1 (Γ\H3 , C) if the generalized character E¯ρ := (−1)i C¯i,ρ i

of G S0 is not trivial. Proof 3.6. Since Γ\H3 is connected and closed, the character Hi (Γ\H3 , C)ρ is trivial for i = 0, 3 and ρ = 1G . Hence the alternated sum E¯ρ is equal to the generalized character ¯ 2 (Γ\H3 , C)ρ − H ¯ 1 (Γ\H3 , C)ρ by Lemma 3.1. If the action of GS0 induces the nontrivial H character, either of H1 (Γ\H3 , C)ρ or H2 (Γ\H3 , C)ρ is nontrivial. Hence the proposition follows from Lemma 3.3

3.2 Character formulas Let r ∈ Aut(G) and ρ ∈ Irrr (G). For θ, g, h ∈ G with

hr

θ ∈ θ we set

ϕrθ (g, h)ρ : C[G/ θ]ρ α → g r αh−1 ∈ C[G/ θ]ρ , Tθr (g, h)ρ := Trace(ϕrθ (g, h)ρ ). We omit the superscript r when it is obvious and the subscript θ if θ = ε. Lemma 3.7. Let r be either r1 or r1 r2 r3 . Suppose Γ is r-normal and ρ ∈ Irrr (G) is nontrivial. Then, for r = r1 −1 )ρ E¯ρ (gr1 ) = −T r1 (g, θE−1 )ρ − T r1 (g, ε)ρ + TθrA1 (g, θB

+TθrD1 (g, θE−1 )ρ + TθrC1 (g, ε)ρ + TθrF1 (g, ε)ρ . and for r = r1 r2 r3 , −1 E¯ρ (gr1 r2 r3 ) = T r1 r2 r3 (g, θA θF θE )ρ − T r1 r2 r3 (g, ε)ρ .

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Proof 3.8. The Lemma follows immediately from the deﬁnition of E¯χ and the direct computation by the formulas in Lemma 2.6 and Lemma 2.8. We shall compute T∗∗ (∗, ∗). To do that we need the following general lemma. Lemma 3.9. Let K be a ﬁnite group. Suppose that r ∈ Aut(K) is of order two and ρ ∈ Irrr (K). Then there exist exactly two irreducible representations ρ¯ and :r ρ¯ of K r which restricts to ρ on K. These satisfy χρ¯(x) + χr ρ¯ (x) = 0 for x ∈ K r \ K. Kr

Kr

σ, ρ =

0 is a component of IndK ρ. Since Proof 3.10. σ ∈ Irr(K r) with ResK Kr ρ is r-invariant, Cliford’s theorem shows IndK ρ is reducible and decomposed into two irreducible representations which restricts to ρ on K. This proves the ﬁrst statement. Kr The second statement follows from ρ¯ + ρ¯ = IndK ρ. Lemma 3.11. Let r and ρ be as in Lemma 3.9 with K = G. Then T r (g, h) = χρ¯(gr)χ∗ρ¯(hr). Proof 3.12. It is immediately veriﬁed that the bi-action of G × G and the action of r on C[G]ρ induces the action of the semidirect product (G × G) r given by the r-action (g, h) → (r g, r h). We denote by σ the representation on C[G]ρ . Considering (G × G) r as a normal subgroup of (G r) × (G r) with index (G×G)r τ = ρ × ρ∗ by 2, we can deﬁne τ ∈ Irr((G × G) r) with ResG×G (Gr)×(Gr)

τ := Res(G×G)r

(¯ ρ × ρ¯∗ ).

Since C[G]ρ is equivalent to ρ × ρ∗ ∈ Irr(G × G), either σ = τ or σ = :τ in view of Lemma 3.9. We shall prove that σ = :τ . Recall that C[G]ρ is a simple component of C-algebra C[G]. Since the action of r induces a C-algebra automorphism of C[G] together with the conjugation by elements of G, the idempotent associated to r-invariant representation ρ is ﬁxed by these actions. Hence we have (G×G)r

σ, 1H H = 0

ResH

where H is the diagonal subgroup in (G r) × (G r). Since 1 1 1 2 2 2

¯ ρ, ρ¯ = |χρ (x)| + |χρ¯(xr)| = |χρ¯(xr)| ,

ρ, ρ + 2|G| x∈G 2 |G| x∈G x∈G we have |G| =

x∈G

|χρ¯(xr)|2 . By deﬁnition of :τ ,

(G×G)r

ResH =

:τ, 1H H =

|χρ (x)|2 − x∈G |χρ¯(xr)|2 1 1 2

ρ, ρ = 0. − |χ (xr)| G ρ ¯ x∈G 2 |G| 1 |H|

x∈G

Hence by (3.3) σ = :τ . Thus σ = τ . In particular, this proves the lemma.

(3.3)

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For θ ∈ G, ρ ∈ Irr(G) and ω ∈ C we set M θ (ω) := {α ∈ C[G]; α · θ = ωα} , Mρθ (ω) := {α ∈ C[G]ρ ; α · θ = ωα} . If g, h ∈ G, r ∈ Aut(G) with hr θ ∈ θ and ρ is r-invariant, we deﬁne isomorphism ψθr (g, h)ρ by ψθr (g, h)ρ : Mρθ α → g r αh−1 ∈ Mρθ . Lemma 3.13. For any θ ∈ G and irreducible representation ρ of G, the projection π : C[G] → C[G/ θ] induces the isomorphism of G-modules π|Mρθ (1) : Mρθ (1) C[G/ θ]ρ . If g, h ∈ G, r ∈ Aut(G) with π|Mρθ ◦ ψθr (g, h)

hr

θ ∈ θ and ρ is r-invariant, we have ϕrθ (g, h) ◦ π|Mρθ =

Proof 3.14. Let n be the order of θ and ω an n-th root of unity. Choosing representative i ¯θi . Clearly {vω,x }x∈G/θ forms x¯ from each coset x θ, we deﬁne vω,x by vω,x := n−1 i=o ω x a basis of M θ (ω). Verifying π(vω,x ) = 0 if ω = 1, it is veriﬁed that dim M θ (1) = :(G/ θ) = dim C[G/ θ] and π induces an isomorphism of M θ (1) onto C[G/ θ]. Since π is G-morphism, this restricts to the homogeneous components. This proves the ﬁrst statement. The second statement follows immediately from the deﬁnition of ϕ’s and ψ’s. Lemma 3.15. Let r and ρ be as in Lemma 3.7 and θ, g, h ∈ G with

hr

θ ∈ θ. Then

1 = χρ¯(gr) χ∗ρ¯(θi hr) n i=0 n−1

Tθr (g, h) where n is the order of θ.

Proof 3.16. In view of Lemma 3.13 we shall compute the trace of action of (g, h)r ∈ (G × G) r on Mρθ . Let H be a subgroup of (G × G) r generated by a := (g, h)r ¯ and b := (1, θ). By assumption we have ab = (1, hr θ)(g, h)r = bδ a for some δ ∈ Z. Let H denote the semidirect product b a with the action of a on b given by b → bδ . Then ¯ by the natural epimorphism, say ξ. Cliﬀord’s theorem H is a homomorphic image of H ¯ gives the following isomorphism as H-module, (G×G)r (G×G)r Mθ,ρ

ξ ∗ ResH σ, lH¯ l ξ ∗ ResH l

¯ which factors through H ¯ → where l runs over all the irreducible representations of H ¯ H/ b a and σ denotes the representation of (G × G) r given by C[G]ρ . Since l’s are all linear, we can compute the character explicitly as follows. (G×G)r Tθr (g, h) = l p∗ ResH C[G]ρ , lH¯ l(a) √ 1 2π m i j i = |H| ¯ ¯ i,j σ(p(b a )) k exp( m k(1 − j) −1) = |H| i σ(p(b a)) where m is the order of a. Then the lemma follows from Lemma 3.11.

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3.3 Proof of Theorem 1.7 Theorem 3.17. Let Γ be an r1 -normal subgroup of ﬁnite index in B4,4,4 and ρ a nontrivial r1 -invariant irreducible representation of G = B4,4,4 /Γ. If −χρ¯(θE−1 r)

3 1 i −1 i −1 χρ¯(θA − χρ¯(r) + θB r) + χρ¯(θD θE r) + χρ¯(θCi r) + χρ¯(θFi r) = 0 4 i=0

for an irreducible representation ρ¯ of G r1 which restricts to ρ on G, then ρ is an irreducible component of G-module H1 (Γ\H3 , C). Proof 3.18. Observe that if θ∗ degenerates in G to an element of order 2 or 1, the character summation formula in Lemma 3.15 still remains valid with n = 4 since it makes the summation twice or four times while the multiple is divided out by the factor 1/4. Computing E¯ρ (gr) by Lemma 3.7, Lemma 3.11 and Lemma 3.15, we have E¯ρ (gr) = χρ¯(gr)L where L is the left hand side of the formula in the statement. Suppose χρ¯(gr) = 0 for any g ∈ G. This implies χρ¯ = χ¯ρ by the second statement of Lemma 3.9, hence ρ¯ = :r1 ρ¯. This contradicts to the ﬁrst statement of the lemma. Hence χρ¯(gr) = 0 for some g ∈ G. Then the result follows from Proposition 3.5. Theorem 3.19. Let Γ be a r1 r2 r3 -normal subgroup of ﬁnite index in B4,4,4 and ρ a nontrivial r1 r2 r3 -invariant irreducible representation of G. If −1 θF θE r1 r2 r3 ) + χρ¯(r1 r2 r3 ) = 0 χρ¯(θA

for an irreducible representation ρ¯ of G r1 r2 r3 which restricts to ρ on G, ρ is an irreducible component of G-module H1 (Γ\H3 , C). Proof 3.20. Computing E¯ρ (gr) by Lemma 3.7 and Lemma 3.11 with r = r1 r2 r3 and applying the same argument to Theorem 3.17 we obtain the theorem.

4

Arithmetic lattices and congruence subgroups

4.1 General notions Let F be a ﬁeld of characteristic = 2 and f a non-degenerate quadratic form on F 4 . Let Of (F ), SOf (F ) denotes the F − rational points in the orthogonal group and the special orthogonal group. For ξ ∈ F 4 with f (ξ) = 0 we denote by rξ ∈ Of (F ) the orthogonal reﬂection Spinorial norm Spf is the unique homomorphism of Of (F ) to F ∗ /(F ∗ )2 which takes reﬂection rξ to f (ξ) mod (F ∗ )2 . Let Ωf (F ) = SOf (F ) ∩ ker Spf .

4.2 Arithmetic lattices Let k be a number ﬁeld, o the ring of integers in k and f a non-degenerate quadratic form on k 4 . Let Of (o) denote the subgroup of Of (k) formed by elements all of whose entries

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are integers. Suppose that v is a real inﬁnite place of k and f induces quadratic form fv at v of type (p, q). Then we have an associated embedding λv of Of (k) into O(p, q : R). In particular, if (p, q) = (3, 1), ker Spf and Ωf (o) embed into ker Sp3,1 (R) = Isom(H3 ) and Ω(3, 1 : R) = Isom0 (H3 ) respectively. The following is derived from the classical theorem due to Siegel. Theorem 4.1. (Siegel) Suppose k = Q is a totally real number ﬁeld and f is a nondegenerate anisotropic quadratic form on k 4 . If f is deﬁnite at all inﬁnite places except for v0 and of type (3,1) at v0 , Γv0 := λv0 (Of (o)) ∩ SO(3, 1 : R) is a cocompact Kleinian group. We say that Γ is an arithmetic lattice of O(3, 1 : R) if Γ is commensurable with Γv0 in Theorem above. Lemma 4.2. Let rA , rB and rC be the reﬂection with respect to plane spanned by Side A, Side B and Side C in Fig. 1. Then (i) Spf (rA ) = Spf (rB ) = Spf (rC ) = 2a−1 (ii) θA = r3 rA , θB = r1 rB , θC = r2 rC . Proof 4.3. See [6] Section 3 for (i). For (ii) observe Fig. 1 or 2. √ Lemma 4.4. B4,4,4 is arithmetic with k = Q( 5), f = a−1 (x2 + y 2 + z 2 ) − t2 and v0 being the embedding which maps a to the positive conjugate. Proof 4.5. Follows from cocompactness, Lemma 2.1 and Lemma 4.2.

4.3 Congruence subgroups For ideal m of o we deﬁne congruence subgroup Of (m) by Of (m) := {g ∈ Of (o); g ≡ 1 mod m} . Clearly Of (m) is an normal subgroup of Of (o). Set Γv0 := λv0 (Of (o) ∩ Ωf (k)), Γm := λv0 (Of (m)) ∩ Ωf0 (R), Γm := λv0 (Of (m) ∩ Ωf (k) := Γv0 ∩ Γm. Note that Γv0 and Γm are of ﬁnite index in Γv0 and Γm, respectively since those groups are ﬁnitely generated by its cocompactness and the spinorial norm maps those groups to the abelian group any non-trivial element of which are of order two. Suppose that p is prime. Reducing the entries modulo p we have injections ιp : Γv0 /Γp −→ SOfp (o/p)

ιp : Γv0 /Γp −→ Ωfp (o/p)

where fp denotes the quadratic form reduced from f modulo p.

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Lemma 4.6. Let Γ be a subgroup of ﬁnite index in Γv0 . Then there exists ﬁnite set PΓ of prime ideals of o containing all prime ideals dividing 2 such that for prime ideal p ∈ PΓ the injection ιp restricts to an isomorphism of Γv0 ∩ Γ/Γp ∩ Γ to Ωfp (o/p). Proof 4.7. The statement follows from Kneser’s strong approximation theorem for simply connected group Ωf . (See [10]; Proposition 3.2.), Lemma 4.8. Let k, f and v0 be as in Lemma 4.4. Suppose p ∈ PB4,4,4 . (1) ιp(B4,4,4 ) = Ωfp (o/p) if and only if |o/p| ≡ ±1 mod 8. (2) ιp(B4,4,4 ) = SOfp (o/p) if and only if |o/p| ≡ ±3 mod 8. Proof 4.9. Let Fp := o/p. Since |SOfp (Fp)/Ωfp (Fp)| = |Fp∗ /(Fp∗ )2 | = 2, ιp(B4,4,4 ) = Ωfp or SOfp by Lemma 4.6. Thus (2) follows from (1). By Lemma 4.6(i) Spf (θA ) = 2a−2 . Hence we have Spfp (θA ) = 2a−2 = 2 mod(Fp∗ )2 and similar formula for other generator θ∗ ’s. (1) follows from the classical fact that 2 is a square in Fp if and only if |Fp| ≡ ±1 mod 8. Lemma 4.10. Let k, p and f be as in Lemma 4.8. Quadratic form fp belongs to the (unique) cogridient class of isotropic quadratic forms or that of anisotropic ones according to −a is a square in Fp := o/p or not. Proof 4.11. The discriminant of fp is −a3 = −a mod(Fp∗ )2 . The lemma follows from the well known fact that the cogridient class of quadratic forms over ﬁnite ﬁeld are classiﬁed by its discriminant.

5

Isotropic case

In Section 5 and Section 6 f always denotes the quadratic form a−1 (x2 + y 2 + z 2 ) − t2 √ over k = Q( 5) and we assume that p ∈ PB4,4,4 . Throughout this section we assume that √ −a = −(1 + 5)/2 is a square in Fp.

5.1 Structure of the split 4-orthogonal group Suppose q is a power of an odd rational prime. Let fq,0 be the isotropic quadratic form 4 over V : = i=1 Fq · ui deﬁned by fq,0 : = u1 u2 + u3 u4 . Let Vij denote the subspace Fq · ui ⊕ Fq · uj . Deﬁne the totally isotropic subspaces of V by V1 : = V12 , V2 : = V34 , W1 : = V14 , W2 : = V23 and the mutually orthogonal hyperbolic subspaces by H1 : = V13 , H2 : = V24 . Let S1 denote the subgroups of Ofq,0 (Fq ) formed by elements which stabilize V1 and V2 . Similarly deﬁne the subgroup S2 for Wi in place of Vi . Obvious computation veriﬁes that the restrictions ResV1 : S1 g → g|V1 ∈ SL2 (V1 ), ResW1 : S1 g → g|W1 ∈ SL2 (W1 ) induce isomorphisms. Hence we have

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Lemma 5.1. S1 ∩S2 coincides with the center of S1 and S2 . The sequence below is exact. 1 −→ S1 ∩ S2 −→ S1 × S2 −→ Ωfq,0 (Fq ) −→ 1 In particular Ωfq,0 (Fq ) SL2 (Fq ) × SL2 (Fq )/ ± (I, I). Let gd be an element of GL2 (Fq ) with det gd ∈ (F∗q )2 . We denote by the element of Out(SL2 (Fq )) represented by the automorphism S → S := gd t S −1 gd−1 of SL2 (Fq ). The following two lemmas follows from the direct computation. Lemma 5.2. SOfq,0 (Fq ) stabilizes S1 and S2 . The action of SOfq,0 (Fq )/Ωfq,0 (Fq ) is generated by the outer automorphism on S1 and S2 identiﬁed with SL2 (Fq ) in the manner of Lemma 1 and the canonical coordinates (u1 , u2 ) or (u1 , u4 ). Lemma 5.3. Elements of Ofq,0 (Fq ) \ SOfq,0 (Fq ) permute S1 and S2 .

5.2 Description of irreducible representations Let q = |Fp|. Then by Lemma 4.10, there exists isomorphism µ of Lp := Fp x ⊕ Fp y ⊕ Fp z ⊕ Fp t onto V in 5.1 with µ∗ fq,0 = fp. Identifying S1 ⊂ Ωfq,0 with SL2 (Fq ) and µ the canonical coordinate (u1 , u2 ), we have an inclusion i1 : SL2 (Fq ) J→ Ωfq,0 →→ Ωfp . Similarly an inclusion i2 := r1 ◦ i1 of SL2 (Fq ) into Ωfp factors through S2 by Lemma 5.3. We denote by p the projection SL2 (Fq ) × SL2 (Fq ) → Ωfp induced from the identiﬁcation and Lemma 5.1.1. Then the following lemma is an immediate consequence of 5.1. Lemma 5.4. (i)The pull back by p induces a one-to-one corresponding of Irr(Ωfp ) with I := {(ρ1 , ρ2 ) ∈ Irr(SL2 (Fq )) × Irr(SL2 (Fq )); ρ1 (−I) × ρ2 (−I) = id} . (ii) r1 acts on I by the permutation of the components. In particular, r1 -invariant irreducible representations of Ωfp correspond to elements of the form (ρ1 , ρ1 ) ∈ I. Lemma 5.5. The action of the non-trivial element of SOfp /Ωfp on Irr(Ωfp ) induces the action on I deﬁned by (ρ1 , ρ2 ) → (ρ1 , ρ2 ) where ∈ Out(SL2 (Fq )) deﬁned in 5.1. Proof 5.6. The identiﬁcation of S2 with SL2 (Fq ) is done by r1 in this subsection and by r0 in 5.1. However by deﬁnition this modiﬁcation does not change the outer automorphism. Then the statement is a consequence of Lemma 5.2. Lemma 5.7. There exists an element of order two in SOfp \ Ωfp denoted by t such that t commutes with r1 and tr1 is a reﬂection. Proof 5.8. Let W be the three dimensional subspace of Lp ﬁxed by reﬂection r1 . Since W is a three dimensional, non-degenerate metric space over Fq , W is of Witt index 1. Hence uniqueness of the Witt index shows that there are mutually orthogonal hyperbolic subspaces H and H such that H ⊂ W and r1 |H is a reﬂection. Choosing reﬂection r1 of

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H with Spfp (r1 )Spfp (r1 ) = 1, we have t = r1 r2 ∈ SOp \ Ωfp of the required property. Fix t1 ∈ SOfp \ Ωp with the properties in Lemma 5.7. Lemma 5.2 and Lemma 5.7 shows that t1 induces the same action on SL2 (Fq ) through i1 and i2 . With the common

2 (Fq ) := SL2 (Fq ) t1 . We also deﬁne the projection by action we let SL pˆ : (SL2 (Fq ) × SL2 (Fq )) t1 (s1 , s2 )δ → i1 (s1 )i2 (s2 )δ ∈ SOfp where t1 acts on SL2 (Fq ) × SL2 (Fq ) by (s1 , s2 ) → (t1 s1 , t1 s2 ) and δ = ε or t1 . We consider

2 (Fq ) × SL

2 (Fq ) via (SL2 (Fq ) × SL2 (Fq )) t1 as a normal subgroup of index two in SL inclusion ˆi : (s1 , s2 )δ → (s1 δ, s2 δ). SOf

ρ ∈ Irr(SOfp ); ResΩfp p ρˆ ∈ Irr(Ωfp )}. The pullback by pˆ Lemma 5.9. (i) Let M1 := {ˆ induces the correspondence of M1 onto 2 2 SL ˆ := (ˆ

2 (Fq )) × Irr(SL

2 (Fq )); (ResSL I ρ1 , ρˆ2 ) ∈ Irr(SL ρ ˆ , Res ρ ˆ ) ∈ I . SL2 1 SL2 2 The inverse image of ρˆ ∈ M1 by the correspondence consists of two elements (ˆ ρ1 , ρˆ2 ) and (:t1 ρˆ1 , :t1 ρˆ2 ) in I. (See 3.9 forthe notion :∗ ). SOfp r1 r1 (ii) I1 := M1 ∩ Irr (SOfp ) = ρˆ ∈ M1 ; ResΩfp ρˆ ∈ Irr (Ωfp ) (SL ×SL )t

Proof 5.10. Suppose ρˆ ∈ M1 . By deﬁnition ResSL22×SL22 1 pˆ∗ ρˆ is irreducible, hence coincides with ρ1 × ρ2 for some ρ1 , ρ2 ∈ Irr(SL2 (Fq )). By the Mackey’s criterion and Lemma 5.2, both ρ1 and ρ2 are t1 -invariant. Thus by Lemma 3.9, we can ﬁnd ρˆ1 and ρˆ2 2 (Fq )) which restrict to ρ1 and ρ2 , respectively. Again by Lemma 3.2.2 one of in Irr(SL 2 ×SL 2 2 SL2 ×SL Res(SL (ˆ ρ1 × ρˆ2 ) or ResSL ρ1 ×:t1 ρˆ2 ) coincides with ρˆ. This shows that (SL2 ×SL2 )t1 (ˆ 2 ×SL2 )t1 2 × SL 2 and (i) follows from Lemma 3.9. the pullback pˆ∗ ρˆ extends to a character of SL SOf

For (ii), ﬁrst note that if ρˆ is r1 -invariant, obviously so is ResΩfp p ρˆ. Suppose conversely SOf

SOf

that ResΩfp p ρˆ is r1 -invariant. Then p∗ ResΩfp p ρˆ = ρ1 × ρ1 for some ρ1 ∈ SL2 (Fq )

2 (Fq )) such that ResSL2 ρˆ1 = ρ1 and by Lemma 1(ii). Then there exists ρˆ1 ∈ Irr(SL SL2 χρˆ(i1 (s1 )i2 (s2 )t) = ±χρˆ1 (s1 t)χρˆ1 (s2 t). Hence Lemma 3 and the deﬁnition of i2 show that r1 χρˆ = χρˆ. This veriﬁes that ρˆ is r1 -invariant. Lemma 5.11. The set of r1 -invariant characters of Irr(SOfp ) is the disjoint union of I1 and the following subsets. SOfp r1 t1 I2 := IndΩfp ρ; ρ ∈ Irr (Ωfp ), ρ ∈ Irr (Ωfp ) , SOf I3 := IndΩfp p ρ; ρ ∈ Irrr1 t1 (Ωfp ), ρ ∈ Irrt1 (Ωfp ) . Proof 5.12. Mackey’s criterion shows that the irreducible characters in Irr(SOfp ) \ M1 are induced from non t1 -invariant irreducible characters of Ωfp . Their r1 -invariance is SOf

SOf

clearly equivalent to that of its restriction to Ωfp . Since ResΩfp p IndΩfp p ρ = ρ + t1 ρ, the

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condition is equivalent to r1 ρ = ρ or r1 t1 ρ = ρ. Since t1 ρ = ρ, these two cases are exclusive. This proves the lemma. Lemma 5.13. The action of r1 r2 r3 on irreducible representations of Ωfp and SOfp is the same as that of r1 . In particular, r1 r2 r3 -invariance of an irreducible representation of G is equivalent to its r1 -invariance. Proof 5.14. Follows from that r2 r3 induces the inner automorphism of G.

5.3 Conjugacy classes By T rk (g) we denote the trace of endmorphism g of L and by T rp(g) the trace of endmorphism g of Lp. 2 ) = 0 for X = A, B, .., F . Lemma 5.15. (i) T rk (θX ) = 2, T rk (θX −1 −1 (ii) T rk (r1 ) = T rk (θE r1 ) = 2. (iii) T rk (r1 r2 r3 ) = T rk (θA θF θE r1 r2 r3 ) = −2.

Proof 5.16. Since θX is elliptic with its axis on xx the action of θX on L ﬁxes twodimensional subspace F spanned by xx . Since f is anisotropic over k, (F, f |F ) is nondegenerate. Hence θX stabilize F and F ⊥ . Since θX is of order four and of determinant 2 is similar. (ii) is proved similarly with 1, T rk (θ|F ⊥ ) = 0 and (i) follows. The proof for θX −1 −1 the determinants of r1 and θE r1 being -1 in this case. Since θA θF θE r1 r2 r3 = rA rB rC by Lemma 2.8 and Lemma 4.2, the element (and r1 r2 r3 ) is a composition of three reﬂections by mutually orthogonal vectors and (iii) follows. Recall that the semisimple conjugacy classes of s ∈ SL2 (Fq ) is determined uniquely by trace T rFq s as an endmorphism of F2q . By Tα , we denote the semisimple conjugacy ¯q. class represented by diagonal matrix tα with entries (α, α−1 ) over algebraic closure F We denote by T4 , only one class of order four and by ±T8 , two classes of order eight. The non-semisimple conjugacy classes are represented by ±u1 and ±ud where ux is the unipotent uppertriangular matrix with the oﬀ-diagonal component x and d is a nonsquare of F∗q . We denote these conjugacy classes by ±U1 and ±Ud , respectively. The following trace formula follows from the direct computation. Lemma 5.17. T rFq (i1 (s1 )i2 (s2 )) = T rFq (s1 )T rFq (s2 ) for s1 , s2 ∈ SL2 (Fq ). Lemma 5.18. Let p be the projection in Lemma 5.2.1 and X be either A, B, .. or F . 2 = p(s1 , s2 ) for some s1 , s2 ∈ T4 . (i) θX (ii) If q ≡ ±1 mod 8 θX = p(s1 , s2 ) for some s1 , s2 ∈ T8 . 2 is nontrivial, hence of order two. This and Lemma Proof 5.19. By Lemma 5.15 (ii) θX −1 2 5.1 show that p (θX ) is of the form (s1 , s2 ) ∈ T4 × T4 or (s1 , s2 ) = ±(I, −I). The comparison of the trace by Lemma 1 and Lemma 2 exclude the latter. This proves (i).

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(ii) is similarly proved. Lemma 5.20. (i) Element i1 (s1 )i2 (s2 )r1 ∈ Ωfp r1 is conjugate in Ωfp r1 to i1 (s)r1 for any element s ∈ SL2 (Fq ) which is conjugate to s1 s2 . (ii) Element i1 (s1 )i2 (s2 )r1 t1 ∈ Ofp is conjugate to i1 (s)r1 t1 in Ofp for any element s ∈ SL2 (Fq ) which is conjugate to s1 t1 s2 . Proof 5.21. By Lemma 5.7 and the deﬁnition of i2 , i1 (s1 )i2 (s2 )r1 is conjugated by i2 (s2 )−1 to i1 (s1 s2 )r1 . Similarly i1 (g)i2 (g) conjugates i1 (s1 s2 )r1 to i1 (g s1 g s2 )r1 . This proves (i). For (ii) replace the latter conjugation by i1 (g)i2 (t1 g). In the sequel we denote ”a ∼ b in G” if a, b ∈ G are conjugate by an element of group G. Lemma 5.22. Suppose that q ≡ ±1 mod 8. Following conjugacy relations holds in Ωfp r1 . (i) θE−1 r1 ∼ r1 ,

i −1 i −1 (ii)θCi r1 ∼ θFi r1 ∼ θA θ B r 1 ∼ θD θE r1 ∼ i1 (T4 )r1 (i = 1, 2, 3)

Proof 5.23. Set θE = i1 (s1 )i2 (s2 ). Since r1 θE = θE−1 by Lemma 2.8 s1 s2 = ±I. The comparison of the trace by Lemma 5.17 and Lemma 5.18(ii) excludes the minus signature and the ﬁrst relation follows from Lemma 5.20. (ii) is similarly proved, using the relations in Lemma 2.4 and Lemma 2.8. The following lemma can be proved similarly. Lemma 5.24. Suppose that q ≡ ±3 mod 8. The following conjugacy relations hold in Ofp . (i)

θE−1 r1 ∼ r1 t1 (ii) θC±1 r1 ∼ θF±1 r1 ∼ i1 (T4 )r1 t1 (iii) θC2 r1 ∼ θF2 r1 ∼ i1 (−I)r1

±1 −1 ±1 −1 2 −1 2 −1 (iv) θA θ B r 1 ∼ θD θE r1 ∼ i1 (T4 )r1 (v) θA θ B r 1 ∼ θD θE r1 ∼ i1 (−I)r1 t1 .

Since r1 r2 r3 acts on G as the conjugation of the corresponding element of Ofp , we may identify G r1 r2 r3 with G r1 by g · r1 r2 r3 → gr2 r3 · r1 . In the sequel we shall adopt the identiﬁcation of G r1 r2 r3 with G r1 −1 θF θE r1 r2 r3 are Lemma 5.25. (i) Suppose that q ≡ ±1 mod 8. Then r1 r2 r3 and θA conjugate to i1 (−I)r1 in Ωfp r1 . (ii) Suppose that q ≡ ±3 mod 8. Then r1 r2 r3 is conjugate to i1 (−I)r1 and −1 θA θF θE r1 r2 r3 is conjugate to i1 (−I)r1 t1 in Ofp .

Proof 5.26. Observe that r2 r3 is of trivial spinorial norm and commutes with r1 and the −1 −1 relation θA θ F θ E r1 r2 r3 θ A θF θE r1 r2 r3 = ε . Then the argument similar to Lemma 5.22 proves the statement by Lemma 5.15 (iii).

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5.4 Characters of ρ¯ Lemma 5.27. (i) Suppose ρ ∈ Irrr1 (Ωfp ) corresponds to (ρ1 , ρ1 ) ∈ I (cf. Lemma 5.4). Two representations ρ¯± of Ωfp r1 which restrict to ρ are given by the formula χρ¯± (i1 (s1 )i2 (s2 )r1 ) = ±χρ1 (s1 s2 ). (ii) Suppose ρ ∈ Irrr1 t1 (Ωfp ) corresponds to (ρ1 , t1 ρ1 ) ∈ I. Two representations ρˇ± of Ωfp r1 t1 which restrict to ρ are given by the formula χρˇ± (i1 (s1 )i2 (s2 )r1 t1 ) = ±χρ1 (s1 t1 s2 ). Proof 5.28. Let V be a virtual representation space of ρ1 . Deﬁne the action of r1 on V ⊗ V by v1 ⊗ v2 → v2 ⊗ v1 . Then this induces the action of (SL2 (Fq ) × SL2 (Fq )) r1 with r1 acting on SL2 (Fq ) × SL2 (Fq ) by (g, h) → (h, g). Hence projecting the action, we have irreducible representation ρ¯+ of Ωfp r1 . To compute the character of ρ¯+ on i1 (s1 )i2 (s2 )r1 , we may conjugate the element to i1 (s1 s2 )r1 by Lemma 5.20. The formula in (i) follows from the direct computation of the trace and Lemma 3.9. For (ii) we deﬁne the action of (g, h) ∈ SL2 × SL2 on V ⊗ V by v1 ⊗ v2 → gv1 ⊗ t1 hv2 and let r1 t1 act on it by the permutation of the components. Then the argument of (i) applies. SOf

Lemma 5.29. Suppose ρˆ ∈ I1 . (cf. Lemma 5.9) Let ρ = ResΩfp p ρˆ and notations ρ¯± , ρˇ± ˆ two irreducible representations ρ¯ˆ± be as in Lemma 5.27. If ρˆ corresponds to (ˆ ρ1 , ρˆ1 ) ∈ I, of Ofp restricting to ρˆ veriﬁes Of ResΩfp r ρ¯ˆ± = ρ¯± , p

1

Of

ResΩfp r p

1 t1

ρ¯ˆ± = ρˇ± .

ˆ Similarly if ρˆ corresponds to (ˆ ρ1 , :t1 ρˆ1 ) ∈ I, Of ResΩfp r ρ¯ˆ± = ρ¯± , p

1

Of

ResΩfp r p

1 t

ρ¯ˆ± = ρˇ∓ .

ˆ 2 we Proof 5.30. Suppose ﬁrst that ρˆ corresponds to (ˆ ρ1 , ρˆ1 ). Replacing SL2 by SL can construct irreducible representation ρ¯ˆ+ as in Lemma 5.27. Then the ﬁrst formula is immediate. To prove the second compute the trace as in Lemma 5.27 and we have the ”right signature” by χρ¯ˆ+ (t1 ) = χρˆ1 (I) > 0. Then the second formula follows immediately. For the other two formulas we modify V ⊗ V to be the virtual representation space of ρˆ1 × :t1 ρˆ1 , giving the opposite signature to the action of t. Then the formula follows similarly. Lemma 5.31. Suppose that ρ ∈ Irr(Ωfp ) is not t1 -invariant. Of

Of

SOf

(i) If ρ is r1 -invariant, ResSOpfp IndΩfp r1 ρ¯± restricts to IndΩfp p ρ ∈ I2 . (ii) If ρ is r1 t1 -invariant,

Of ResSOpfp

p

Of

SOf

IndΩfp r1 ρˇ± restricts to IndΩfp p ρ ∈ I3 . p

Proof 5.32. By the associativity of induced modules, Of

Of

SOf

ResSOpfp IndΩfp r1 ρ¯+ , IndΩfp p ρSOfp = 0. p

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M. Toda / Central European Journal of Mathematics 2(2) 2004 218–249 Of

Of

SOf

Hence ResSOpfp IndΩfp r1 ρ¯+ contains IndΩfp p ρ as its irreducible component. p Since these representations have the same virtual dimension 2χρ (ε), they coincide. This proves (i). (ii) is similarly proved.

5.5 Computation For irreducible representation ρ of G set Nρ := − χρ¯(θE−1 r1 ) − χρ¯(r1 ) i −1 i −1 θB r1 ) + χρ¯(θD θE r1 ) + χρ¯(θCi r1 ) + χρ¯(θFi r1 ) + 14 3i=0 χρ¯(θA −1 θF θE r2 r3 r1 ) Mρ : = χρ¯(r2 r3 r1 ) + χρ¯(θA where ρ¯ is an irreducible representation of G r1 which restricts to ρ. Proposition 5.33. Suppose that q ≡ ±1 mod 8 and ρ is a r1 -invariant irreducible representation of G = Ωfp with p∗ ρ = (ρ1 , ρ1 ) ∈ I. Then Nρ = |−χρ1 (I) + χρ1 (−I) + 2χρ1 (T4 )| , Mρ = 2χρ1 (I). Proof 5.34. Applying Lemma 5.22 and Lemma 5.27(i), we obtain the ﬁrst formula by the direct computation. Similarly the second formula follows from Lemma 5.25(i) and Lemma 5.27(i).

1SL2

St

Hn

Dm

G± 1

G± 2

I

1

q

q+1

q−1

q+1 2

q−1 2

−I

1

q

(−1)n (q + 1)

(−1)m (q − 1)

(−1)ν q+1 2

U1

1

0

1

−1

κ±

λ±

Ud

1

0

1

−1

κ∓

λ∓

−U1

1

0

(−1)n

−(−1)m

(−1)ν κ±

(−1)ν λ±

−Ud

1

0

(−1)n

−(−1)m

(−1)ν κ∓

(−1)ν λ∓

T αi

1

1

−ni eni q−1 + eq−1

0

(−1)i

0

0

−(−1)j

Tβ j

1

−1

0

−(emj q+1

+

e−mj q+1 )

√

i = 0 ∈ Z/(q − 1)Z, j = 0 ∈ Z/(q + 1)Z, ek := exp( 2π k −1 ), ν := √ √ 1± (−1)ν q −1± (−1)ν q ± ± , λ := . κ := 2 2

(−1)ν

q−1 2

q−1 2 ,

ν :=

q+1 2 ,

Table 1 The character table of SL2 (Fq ).

Computation I In Table 1 α denotes a generator of F∗q and β denotes a element of F∗q2 of order q + 1 with β + β −1 ∈ Fq . If q ≡ 1 mod 4, T4 is rationally diagonalized, hence

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239

coincides with T q−1 . If q ≡ −1 mod 4, it is a nonsplit semisimple class, hence coincides α 4 q+1 with T 4 . According to this dichotomy we can compute Nρ based on Proposition β 5.33.(See Table 2) ρ1

q ≡ 1 mod 4

q ≡ −1 mod 4

St

2

2

Hn (n even)

4

0

2(q+1)

2(q+1)

0

4

2(q-1)

2(q-1)

G± 1

2

q+1

G± 2

q-1

2

(n odd) Dm (m even) (m odd)

Table 2 The value of Nρ .

Proposition 5.35. Suppose that q ≡ ±3 mod 8 and ρˆ is a r1 -invariant irreducible representation of G = SOfp . ˆ then ρ1 , ρˆ1 ) ∈ I, (i) If ρˆ ∈ I1 and pˆ∗ ρˆ extends to (ˆ Nρˆ = −χρ1 (I) + χρ1 (−I) + 2χρ1 (T4 ). Mρˆ = 2χρˆ(I) ˆ then Nρˆ = Mρˆ = 0. (iii) If ρˆ = IndSOfp ρ ∈ ρ1 , :t1 ρˆ1 ) ∈ I, (ii) If ρˆ ∈ I1 and pˆ∗ ρˆ extends to (ˆ Ωfp SOf

I2 for ρ ∈ Irr(Ωfp ) with p∗ ρ = (ρ1 , ρ1 ) ∈ I or if ρˆ = IndΩfp p ρ ∈ I3 for ρ ∈ Irr(Ωfp ) with p∗ ρ = (ρ1 , t1 ρ1 ) ∈ I, then Nρˆ = −χρ1 (I) + χρ1 (−I) + 2χρ1 (T4 ), Mρˆ = 2χρ1 (I). Proof 5.36. (i) and (ii) are consequences of Lemma 5.24, Lemma 5.25 and Lemma 5.29. (iii) and (iv) follow from Lemma 5.24, Lemma 5.25 and Lemma 5.31. Computation II The action of outer automorphism permutes the conjugacy classes ±U1 and ±Ud and ﬁxes all the semisimple classes. Hence by Lemma 5.5, r1 , t1 -invariant irreducible representations of Ωfp correspond to (ρ1 , ρ1 ) with ρ1 = St, Hn ’s and Dm ’s according to Table 1. These representation extend to the irreducible representations in I1 , (ρ1 , ρ1 ) or (ρ1 , :t1 ρ1 ). For the former case, the result in Table 2 for the corresponding ρ1 is carried over by Proposition 5.35(i). For the latter case, the formula automatically − gives zero. It follows from Table 1 that the action of t1 permutes G+ i and Gi . Hence SOf + the irreducible representations in I2 are IndΩfp p p∗ (G+ i × Gi ) (i = 1, 2). Those in I3 are SOf

− IndΩfp p p∗ (G+ i × Gi ) (i = 1, 2) Again the formulas in Proposition 5.35(iii) show that the result in Table 2 for the corresponding ρ1 is carried over.

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Theorem 5.37. Suppose that p ∈ PB4,4,4 . Nontrivial r1 -invariant irreducible representation ρ of G is an irreducible component of H1 (Γp\H3 , C), unless q ≡ ±3 mod 8 and ˆ ρ ∈ I1 corresponds to (ˆ ρ1 , :t1 ρˆ1 ) ∈ I. Proof 5.38. Follows from Theorem 3.19, Proposition 5.33 and Proposition 5.35. Remark 5.39. For the proof of Theorem 5.37 Nρ plays no role. (see Remark following Theorem 6.5.1)

6

Anisotropic case

In this section we assume that −a is a non-square element of Fp.

6.1 Structure of the non-split 4-orthogonal group Let fq,1 be the anisotropic quadratic form on W := ⊕4i=1 Fq vi deﬁned by fq,1 := v1 v3 + (v22 − dv42 )/2 where d is a non-square element in F∗q . Extending the base ﬁeld to Fq2 /Fq , we take the new bases {ui }4i=1 by 1 1 u 1 = v 1 , u2 = v 2 + √ v 4 , u3 = v 3 , u4 = v 2 − √ v 4 d d

(6.1)

so that the base ﬁeld extension of fq,1 is represented with respect to the new basis as fq,1 = u1 u3 + u2 u4 , which is of the same form as fq2 ,0 in the notion of 5.1. Lemma 6.1. Let F0 be the Frobenius morphism of W ⊗Fq Fq2 over V = ⊕i Fq ui . (i) Frobenius morphism F1 of W ⊗Fq Fq2 over W acts on End(W ⊗Fq Fq2 ) by α → where r0 is the reﬂection deﬁned in 5.1. (ii) Let S1 and S2 be the subgroups of Ωfq2 ,0 deﬁned in 5.1. Then we have

Ωfq,1 = g ∈ Ωfq2 ,0 ; g = s

F1

s with s ∈ S1

F0 r0

α

In particular, i : SL2 (Fq2 ) s → i1 (s)F1 i1 (s) ∈ Ωfq,1 induces the isomorphism of P SL2 (Fq2 ) onto Ωfq,1 . (iii) SOfq,1 = Ωfq,1 × ± I4 where I4 denotes the identity of End(W ⊗Fq Fq2 ). (iv) The action of r0 ∈ Ofq,1 \ SOfq,1 on Ωfq,1 induces the action of canonical Frobenius morphism F of Fq2 /Fq on P SL2 (Fq2 ) through isomorphism i in (ii). Proof 6.2. By (6.1), F1 = F0 · r0 , hence (i). By deﬁnition S1 and S2 are stabilized by F0 and Ωfq,1 = ΩFf 12 ∩ Ker Spfq,1 If s1 ∈ S1 , s2 ∈ S2 and s = s1 s2 ∈ ΩFf 12 , it follows from q ,0

(i) that s2 = ±F1 s1 . Hence we have

q ,0

ΩFf 12 = g ∈ Ωfq2 ,0 ; g = δs1 F1 s1 where s1 ∈ S1 and δ ∈ Z(Ωfq2 ,0 ). . q ,0

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Let H ⊂ ΩFf 12

q ,0

241

denote the subgroup of elements with δ = I4 . Then H is isomorphic

to P SL2 (Fq2 ) and ΩFf 12

q ,0

= H × ±I4 . Since H is a simple group and Spfq,1 (−I4 ) is

nontrivial, Ker Spfq,1 ∩ ΩFf 12 = H. This proves (ii) and (iii). (iv) follows immediately q ,0 from our description of Ωfq,1 . It follows from Lemma 4.10 that we may identify Lp with W by an isometry as in 5.2. Fix any such isometry and denote it by ν. ν induces the isomorphisms of the orthogonal groups. Lemma 6.3. The action of r1 on Ωfp is conjugated by an element of Ωfp to the action induced by r0 . In particular,there is an element of g ∈ Ωfq,1 such that j := g ◦ ν ◦ i induces an isomorphism of P SL2 (Fq2 ) F to Ωfp r1 taking F to r1 . Proof 6.4. It is suﬃcient to show both reﬂections have the same spinorial norm. In fact, we have Spfp (r1 ) = a−1 , Spfq,1 (r0 ) = −d. The last statements follows immediately from Lemma 1. Lemma 6.5. (i) Suppose that q ≡ ±1 mod 8. Then Irr(G) is identiﬁed with J := {ρ ∈ Irr(SL2 (Fq2 )); ρ(−I) = id.} . (ii) Suppose that q ≡ ±3 mod 8. Then Irr(G) is identiﬁed with ˆ := {ˆ J ρ ∈ Irr(SL2 (Fq2 ) × Z2 ); ρˆ(−I, 0) = id.} . (iii) r1 -invariant irreducible representations of G are identiﬁed with F -invariant repˆ resentations of J or J. Proof 6.6. (i) and (ii) follow immediately from Lemma 6.1(ii) and (iii). (iii) follows from Lemma 6.1(iii) and Lemma 6.3.

6.2 General formula for irreducible characters of Z2 -extensions In this subsection K denotes a general ﬁnite group and r an automorphism of order two of K. Suppose that ρ ∈ Irrr K. Let V be a virtual representation space of ρ. For l ∈ EndC (V ) set 1 f (l) = ρ(g) · l · ρ(r g −1 ) ∈ EndC (V ). |K| g∈K Lemma 6.7. Suppose ρ¯ is an irreducible representation of K r which restricts to ρ. Then for any l ∈ EndC (V ), f (l) ∈ C¯ ρ(r). Proof 6.8. It follows from Schur’s lemma and r-invariance of ρ that subspace J of End(V ) deﬁned by J := {l ∈ EndC (V ); ρ(g) · l = l · ρ(r g) for any g ∈ K} is one dimen-

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sional. Hence J = C¯ ρ(r). Since f (l) · ρ(h) = ρ(r h) · f (l) by deﬁnition we have f (l) ∈ C¯ ρ(r). Lemma 6.9. Suppose that Trace(f (idV )) = 0. Then for any θ ∈ K there holds χρ¯(θr) = (

1 χρ (ε) ) 2 Trace(f (ρ(θ))). Trace(f (idV ))

where ρ¯ is a suitably chosen irreducible representation of K r restricting to ρ according to the choice of the square root. Proof 6.10. By Lemma 6.7, we have c¯ ρ(r) = f (idV ) for some c ∈ C. Since r is of order two, we have by deﬁnition c2 idV =

1 1 r r −1 r −1 ρ(g g )ρ(h h ) = ρ( h (r h−1 g r (r h−1 g)−1 )). 2 2 |K| g, h∈K |K| g, h∈K

Taking the trace of the equation, we have c2 χρ (ε) = Trace(f (idV )). Hence we can determine the constant c and proved the lemma for θ = 1. For general θ we compute as

ρ(g) 1 1 r −1 Trace(f (ρ(θ))) = |K| ρ(θ r g −1 g) = |K| g∈K Trace g∈K χρ (θ g g) (idV )) = Trace(ρ(θ) · f (idV )) = Trace(f χρ¯(θr). χρ (ε)

We deﬁne the action L of K on itself by L(g) : K θ → gθ r g −1 ∈ K and call it r-conjugation. We denote the r-conjugate orbit of θ ∈ K by Lθ and call it r-conjugacy class of θ. Lr denotes the set of r-conjugacy classes in K and DK (θ) the isotropy group of θ ∈ K by the r-conjugation. The following lemmas are immediate. Lemma 6.11. Let CK be the set of conjugacy classes of K. Then Trace(f (ρ(θ))) :=

1 |c ∩ Lθ | ρ(c). |DK (θ)| c∈C K

Lemma 6.12. Suppose θ ∈ K. Then subset Lθ r = {lr; l ∈ Lθ } of K r is the K-orbit of θr by the conjugation. The map Lr Lθ → Lθ r identiﬁes Lr with the set of all the K-orbits in K r \ K. Moreover, it induces the one-to-one corresponding of Lr / r with the set of conjugacy classes of K r in K r \ K.

6.3 F -conjugacy class in SL2 We shall specialize the argument in 6.2 to K = SL2 (Fq2 ) and r = F , the canonical Frobenius morphism. We denote by S the geometric point SL2 (Fq ) of algebraic group

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243

2

SL2 over Fq . In particular K = S F . Let tα denotes the diagonal matrix with entry (α, α−1 ). We set

        ∗ 1 β  T := tα ∈ S; α ∈ Fq . , V := v ∈ S; v = ±   , β ∈ Fq . ,     1           ∗ α β   0 1 B := b ∈ S; b = ±   , α ∈ Fq , β ∈ Fq . , J :=   , N := T · J     α−1 −1 0 Lemma 6.13. (i) For each non-central element s of S there exists the unique maximal abelian subgroup of S containing s and it coincides with CS (s). (ii) If s is semisimple and not central, CS (s) is conjugate to T in S. (iii) If s is not semisimple, CS (s) is conjugate to V in S. (iv) NS (T ) = N , NS (V ) = B where NS (·) denotes the normalizer. (v) Any F -stable torus is conjugate by an element of S F either to the F -split torus T or the nonsplit split torus which is unique up to S F conjugate. (vi) Any F -stable conjugate of V is conjugated to each other by an element of S F . Proof 6.14. All the statement can be veriﬁed directly. Deﬁne the automorphism of variety S by n : S θ → θ F θ ∈ S. It is readily veriﬁed that n induces a map of set LF of F -conjugacy classes to set CK of conjugacy classes. Lemma 6.15. Let z ∈ {±I} be a central element of S. Then n−1 (z) ∩ K coincides with a F -conjugacy class of K. In particular, n−1 (I) ∩ K = LI and n−1 (−I) ∩ K = Lθ4 for any F -rational element θ4 of order four. Proof 6.16. It is readily veriﬁed that n−1 (z) is stable under the F -conjugation by ele2 ments of K = S F . Suppose that x, y ∈ n−1 (z). By the Lang-Steinberg theorem, we can 2 ﬁnd ζ ∈ S such that y = ζx F ζ −1 . Since z = y F y = ζz F ζ −1 and z is central, ζ ∈ K. This proves the lemma. Lemma 6.17. Let z be as in Lemma 6.15. For any element x of n−1 (z), CK (x) is F -stable. Proof 6.18. Since x F x = z, CK (x) = CK (F x) = F CK (x). Lemma 6.19. Let Tα and ±U∗ be the conjugacy classes of K = SL2 (Fq2 ) described in

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5.3 (replacing the base ﬁeld with Fq2 ). Suppose    q(q + 1)    |LI ∩ Tα | = q(q − 1)     0

|LI ∩ ±Uβ | =

that α2 = 1. Then if α

F0

α = 1,

if α

F0

α−1 = 1,

otherwise.

   q2 − 1

if 4 | q + 1 and β = 1 or if 4 | q − 1 and β = d,

 0

otherwise.

Proof 6.20. Suppose x is an element of LI ∩ Tα . Then Lemma 6.13(ii) and Lemma 6.17 show that T1 := CS (x) is F -stable. Then by Lemma 6.13(v) and Lemma 6.15, either T1 is F -split and α F α = 1 or T1 is nonsplit and α F α−1 = 1. Then the ﬁrst formula follows from Lemma 6.13(i) since any noncentral semisimple conjugacy class intersects with a torus at two elements. The second formula follows similarly from Lemma 6.13(vi). Lemma 6.21. Suppose that θ4 ∈ K is     q(q + 1)   |Lθ4 ∩ Tα | = q(q − 1)     0

an F -rational element of order four. Then if α

F0

α = −1,

if α

F0

α−1 = −1, |Lθ4 ∩ ±Uβ | = 0.

otherwise.

Proof 6.22. The ﬁrst formula follows from Lemma 6.15 and the same argument as Lemma 6.23. For the second formula we just have to note that equation v F v = −I can not hold true for v ∈ V since F preserves the unipotent elements of SL2 . Lemma 6.23. For x ∈ LI ∪ Lθ4 |DK (x)| = q(q 2 − 1). Proof 6.24. By the Lang-Steinberg Theorem we can choose y ∈ S with x = y The direct computation shows that DK (x) = y (S F ) and the Lemma follows.

F −1

y .

6.4 Conjugacy classes Lemma 6.25. Let θ4 be as in Lemma 6.15. Suppose that g ∈ Ωfp veriﬁes g r1 g = I4 . Then gr1 ∈ Ωfp r1 is conjugate to r1 in Ωfp r1 if T rF2q (gr1 ) = 2 and to i(θ4 )r1 if T rF2q (gr1 ) = −2. Proof 6.26. Let g = j(s) for s ∈ SL2 (Fq2 ) and j the projection in Lemma 6.3. Then s F s = ±I by assumption. Lemma 6.12 and Lemma 6.15 show that gr1 is conjugate to r1 if s F s = I and to j(θ4 )r1 if s F s = −I. Then the distinction by T rF2q (gr1 ) is directly veriﬁed.

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Lemma 6.27. Any element of order four in Ωrf1p is conjugate to each other in Ωrf1p . Proof 6.28. Suppose that g = j(s) is an element of order four in Ωrf1p . Then F s = ±s and s ∈ SL2 (Fq2 ) is of order eight. Hence T1 := CK (s) is an F -stable torus. It is immediate that T1 is F -split if q ≡ 1 or −3 mod 8 and nonsplit otherwise. The lemma follows since any element of order eight in a torus is conjugate up to center. Lemma 6.29. Suppose that q ≡ ±1 mod 8 and θ8 is an element of order eight in SL2 (Fq2 ) with F θ8 = ±θ8 .Then the following conjugacy relations in Ωfp r1 hold. 2 −1 2 −1 (i) θE−1 r1 ∼ r1 (ii) θC2 r1 ∼ θF2 r1 ∼ θA θ B r 1 ∼ θD θE r1 ∼ r2 r3 r1 ∼ j(θ4 )r1 −1 ±1 ±1 ±1 −1 ±1 −1 (iii)θA θF θE r2 r3 r1 ∼ j(θ4 )r1 (iv) θC r1 ∼ θF r1 ∼ θA θ B ∼ θD θE r1 ∼ j(θ8 )r1 Proof 6.30. For element gr1 in the statements (i), (ii) and (iii), verify that g r1 g = I by Lemma 2.4 and Lemma 2.8. Lemma 5.22 computes the traces of these elements, extending the base ﬁeld to Fq2 . Then (i),(ii) and (iii) follow from Lemma 6.25. Since θC and θF are of order four by Lemma 5.18 and they are ﬁxed by r1 , they are conjugate to j(θ8 ) in Ωfp by Lemma 6.27, hence the statement for θC r1 and θF r1 follows. Similarly the statement for θD θE−1 r1 follows from (i) and the commutativity of θD and θE−1 r1 . The −1 argument also applies to θD θB r1 . Lemma 6.31. Suppose that q ≡ ±3 mod 8. The following conjugacy relations in Ofp hold (i) θE−1 r1 ∼ −I4 · j(θ4 )r1 (ii) θC2 r1 ∼ θF2 r1 ∼ j(θ4 )r1 (iii) r2 r3 r1 ∼ j(θ4 )r1 −1 2 −1 2 −1 (iv) θA θ B r 1 ∼ θD θE r1 ∼ −I4 · r1 . (v) θA θF θE r1 r2 r3 ∼ −I4 · r1 . ±1 ±1 ±1 −1 ±1 (vi) θC r1 ∼ θF r1 ∼ −I4 · j(θ8 )r1 . (vii) θA θB r1 ∼ θD θE r1 ∼ j(θ8 )r1 . Proof 6.32. All the statement follow from Lemma 6.1(iii), Lemma 6.25, Lemma 6.27 and the corresponding argument in the proof of Lemma 6.29.

6.5 Computation Lemma 6.33. Let the notions be as in Table 1. F -invariant irreducible representations of SL2 (Fq2 ) are St, G± 1 and Hn (n ∈ (q ± 1)Z). Proof 6.34. directly veriﬁed. Lemma 6.35. Suppose that q ≡ ±1 mod 8. Let ρ ∈ Irrr1 (G) and σ ∈ J the F -invariant irreducible representation of SL2 (Fq2 ) corresponding to ρ in terms of Lemma 6.5. Then for the irreducible representation σ ¯ of SL2 (Fq2 ) F restricting to σ, we have Nρ = | − χσ¯ (F ) + χσ¯ (θ4 F ) + 2χσ¯ (θ8 F )|, Mρ = 2|χσ¯ (θ4 F )|. Proof 6.36. The formulas follow immediately from Lemma 6.29.

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Lemma 6.37. Suppose that q ≡ ±3 mod 8. Let ρ ∈ Irr r1 (G), σ × τ ∈ IrrF (SL2 (Fq2 ) × Z2 ) correspond to ρ in terms of Lemma 6.5 and σ ¯ × τ an irreducible representation of 2 (SL2 (Fq ) F ) × Z2 which restricts to σ × τ . (i) If τ = 1Z2 , Nρ = 2|χσ¯ (θ8 F )|, Mρ = |χσ¯ (F ) + χσ¯ (θ4 F )|. (ii) If τ = −1Z2 , Nρ = Mρ = |χσ¯ (θ4 F ) − χσ¯ (F )|. Proof 6.38. The formulas follow immediately from Lemma 6.31. Computations By Lemma 6.19 and Lemma 6.23 we can compute the formula of Trace(f (I)) in Lemma 6.9. In fact we have Trace(f (I)) = 0 for any F -invariant representations of SL2 (Fq2 ) in Lemma 1. Hence according to the formula in Lemma 6.17 and Table 1 we can compute values σ ¯ (F ) and σ ¯ (θ4 F ), the result of which is found in Table 3. σ

σ ¯ (F )

σ ¯ (θ4 F )

St H(q+1)k (k = 0 even) (k odd) H(q−1)l (l = 0 even) (l odd)

q q+1 q+1 q−1 q−1

q q+1 −q q−1 −(q − 1)

q 1

−1 −q

1 q

q 1

if q + 1 ≡ 0 mod 4 G+ 1 G− 1 if q − 1 ≡ 0 mod 4 G+ 1 G− 1

Table 3 The value of σ ¯ (F ) and σ ¯ (θ4 F ).

Theorem 6.39. Suppose that −a is a non-square in Fp. Let ρ, σ, σ ¯ and τ be as in Lemma 6.35 or Lemma 6.37. Then ρ appears as an irreducible component of H1 (Γ\H3 , C) unless q ≡ ±3 mod 8 and σ × τ is one of H(q−1)l × 1Z2 (l: odd), St × −1Z2 and H(q±1)k × −1Z2 (k: even). Proof 6.40. It follows from the formulas for Mρ in Lemma 6.35 and Lemma 6.37 and the result of computations in Table 3 that Mρ = 0 except for the irreducible representations in the last part of the statement. Hence the theorem follows from Theorem 3.19. Remark 6.41. The author could not determine the value at θ8 F of the character. If it is ¯ (q−1)l veriﬁes the criterion of Theorem 3.17 and it would be the only case nonzero, σ = H

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that the action of r1 gives us more information than that of r1 r2 r3 in our formulation.

7

Remarks

7.1 A result for a general case Lemma 7.1. Let Γ be a cocompact Kleinian group, r an automorphism of Γ of order two and Γ0 a maximal r-normal subgroup of ﬁnite index in Γ. Then G := Γ/Γ0 is either (a) a simple group or (b) the direct product of two copies of a simple group and r acts on G by the permutation of the factors of the direct product. Proof 7.2. Suppose that G is not simple. Let N1 be a maximal normal subgroup of G. Set N2 := r N1 . Then N1 ∩ N2 = {ε} and G = N1 N2 since Γ0 is maximal r-normal and N1 is maximal normal. Since N1 is maximal, N2 N1 is a simple group. Then validity of Schreier’s conjecture (see [5] Theorem 1.46) and the Eilenberg-MacLane theory of the extension of groups (see e.g. [9] Theorem 8.8 in Chapter IX) show that G N1 × N1 provided N1 is non-abelian. In the abelian case, the lemma is obvious. The method in Section 5 readily implies the following general statement for r1 r2 r3 simple quotient of type (b) in Lemma 7.1. Theorem 7.3. Let Γ0 be a maximal r1 r2 r3 -normal, but not maximal normal subgroup of ﬁnite index in B4,4,4 and G := B4,4,4 /Γ0 . Then all the nontrivial r1 r2 r3 -invariant irreducible representation are irreducible components of G-module H1 (Γ0 \H3 , C). Proof 7.4. By assumption G is of type (b) in Lemma 7.1 with r = r1 r2 r3 . We may assume that G = K × K and r acts on G by G (k1 , k2 ) → (k2 , k1 ) ∈ G. Then we can trace the arguments in Section 5 to obtain the followings, (1) It is immediate that any r-invariant irreducible representation of G is of form ρ1 × ρ1 for some ρ1 ∈ Irr(K). (2) If gr ∈ G r is of order two, g is of form (k, k −1 ) ∈ K ×K = G. Hence in particular such gr’s are all conjugate. (3) The computations in 5.4 shows that ρ¯(r) = ρ¯1 (ε) if ρ = ρ1 × ρ1 . Then the theorem follows from (1), (2), (3) and Theorem 3.19.

7.2 The growth of the ﬁrst Betti number To show the existence of congruent coverings of arbitrary large ﬁrst Betti number, Millson gives in [10] a lower bound for the ﬁrst Betti number of congruence subgroup Γm in terms of the number of distinct primes in the decomposition of ideal m. For our particular case, B4,4,4 , the results in Section 5 and Section 6 give a lower bound in terms of the norm of the prime ideals. In particular we have

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√ Proposition 7.5. Let p be a prime ideal of Q( 5) and Γp the congruence subgroup of B4,4,4 associated to p. Then lim inf q→∞

b1 (Γp\H3 ) > 0, q4

lim sup q→∞

b1 (Γp\H3 ) <∞ q6

where q is the norm of p and b1 (·) denotes the ﬁrst Betti number of the topological space. Proof 7.6. We shall carry over the notions in Section 4. The second inequality is obvious since the dimension of our 1-chain module is of order O(q 6 ). To prove the ﬁrst inequality ¯ 1,ρ (gr1 ) − H ¯ 2,ρ (gr1 )| = |¯ recall that by deﬁnition of Nρ in 5.5 |H ρ(gr1 )|Nρ . Suppose that ¯ ¯ H1,ρ = n1 ρ¯ + n2 :r1 ρ¯ and H1,ρ = n1 ρ¯ + n2 :r1 ρ¯. Then Nρ = |(n1 − n2 ) − (n1 − n2 )|. Since H1,ρ = H2,ρ by Lemma 3.3, |Nρ |ρ(ε) ≤ 2H1,ρ (ε). Thus we have b1 (Γp\H3 ) ≥ ρ |Nρ |ρ(ε). Computing the right hand side by the results in 5.5 and 6.5, we have the ﬁrst inequality.

7.3 A conclusion from group theory Theorem 7.7. Let Γ be a subgroup of ﬁnite index in B4,4,4 and TΓ the subgroup of Γ √ deﬁned in 1.1. If there exists prime ideal p ∈ PB4,4,4 of Q( 5) such that the image of TΓ in G := B4,4,4 /Γp does not contain any noncentral normal subgroup of G, Γ\H3 has a ﬁnite-sheeted cover of positive ﬁrst Betti number. Sketch of the Proof 7.8. We have the complete list of subgroups of SL2 (Fq ) by Dickson [3]. If −a is nonsquare in Fp, we can directly verify that the Steinberg character St (or St × 1Z2 if q ≡ ±3 mod 8) has nontrivial ﬁxed point on any proper subgroup of SL2 (Fq2 ). Then in view of Theorem 6.5.1 and Proposition 1.1 this case is done. If −a is square, we need the following lemma, which is an immediate consequence of Mackey’s formula. Lemma 7.9. Let K a ﬁnite group, L1 and L2 be subgroups of K. Then subgroup L1 ×L2 of K × K has nontrivial ﬁxed point in σ := 1=ρ∈Irr(K) ρ × ρ if and only if |L1 \K/L2 | =

1 By Lemma 7.9 and Dickson’s classiﬁcation we can verify that σ has nontrivial ﬁxed points set on proper subgroups of SL2 (Fq ) × SL2 (Fq ) other than some exceptional cases. In view of Theorem 5.37 and Proposition 1.5 we can apply this consequence to prove the theorem for q ≡ ±1 mod 8 since the exceptional subgroup is excluded by the fact that the image of TΓ is generated by the elements of order two or four. The modiﬁed argument also applies to the case of q ≡ ±3 mod 8

References [1] A. Borel: “Commesurability classes and Volumes of hyperbolic 3-manifolds“, Journ. Ann. Scuola Norm. Sup. Pisa, Vol. 8, (1981), pp. 1–33.

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[2] R.W. Carter: Finite groups of Lie type, Conjugacy classes and complex character, John Wiley and Sons, Chichester-New York-Brisbane-Tronto-Singapore, 1985. [3] L.E. Dickson: Linear groups with an exposition of the Galois ﬁeld theory, Dover Publ., New York, 1958. [4] L.E. Digne and J. Michel: Representation of ﬁnite groups of Lie type, London Math. Soc., London, 1991. [5] D. Gorenstein: Finite simple groups An introduction to their classiﬁcation, Plenum Press, New York, 1982. [6] H.M. Hilden, M.T. Lozano and J.M. Montesinos: “On the Borromean orbifolds: Geometry and arithmetics“, In: B. Apanasov, W.D. Neuman, A.W. Reid and L. Siebenmann (Eds.): Topology ’90, de Gruyter, Berlin, 1992, pp. 133–167. [7] H.M. Hilden, M.T. Lozano and J.M. Montesinos: “On the universal groups of the Borromean rings“, In: B. Apanasov, W.D. Neuman, A.W. Reid and L. Siebenmann (Eds.): Proceedings of the 1987 Siegen conference on Diﬀerential Topology, Springer Verlag, Berlin, 1988, pp. 1–13. [8] N. Jacobson: “Basic Algebra I, II“, 2nd Ed., Freeman, New York, 1985. [9] S. MacLane: Homology, Springer Verlag, Berlin, 1967. [10] J.J. Millson: “On the ﬁrst Betti number of a constant negatively curved manifold“, Jour. Ann. of Math., Vol. 104, (1976), pp. 235–247. [11] J.G. Ratcliﬀe: Foundation of hyperbolic manifolds, Springer, New York, 1994. [12] A.W. Reid: Arithmetic Kleinian groups and their Fuchsian subgroups, Thesis (PhD), University of AberdeenUniversity of Aberdeen, 1985. [13] M. Suzuki: Group Theory I, II, Springer Verlag, New York, 1985. [14] W.P. Thurston: “Three dimensional manifolds, Kleinian groups and hyperbolic geometry“, Jour. Bull. of AMS, Vol. 6, No. 3, (1982). pp. 357–381.

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Quasilinearization for the periodic boundary value problem for hybrid diﬀerential equation L.M. Hall1∗ , S.G. Hristova2† 1

University of Missouri-Rolla, Rolla, MO 65409-0020, USA 2 Denison University, Granville, OH 43023, USA

Received 18 September 2003; accepted 18 March 2004 Abstract: The method of quasilinearization for a periodic boundary value problem for nonlinear hybrid diﬀerential equations is studied. It is shown that the convergence is quadratic. c Central European Science Journals. All rights reserved. Keywords: hybrid diﬀerential equations, periodic boundary value problem, quasilinearization, quadratic convergence MSC (2000): 34K10, 34B15, 34K25

1

Introduction

Hybrid systems combine both continuous components, which are governed by diﬀerential equations, and digital components like computers, sensors, and actuators controlled by programs([2], [5], [11]). However, the diﬃcult ﬁnding of solutions of hybrid diﬀerential equations in closed form requires the proving of diﬀerent approximate methods. There are many eﬀective approximate methods which use upper and lower solutions to the given problem([3], [4]). One of them is the method of quasilinearization. The ﬁrst results for this method were obtained by Bellman and Kalaba ([1]). In recent years the method has been applied to a variety of problems by V.Lakshmikantham and his scholars([6], [7], [8], [9], [10]). The main advantages of this method are the practicality of ﬁnding successive approximations of the unknown solution as well as the quadratic convergence. ∗ †

E-mail: [email protected] E-mail: [email protected]

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In the current paper we will prove this method for the periodic boundary value problem for a nonlinear hybrid diﬀerential equation. We will also show that the convergence is quadratic.

2

Preliminary notes and deﬁnitions

Let the points tk ∈ [0, T ], k = 0, 1, 2, . . . , p + 1 be ﬁxed such that t0 = 0, tp+1 = T , and tk < tk+1 . Consider the following boundary value problem for the nonlinear hybrid diﬀerential equation (PBVP) x = f (t, x(t), Λk (x(tk ))) for

t ∈ (tk , tk+1 ], k = 0, 1, . . . , p,

x(0) = x(T ),

(1) (2)

where x ∈ R, f : [0, T ] × R × R → R, T = const > 0, Λk : R → R, k = 0, 1, . . . , p. The function α(t) ∈ C 1 ([0, T ], R) is called a lower solution of the PBVP (1),(2) if the following inequalities are satisﬁed: α(t) ≤ f (t, α(t), Λk (α(tk ))) for t ∈ (tk , tk+1 ], k = 0, 1, . . . , p,

(3)

α(0) ≤ α(T ).

(4)

Analogously, the deﬁnition of an upper solution of the PBVP (1),(2) can be introduced if the inequalities in (3),(4) hold true in opposite directions. Let the functions α, β ∈ C([0, T ]), R) be such that α(t) ≤ β(t). Consider the sets: S(α, β) = {u ∈ C([0, T ], R) : α(t) ≤ u(t) ≤ β(t) for t ∈ [0, T ]} Ω(α, β) = {(t, x) ∈ [0, T ] × R : α(t) ≤ x ≤ β(t)}. Lemma 2.1. Let the following conditions be fulﬁlled: 1. The functions g1 ∈ C([0, T ], R), g2 ∈ C([0, T ], (−∞, 0]) and g2 ≡ 0 for t ∈ [0, T ]. 2. The inequality N T eM h ≤ 1

(5)

holds, where M = max{|g1 (t)| : t ∈ [0, T ]} ≥ 0, N = max{|g2 (t)| : t ∈ [0, T ]} > 0, h = max{tk+1 − tk , k = 0, 1, . . . , p}. 3. The function u ∈ C 1 ([0, T ], R) satisﬁes the inequalities u ≤ g1 (t)u(t) + g2 (t)u(tk ) for t ∈ (tk , tk+1 ], k = 0, 1, . . . , p,

(6)

u(0) ≤ 0

(7)

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Then the function u(t) is nonpositive in the interval [0, T ]. Proof. Consider the following two cases: Case 1.Let the inequalities (6) and (7) be strict. Deﬁne the function v : [0, T ] → R by the equality v(t) = u(t)eM t for t ∈ [0, T ]. The function v(t) satisﬁes the inequalities v (t) = u (t)eM t + M u(t)eM t < g1 (t)v(t) + g2 (t)eM t u(tk ) + M v(t) = g2 (t)eM (t−tk ) v(t) + {g1 (t) + M }v(t)

for t ∈ (tk , tk+1 ],

v(0) = u(0) < 0. We shall prove that v(t) ≤ 0 for t ∈ [0, T ]. If it does not hold then there exists a point η ∈ (0, T ) such that v(η) = 0 and v(t) < 0 for t ∈ [0, η) and v (η) ≥ 0. Denote min{v(t) : t ∈ [0, η)} = −λ < 0, where λ = const > 0. Then there exist points ζ ∈ [0, η) and τ ∈ (ζ, η) such that v(ζ) = −λ and the equality v(η) − v(ζ) = v (τ )(η − ζ)

(8)

holds. Let τ ∈ (tl , tl+1 ]. Then from the inequality (6) it follows that v (τ ) < g2 (τ )eM (τ −tk ) v(tk ) + {g1 (τ ) + M }v(τ ) ≤ g2 (τ )eM (τ −tk ) v(tk ) ≤ −g2 (τ )eM (τ −tk ) λ ≤ N eM h λ.

(9)

From the inequality (9) and equality (8) it follows the inequality 1 < N T eM h which contradicts to the inequality (5). Case 2. Let at least one of the inequalities (6) or (7) not be strict. We consider the function w : [0, T ] → R deﬁned by the equality w(t) = u(t) − "eAt for t ∈ [0, T ], where " > 0 is an arbitrary number, A = 2M + N e−M h . According to the inequality (6) we obtain that w (t) ≤ g1 (t)u(t) + g2 (t)u(tk ) − "AeAt < g1 (t)w(t) + g2 (t)w(tk ) − "M eAt < g1 (t)w(t) + g2 (t)w(tk )

for t ∈ (tk , tk+1 ].

From the deﬁnition of the function w(t) and the inequality (7) it follows that w(0) < 0. According to the case 1 it follows the validity of the inequality w(t) ≤ 0 for t ∈ [0, T ]. Taking into the limit as " → 0 in the deﬁnition of the function w(t) we obtain that u(t) ≤ 0 for t ∈ [0, T ]. Lemma 2.2. Let the following conditions be fulﬁlled: 1. The functions g1 ∈ C([0, T ], R), g2 ∈ C([0, T ], (−∞, 0]) are such that g2 (t) ≡ 0 for t ∈ [0, T ] and 0T g1 (s)ds < 0.

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2. The inequality (5) holds. 3.The function u ∈ C 1 ([0, T ], R) satisﬁes the inequality (6) and u(0) ≤ u(T ). Then the function u(t) is nonpositive on the interval [0, T ]. Proof. We consider the following two cases: Case 1. Let u(0) ≤ 0. Then according to Lemma 2.1 it follows the validity of the inequality u(t) ≤ 0 for t ∈ [0, T ]. Case 2. Let u(0) > 0. Then u(T ) > 0. We deﬁne the function v : [0, T ] → R as v(t) = u(t)exp(− 0t g1 (s)ds) for t ∈ [0, T ]. The function v(t) satisﬁes the following inequalities v (t) ≤ g2 (t)u(tk )exp(−

t 0

g1 (s)ds) for t ∈ (tk , tk+1 ],

v(0) = u(0) ≤ u(T ) < u(T )exp(−

(10)

T

g1 (s)ds) = v(T ).

0

From the inequality u(0) > 0 it follows that v(0) > 0. Case 2.1. Let v(t) ≥ 0 for t ∈ [0, T ]. Therefore, u(t) ≥ 0 for t ∈ [0, T ]. Using inequality (10) it follows that v (t) ≤ 0 for t ∈ [0, T ], i.e. the function v(t) is nonincreasing and v(0) ≥ v(t) ≥ v(T ) for t ∈ [0, T ]. Therefore, v(0) = v(T ) = v(t) = K = const. This contradicts inequality (10). Case 2.2. Let there exists a point η ∈ (0, T ] such that v(η) < 0. We deﬁne min{v(s) : s ∈ [0, T ]} = −λ < 0, where λ = const > 0, then there exists a point ξ ∈ (0, T ) such that v(ξ) = −λ. Therefore, v(T ) − v(ξ) = v (ζ)(T − ξ)

(11)

where ζ ∈ (ξ, T ), ζ ∈ (tl , tl+1 ], 0 < l < p + 1. From (10) it follows the inequality

v (ζ) ≤ g2 (ζ)u(tl )exp(−

ζ 0

= g2 (ζ)v(tl )exp(−

g1 (s)ds)

ζ tl

g1 (s)ds).

(12)

From inequality (12) and v(tl ) ≥ −λ it follows that ζ

v(tl )exp(−

tl

g1 (τ )dτ ) ≥ −λexp(−

ζ tl

g1 (τ )dτ ) ≥ −λeM h .

(13)

From inequalities (12) and (13) we obtain v (ζ) ≤ −g2 (ζ)λeM h ≤ N λeM h .

(14)

Inequalities (11) and (14) and the choice of the point ζ show that the following inequalities λ ≤ N eM h (T − ξ)λ < N eM h λT. hold. Inequality (15) contradicts the inequality (5).

(15)

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Main results

We will prove the method of quasilinearization for ﬁnding approximate solutions of the periodic boundary value problem for a nonlinear hybrid diﬀerential equation. We will prove that the convergence of the successive approximations is quadratic. Theorem 3.1. Let the following conditions hold: 1. The functions α0 (t), β0 (t) are lower and upper solutions to the PBVP (1),(2) and α0 (t) ≤ β0 (t) for t ∈ [0, T ]. 2. The functions Λk ∈ C(R, R) are increasing and there exist constants Lk > 0 such that for x, y ∈ R, x ≤ y the inequalities Λk (x) − Λk (y) ≥ Lk (x − y),

k = 0, 1, . . . , p

hold. 3. There exist functions F, g ∈ C 0,2,2 (Ω(α0 , β0 ), R) such that F (t, x, y) = f (t, x, y) + g(t, x, y), Fxx (t, x, y) ≥ 0, Fxy (t, x, y) ≥ 0, Fyy (t, x, y) ≥ 0, gxx (t, x, y) ≥ 0, gxy (t, x, y) ≥ 0, gyy (t, x, y) ≥ 0, p t k+1 k=0 tk

[Fx (s, β0 (s), Λk (β0 (tk )) − gx (s, α0 (s), Λk (α0 (tk ))]ds < 0,

Fy (t, β0 (t), Λk (β0 (tk )) ≤ gy (t, α0 (t), Λk (α0 (tk )), k = 0, 1, . . . , p N T eM h ≤ 1, where M = max{|Fx (t, β0 (t), β0 (t − h)) − gx (t, α0 (t), α0 (t − h))| : t ∈ [0, T ]}, N = max{gy (t, α0 (t), α0 (t − h)) − Fy (t, β0 (t), β0 (t − h)) : t ∈ [0, T ]}. ∞ Then there exist two sequences of functions {αn (t)}∞ 0 and {βn (t)}0 such that: a. The sequences are increasing and decreasing correspondingly. b. The functions αn (t) are lower solutions and the functions βn (t) are upper solutions of the PBVP (1), (2). c. Both sequences are uniformly convergent in the interval [0, T ] to the unique solution of the PBVP (1),(2) in S(α0 , β0 ). d. The convergence is quadratic.

Proof. From condition 2 of theorem 3.1 it follows that for (t, x1 , y1 ), (t, x2 , y2 ) ∈ Ω(α0 , β0 ) and x1 ≥ x2 , y1 ≥ y2 the inequalities f (t, x1 , y1 ) ≥ f (t, x2 , y2 ) + Fx (t, x2 , y2 )(x1 − x2 ) + Fy (t, x2 , y2 )(y1 − y2 ) +g(t, x2 , y2 ) − g(t, x1 , y1 ),

(16)

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255

f (t, x2 , y2 ) ≤ f (t, x1 , y1 ) + Fx (t, x2 , y2 )(x2 − x1 ) + Fy (t, x2 , y2 )(y2 − y1 ) +g(t, x1 , y1 ) − g(t, x2 , y2 )

(17)

hold. We consider the linear hybrid diﬀerential equation x (t) = f (t, α0 (t), Λk (α0 (tk ))) + Q0 (t)(x(t) − α0 (t)) +Lk q0k (t)(x(tk ) − α0 (tk )) for t ∈ (tk , tk+1 ], k = 0, 1, . . . , p,

(18)

where Qk0 (t) = Fx (t, α0 (t), Λk (α0 (tk ))) − gx (t, β0 (t), Λk (β0 (tk ))), q0k (t) = Fy (t, α0 (t), Λk (α0 (tk ))) − gy (t, β0 (t), Λk (β0 (tk ))), k = 0, 1, . . . , p. The PBVP (18),(2) has a unique solution α1 (t). We will prove that α1 (t) ∈ S(α0 , β0 ). It is easy to show that α0 (t) ≤ α1 (t) . We will prove that α1 (t) ≤ β0 (t). We deﬁne the function u(t) = α1 (t) − β0 (t). According to the deﬁnition of the functions α1 and β0 , and the inequality (16), we obtain the inequalities u (t) ≤ f (t, α0 (t), Λk (α0 (tk ))) + Q0 (t)(α1 (t) − α0 (t)) +Lk q0k (t)(α1 (tk ) − α0 (tk )) − f (t, β0 (t), Λk (β0 (tk )) ≤ Qk0 (t)(α1 (t) − β0 (t)) + Qk0 (t)(β0 (t) − α0 (t)) + Lk q0k (t)(α1 (tk ) − β0 (tk )) +Lk q0k (t)(β0 (tk ) − α0 (tk )) − Fx (t, α0 (t), Λk (α0 (tk )))(β0 (tk ) − α0 (tk ))

−Fy (t, α0 (t), Λk (α0 (tk )))(Λk (β0 (tk )) − Λk (α0 (tk ))) + g(t, β0 (t), Λk (β0 (tk ))) −g(t, α0 (t), Λk (α0 (tk ))) − gx (t, β0 (t), Λk (β0 (tk )))(β0 (tk ) − α0 (tk )) −gy (t, β0 (t), Λk (β0 (tk )))(Λk (β0 (tk )) − Λk (α0 (tk ))) ≤ Qk0 (t)u(t) + Lk q0k (t)u(tk ) +Lk q0k (t)(β0 (tk ) − α0 (tk )) − Lk q0k (t)Λk (β0 (tk )) − Λk (α0 (tk ))) ≤ Qk0 (t)u(t) + Lk q0k (t)u(tk ) for t ∈ (tk , tk+1 ], k = 0, 1, . . . , p u(0) ≤ u(T ).

(19) (20)

According to Lemma 1 and inequalities (19) and (20) it follows that u(t) ≤ 0 which proves the inequality α1 (t) ≤ β0 (t). We consider the linear hybrid diﬀerential equation x (t) = f (t, β0 (t), Λk (β0 (tk ))) + Qk0 (t)(x(t) − β0 (t)) +Lk q0 (t)(x(tk ) − β0 (tk )) for t ∈ (tk , tk+1 ).

(21)

There exists a unique solution β1 (t) of the PBVP (21), (2). The inclusion β1 (t) ∈ S(α0 , β0 ) is valid. We will prove that α1 (t) ≤ β1 (t) for t ∈ [0, T ]. Deﬁne the function u(t) = α1 (t) − β1 (t) for t ∈ [0, T ]. From the choice of the functions α1 (t) and β1 (t) and the inequality (22) we obtain that the function u(t) satisﬁes the

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inequalities u = f (t, α0 (t), Λk (α0 (tk ))) − f (t, β0 (t), Λk (β0 (tk ))) + Qk0 (t)u(t) +Qk0 (t)(β0 (t) − α0 (t)) + Lk q0k (t)(α1 (tk )) − β1 (tk ))

(22)

+Lk q0k (t)(β0 (tk )) − α0 (tk )) ≤ Qk0 (t)u(t) + Lk q0k (t)u(tk ) for t ∈ (tk , tk+1 ), u(0) = u(T ).

(23)

According to Lemma 2.2 and inequalities (22) and (23), it follows that the inequality u(t) ≤ 0, t ∈ [0, T ] holds, i.e. α1 (t) ≤ β1 (t). The function α1 (t) is a lower solution of the PBVP(1), (2). Indeed, for t ∈ (tk , tk+1 ) the inequalities α1 ≤ f (t, α1 (t), Λk (α1 (tk ))) + Fx (t, α0 (t), Λk (α0 (tk )))(α0 (t) − α1 (t))

+Fy (t, α0 (t), Λk (α0 (tk )))(glk (α0 (tk ) − Λk (α1 (tk ))) + g (t, α1 (t), Λk (α1 (tk ))) −g(t, α0 (t), Λk (α0 (tk ))) + Qk0 (t)(α1 (t) − α0 (t))

+Lk q0k (t)(α1 (tk )

(24)

− α0 (tk ))

≤ f (t, α1 (t), Λk (α1 (tk ))) hold. From inequality (24), and the periodic boundary conditions of the function α1 (t), it follows that the function α1 (t) is a lower solution of the PBVP (1), (2). Analogously, it can be shown that the function β1 (t) is an upper solution of the PBVP (1),(2). ∞ In this way we can construct two sequences of functions {αn (t)}∞ 0 and {βn (t)}0 , αn , βn ∈ S(αn−1 , βn−1 ). The function αn+1 (t) is the unique solution of the periodic boundary value problem for the linear hybrid diﬀerential equation x = f (t, αn (t), Λk (αn (tk ))) + Qkn (t)(x − αn (t)) +Lk qnk (t)(x(tk ) − αn (tk )) for t ∈ (tk , tk+1 ), x(0) = x(T ),

(25) (26)

and the function βn+1 (t) is the unique solution of the periodic boundary value problem x (t) = f (t, βn (t), Λk (βn (tk ))) + Qkn (t)(x − βn (t)) +Lk qnk (t)(x(tk ) − βn (tk )) for t ∈ (tk , tk+1 ), x(0) = x(T ), where Qkn (t) = Fx (t, αn (t), Λk (αn (tk ))) − gx (t, βn (t), Λk (βn (tk ))), qnk (t) = Fy (t, αn (t), Λk (αn (tk ))) − gy (t, βn (t), Λk (βn (tk ))). According to condition 3 the inequalities qnk (t) ≤ 0 hold.

(27) (28)

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257

Just as in the case n = 0 it can be shown that the functions αn (t) and βn (t) are lower and upper solutions of the PBVP (1), (2) and the inequalities α0 (t) ≤ α1 (t) ≤ . . . ≤ αn (t) ≤ βn (t) ≤ . . . ≤ β0 (t)

(29)

hold. ∞ Therefore the sequences {αn (t)}∞ 0 and {βn (t)}0 are uniformly bounded and equicontinuous in the interval [0, T ] and uniformly convergent. Denote limn→∞ αn (t) = u(t), limn→∞ βn (t) = v(t). From uniform convergence and the deﬁnition of the functions αn (t) and βn (t) the validity of the following inequalities is assured α0 (t) ≤ u(t) ≤ v(t) ≤ β0 (t).

(30)

Taking the limit of equalities (25)-(28) we obtain that the functions u(t) and v(t) are solutions of the PBVP (1),(2) and therefore u(t) = v(t). We will now prove that the convergence is quadratic. Deﬁne the functions an+1 (t) = u(t) − αn+1 (t) and bn+1 (t) = βn+1 (t) − u(t), t ∈ [0, T ]. Both functions are nonnegative. For t ∈ (tk , tk+1 ) we obtain the inequalities an+1 ≤ Qkn (t)an+1 (t) + Lk qn (t)an+1 (tk ) +[Fx (t, u(t), Λk (u(tk ))) − gx (t, αn (t), Λk (αn (tk ))) − Qkn (t)]an (t) +[Fy (t, u(t), Λk (u(tk ))) − gy (t, αn (t), Λk (αn (tk ))) − Lk qnk (t)]an (tk ) = Qkn (t)an+1 (t) + Lk qnk (t)an+1 (tk ) + Fxx (t, ξ1 , ξ2 )a2n (t) +Fxy (t, ξ1 , ξ2 )an (t)(Λk (u(tk )) − Λk (αn (tk )))

(31)

+gxx (t, η1 , η2 )an (t)(βn (t) − αn (t)) +gxy (t, η1 , η2 )an (t)(Λk (βn (tk )) − Λk (αn (tk ))) +Fxy (t, ξ3 , ξ4 )(Λk (u(tk )) − Λk (αn (tk )))an (t) +Fyy (t, ξ3 , ξ4 )(Λk (u(tk )) − Λk (αn (tk )))2 +gyy (t, η3 , η4 )(Λk (u(tk )) − Λk (αn (tk )))(Λk (βn (tk )) − Λk (αn (tk ))) +gxy (t, η3 , η4 )(Λk (u(tk )) − Λk (αn (tk )))(βn (t) − αn (t))

where u(t) ≤ ξi ≤ αn (t), u(tk ) ≤ ξlk ≤ αn (tk ), αn (t) ≤ ηi ≤ βn (t), αn (tk ) ≤ ηl ≤ βn (tk ), i = 1, 3, l = 2, 4, k = 0, 1, . . . , p. It is easy to verify the following inequalities 1 3 an (t)(βn (t) − αn (t)) = an (bn + an ) ≤ b2n (t) + a2n (t), 2 2 1 3 2 2 an (t)(βn (tk ) − αn (tk )) ≤ ||bn || + ||an || , 2 2 1 3 2 a(tk ))(βn (t) − αn (t)) ≤ ||bn || + ||an ||2 , 2 2 1 3 an (tk ))(βn (tk ) − αn (tk )) ≤ ||bn ||2 + ||an ||2 2 2

(32)

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L.M. Hall and S.G. Hristova / Central European Journal of Mathematics 2(2) 2004 250–259

where ||a|| = max{|a(t) : t ∈ [0, T ]}. From inequalities (31) and (32), and qnk (t) ≤ 0, it follows that for t ∈ (tk , tk+1 ] the inequalities an+1 (t) ≤ Qkn (t)an+1 (t) + Lk qnk (t)an+1 (tk ) + σnk (t) ≤ Qkn (t)an+1 (t) + σnk (t),

(33)

hold, where 3 3 σnk (t) = [Fxx (t, ξ1 , ξ2 ) + gxx (t, η1 , η2 ) + Lk gxy (t, η1 , η2 ) 2 2 3 +Lk Fxy (t, ξ1 , ξ2 ) + Lk Fxy (t, ξ3 , ξ4 ) + Lk gxy (t, η3 , η4 ) 2 3 + (Lk )2 gyy (t, η3 , η4 ) + (Lk )2 Fyy (t, ξ3 , ξ4 )]||an ||2 2 1 + [gxx (t, η1 , η2 ) + Lk gxy (t, η1 , η2 ) + Lk gxy (t, η3 , η4 ) 2 +(Lk )2 gyy (t, η3 , η4 )]||bn ||2 .

(34)

From the periodic condition for the functions u(t) and αn (t) we obtain, an+1 (0) = an+1 (T ).

(35)

From inequality (33) and equation (35) we obtain the following estimate of the function an+1 (t): an+1 (t) ≤ (an+1 (0) + t

+ t kt

+ tk

Σk−1 i=0

ti+1 ti

σnk (s) exp(−

s tk

σni (s) exp(−

s ti

Qin (τ )dτ )ds

Qkn (τ )dτ )ds) exp(Σk−1 i=0

ti+1 ti

Qin (s)ds

Qkn (s)ds), t ∈ (tk , tk+1 ],

(36)

where an+1 (0) ≤ [1 −

exp Σpi=0

×Σpi=0

s ti

ti+1 ti

Qin (s)ds)]−1

σni (s) exp(−

ti+1 ti

Qin (τ )dτ )ds.

(37)

From the properties of the functions F (t, x, y) and g(t, x, y), the deﬁnition (34) of the σ(t) and inequalities (36) and (37) it follows that there exist constants λ1 > 0 and λ2 > 0 such that ||an+1 || ≤ λ1 ||an ||2 + λ2 ||bn ||2 . (38) Similarly it can be shown that there exists constants µ1 > 0 and µ2 > 0 such that ||bn+1 || ≤ µ1 ||bn ||2 + µ2 ||an ||2 .

(39)

Inequalities (38) and (39) prove that the convergence is quadratic.

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259

References [1] R. Bellman and R. Kalaba: Quasilinearization and Nonlinear Boundary Value Problems, Elsivier, New York, 1965. [2] F.H. Clark, Yu.S. Ledayaev, R.I. Steru and P.R. Wolenski: Nonsmooth Analysis and Control Theory, Springer Verlag, New York, 1998. [3] L.J. Grimm and L.M. Hall: “Diﬀerential Inequalities and Boundary Problems for Functional-Diﬀerential Equations“, In: Simposium on Ordinary Diﬀerential Equations, Lecture Notes in Mathematics, Vol. 312, Springer Verlag, Berlin, pp. 41–53. [4] G. Ladde, V. Lakshmikantham and A. Vatsala: Monotone Iterative Techniques for Nonlinear Diﬀerential Equations, Pitman, Belmonth, 1985. [5] V. Lakshmikantham, “Extension of the method J.Optim.Theor.Applic., Vol. 82, (1994), pp. 315–321.

of

quasilinearization“,

[6] V. Lakshmikantham and X.Z. Liu: “Impulsive hybrid systems and stability theory“, Intern.J. Nonlinear Diﬀ.Eqns, Vol. 5, (1999), pp. 9–17. [7] V. Lakshmikantham and S. Malek: Generalized quasilinearization, Nonlinear World, 1, (1994), 59-63. [8] V. Lakshmikantham and J.J. Nieto: “Generalized quasilinearization for nonlinear ﬁrst order ordinary diﬀerential equations“, Nonlinear Times and Digest, Vol. 2, (1995), pp. 1–9. [9] V. Lakshmikantham, N. Shahzad and J.J. Nieto: “Method of generalized quasilinearization for periodic boundary value problems, Nonlinear Analysis, Vol. 27, (1996), pp. 143–151. [10] V. Lakshmikantham and A.S. Vatsala: Generalized Quasilinearization for Nonlinear Problems, Kluwer Academic Publishers, 1998. [11] A. Nerode and W. Kohn: Models in Hybrid Systems, Lecture Notes in Computer Science, Vol. 36, Springer Verlag, Berlin, 1993.

CEJM 2(2) 2004 260–271

The Dirichlet problem for Baire–one functions Jiˇr´ı Spurn´ y∗ Faculty of Mathematics and Physics, Charles University, Sokolovsk´a 83, 186 75 Praha 8, Czech Republic

Received 6 October 2003; accepted 1 April 2004 Abstract: Let X be a compact convex set and let ext X stand for the set of all extreme points of X. We characterize those bounded function deﬁned on ext X which can be extended to an aﬃne Baire–one function on the whole set X. c Central European Science Journals. All rights reserved. Keywords: abstract Dirichlet problem, Baire–one functions MSC (2000): 46A55, 26A21

1

Introduction

Let f be a function deﬁned on the set ext X of all extreme points of a compact convex set X. There is a number of papers devoted to the question under which conditions the function f can be extended to a continuous aﬃne function deﬁned on the whole set X (see [2], [7] and [9]). The problem of ﬁnding this extension is sometimes called the abstract Dirichlet problem. The most general result in this direction due to E.M. Alfsen (see [3]) can be found in [1, Theorem II. 4.5]). Functions on ext X, which admit a solution to the abstract Dirichlet problem, are characterized by means of the upper and lower envelopes. The precise formulation is the following: Let f be a continuous function on ext X. Then f can be extended to an aﬃne continuous function deﬁned on X if and only if f ∗ = f∗ on ext X and µ(f ∗ ) = ν(f ∗ ) for every couple µ, ν of maximal probability measures on X with the same barycenter. ∗

E-mail: [email protected]ﬀ.cuni.cz

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261

We recall that the upper envelope f ∗ of a bounded function f on ext X is deﬁned as f ∗ := inf{h : h is aﬃne continuous on X and h ≥ f on ext X} , and f∗ := −(−f )∗ . The aim of our paper is a proof of an analogue of this result for aﬃne Baire–one functions. As in the aforementioned result we characterize those bounded functions deﬁned on ext X which admit an aﬃne Baire–one extension on the whole set X. We remark that the results of the paper are formulated in a more general context of function spaces. This framework required a distinction between various classes of “abstract” aﬃne Baire–one function as we explain later.

2

Notation and some basic facts

All topological space will be considered as Hausdorﬀ. If K is a compact space, we denote by C(K) the space of all continuous functions on K. We will identify the dual of C(K) with the space M(K) of all Radon measures on K. Let M1 (K) denote the set of all probability Radon measures on K and let εx stand for the Dirac measure at x ∈ K. We always consider the space M(K) endowed with the weak–star topology. If K is a topological space, we write B(K) for the space of all bounded Baire functions on K, i.e., the smallest space containing C(K) and closed with respect to taking pointwise limits of sequences. The space of all Baire–one functions (the space of pointwise limits of sequences of continuous functions) on K is denoted by B1 (K). If F is a family of functions on a space K, we denote by F b the family of all bounded elements of F. If f is a Baire–one function than it is easy to see that there are functions un , ln , n ∈ N, such that un , −ln , n ∈ N, are upper semicontinuous, un f and ln f on K (see e.g. [12, Ex. 3.G.1]). We will also need the fact that a real–valued function f on a normal topological space is of the ﬁrst Baire–class if and only if f −1 (U ) is an Fσ –set for any open set U ⊂ R (see e.g. [12, Ex. 3.A.1]). Throughout the paper we will consider a function space H on a compact space K. By this we mean a (not necessarily closed) linear subspace of C(K) containing the constant functions and separating the points of K. Let Mx (H) be the set of all H-representing measures (or brieﬂy representing measures) for x ∈ K, i.e., 1 Mx (H) := {µ ∈ M (K) : f (x) = f dµ for any f ∈ H}. K

If µ ∈ Mx (H), we say that x is a barycenter of µ and denote x = r(µ). The set ChH K := {x ∈ K : Mx (H) = {εx }} is called the Choquet boundary of H. It may be highly irregular from the topological point of view but it is a Gδ –set if K is metrizable. We introduce the following main examples of function spaces.

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(a) In the “convex case”, the function space H is the linear space Ac (X) of all continuous aﬃne functions on a compact convex subset X of a locally convex space. In this example, the Choquet boundary of Ac (X) coincides with the set of all extremal points of X and is denoted by ext X. Thus the barycenter of a probability measure µ on X is a unique point r(µ) ∈ X for which f (r(µ)) = X f dµ for any f ∈ Ac (X), i.e., µ is Ac (X)–representing x. (b) In the “harmonic case”, U is a bounded open subset of the Euclidean space Rm and the corresponding function space H is H(U ), i.e., the family of all continuous functions on U which are harmonic on U . In the harmonic case, the Choquet boundary of H(U ) coincides with the set ∂ reg U of all regular points of U . We deﬁne the space A(H) of all H-aﬃne functions as the family of all bounded Borel functions on K satisfying the following barycentric formula: f (x) = f dµ for each x ∈ K and µ ∈ Mx (H) . K

Further, let Ac (H) be the family of all continuous H–aﬃne functions on K. It is easy to deduce that Ac (H) coincides with H both in the “convex” and “harmonic” case. We write B1 (H) for the set of all pointwise limits of sequences from H and by B1b (H) we understand the set of bounded elements from B1 (H). We denote by B1bb (H) the family of all functions on K which are pointwise limits of a bounded sequence of functions from H. Obviously we have the following inclusion B1bb (H) ⊂ A(H) ∩ B1b (K) , but the converse need not hold (see [10, Example 5.5]). An upper bounded Borel function f is called H–convex if f (x) ≤ µ(f ) for any x ∈ K and µ ∈ Mx (H). Let Kc (H) denote the family of all continuous H–convex functions on K. We notice that the space Kc (H) − Kc (H) is uniformly dense in C(K) due to the lattice version of the Stone–Weierstrass theorem. The convex cone Kc (H) determines a partial ordering ≺ (called the Choquet ordering) on the space M+ (K) of all positive Radon measures on K: µ≺ν

iﬀ µ(f ) ≤ ν(f ) for each f ∈ Kc (H) .

Lemma I.4.7 in [1] implies that for any measure µ ∈ M1 (K) there exists a measure ν such that µ ≺ ν and ν is maximal in the Choquet ordering. If we take µ to be the Dirac measure εx in a point x ∈ K, we obtain that for any point x ∈ K there exists a maximal measure ν such that f (x) = ν(f ) for every f ∈ H. This is the content of the famous Choquet–Bishop–de-Leeuw theorem [1, Theorem I.4.8]. A signed measure µ ∈ M(K) is said to be a boundary measure if its total variation |µ| is maximal. A measure µ is boundary if and only if both its positive part µ+ and its negative part µ− is maximal. Also µ − ν is a boundary measure provided µ, ν are positive maximal measures. If K is metrizable, then a measure µ is boundary if and only if |µ|(K \ ChH K) = 0. In nonmetrizable spaces every boundary measure µ ∈ M(K)

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263

satisﬁes |µ|(G) = 0 for any Gδ –set disjoint from ChH K (see [5, Lemma 27.14]) and µ(B) = 0 for any Baire set B ⊂ K \ ChH K (see [1, Corollary I.4.12 and the subsequent Remark]. If, for every x ∈ K, a maximal measure representing x is uniquely determined, we say that H is a simplicial function space. In the “convex case” it is equivalent to say that X is a Choquet simplex (see [1, Theorem II.3.6]). We denote the unique maximal measure representing a point x ∈ K by δx . For a simplicial function space H we also deﬁne an operator T by T f (x) = δx (f ) ,

f ∈ Bb (K) .

It is well–known (see [10, Proposition 6.1]) that T f ∈ A(H) for any bounded Baire function f on K. Moreover, T f (x) = f ∗ (x) = inf{h(x) : h ≥ f, h ∈ H} ,

x∈K,

for every H–convex bounded upper semicontinuous function f on K (see [4, Theorem 3.1]). Note also that T f (x) = f (x) for every x ∈ ChH K and every f ∈ Bb (K). We write H⊥ for the space of all Radon measures µ on K which satisfy µ(h) = 0 for every h ∈ H. It follows from [4, Corollary 3.5] that H is simplicial if and only if there is no nonzero boundary measure µ ∈ (Ac (H))⊥ . If ϕ : X → Y is a continuous mapping of a compact space X onto a compact space Y and µ is a signed measure on X, its image ϕµ ˜ ∈ M(Y ) is deﬁned as ϕµ(B) ˜ := µ(ϕ−1 (B)) ,

B a Borel subset of Y .

According to [8, Theorem 12.46], for any bounded Borel function g on Y holds ϕµ(g) ˜ = µ(g ◦ ϕ) .

3

(1)

The Dirichlet problem for Baire–one functions

We are going to provide a necessary and suﬃcient condition which ensures that a bounded function f deﬁned on ChH K can be extended to an H–aﬃne Baire–one function h (Theorem 3.1). A more restrictive assumption imposed on f enables us to ﬁnd an extension h which is even contained in B1bb (H) (see Theorem 3.8). As in [1, Theorem II.4.] mentioned in the introduction, extendable Baire–one functions deﬁned on ChH K are characterized by means of the upper and lower envelopes. Unlike continuous extensions, we need to employ envelopes generated by H–aﬃne Baire– one functions. This leads us to the following deﬁnitions. For a bounded function f on ChH K and x ∈ K, we set fˆ(x) := inf{h(x) : h ≥ f on ChH K, h ∈ B1b (K) ∩ A(H)} , fˇ(x) := sup{h(x) : h ≤ f on ChH K, h ∈ Bb (K) ∩ A(H)} . 1

and

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Theorem 3.1. Let f be a bounded function on ChH K. Then the following two conditions are equivalent: (i) there is an H–aﬃne Baire–one function h on X such that f = h on ChH K; (ii) fˆ = fˇ on ChH K, fˆ is a Baire–one function on ChH K and µ(fˆ) = ν(fˆ) for every x ∈ K and every couple of maximal measures µ, ν ∈ Mx (H). Moreover, if this H–aﬃne Baire–one extension of f exists, it is uniquely determined. We start with the following well–known minimum principle for Baire H–concave functions. Proposition 3.2. Let f be an H–concave Baire function on K such that f ≥ 0 on ChH K. Then f ≥ 0 on K. Proof. Let f be an H–concave Baire–one function on K which is positive on the Choquet boundary ChH K. Suppose that f (x) < 0 for some x ∈ K. Then L := {y ∈ K : f (y) ≤ f (x)} is a Baire set not intersecting ChH K. According to [1, Corolary I.4.12 and the subsequent Remark], µ(L) = 0 where µ is a maximal measure representing x. Then the following inequalities f (x) ≥ µ(f ) =

f dµ > K\L

f (x) dµ = f (x) K\L

yield a contradiction and concludes the proof.

Before the proof of Theorem 3.1 we establish the following useful proposition which may be interesting on its own right. Note that it is a more general version of [15, Theorem 3.3]. Proposition 3.3. Let f be a bounded Borel function on K such that µ(f ) = f (x) for every x ∈ K and µ ∈ M1 (ChH K) ∩ Mx (H). Then f is H–aﬃne. Proof. Set L := {µ ∈ M1 (ChH K) : there exists x ∈ K such that r(µ) = x} . Then L is a compact subset of M1 (ChH K) and the mapping r : L → K is continuous and surjective. For any bounded Borel function g on ChH K we deﬁne g˜ : ChH K → R as g˜(µ) := µ(g), µ ∈ L. It is well–known that g˜ is a bounded Borel function on L (see e.g. [15, Proposition 3.2]). Let µ be an H–representing measure for x ∈ K. We need to prove that µ(f ) = f (x). Let {µi }i∈I be a net of molecular measures such that µi → µ, i.e.,

µi =

ni k=1

αki εxik

,

ni ∈ N ,

αki

∈ (0, 1) ,

ni k=1

αki = 1 , and xik ∈ K for k = 1, . . . , ni .

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265

For every xik we choose a measure λik ∈ L with r(λik ) = xik and deﬁne λi :=

ni

αki λik

and

Λi :=

k=1

ni

αki ελik .

k=1

Then λi ∈ M1 (ChH K) and Λi ∈ M1 (L) for i ∈ I. By passing to a subnet if necessary we may assume that there are probability measures λ ∈ M1 (ChH K), Λ ∈ M1 (L) such that λi → λ and Λi → Λ. We claim that r(λ) = x ,

r˜Λ = µ ,

Λ(f˜) = λ(f ) .

and

(2)

Indeed, for a function h ∈ H we have λ(h) = lim λi (h) = lim µi (h) = µ(h) = h(x) . i

i

Thus r(λ) = x as required. If g is a continuous function on K, then

ni αki ελik (g ◦ r) r˜Λ (g) = Λ(g ◦ r) = lim i

= lim

ni

i

αki (g

◦

k=1

r)(λik )

= lim i

k=1

ni

αki g(xik )

k=1

= lim µi (g) = µ(g) , i

which proves r˜Λ = µ. Since L ⊂ M1 (ChH K), we may regard Λ as a probability measure on M1 (ChH K). Further, for a continuous function g on ChH K we have Λ(˜ g ) = lim i

= lim i

ni

αki ελik (˜ g)

= lim i

k=1

ni k=1

ni

αki g˜(λik )

k=1

αki λik (g) = lim λi (g) = λ(g) i

= g˜(λ) . Thus Λ is a Ac (M1 (ChH K))–representing measure for λ, in other words, λ is a barycenter of Λ (we remind that M1 (ChH K) is a compact convex set and every continuous aﬃne function on M1 (ChH K) is of the form µ → µ(g) for some continuous function g on ChH K). According to [15, Proposition 3.1], Λ(˜ g ) = λ(g) for every bounded Borel ˜ function g on ChH K, in particular we have Λ(f ) = λ(f ). This concludes the proof of (2). Now we are able to ﬁnish the proof of the proposition. By combining (1) and (2)

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together with the assumption on the function f we get µ(f ) = (˜ rΛ)(f ) = Λ(f ◦ r) = (f ◦ r)(ν) dΛ(ν) L f˜(ν) dΛ(ν) = f˜(ν) dΛ(ν) = L

M1 (ChH K)

= Λ(f˜) = λ(f ) = f (x) . Hence f is H–aﬃne and the proof is ﬁnished.

Now we are ready for the proof of Theorem 3.1. Proof (of Theorem 3.1). The implication (i) =⇒ (ii) is almost obvious. If h is a Baire–one H–aﬃne function on K satisfying h = f on ChH K then h = fˆ = fˇ on ChH K. Obviously, h(x) = µ(fˆ) = ν(fˆ) for each couple of maximal measures µ, ν ∈ Mx (H). For the proof of the converse implication (ii) =⇒ (i), let f be a bounded Baire–one function on ChH K possesing the properties described in (ii). Let g stand for the common value of fˆ and fˇ on ChH K. Claim 3.4. There exists a bounded sequence {kn } of H–concave continuous functions on K such that kn → g on ChH K. Proof. Let {ln } and {un } be bounded sequences of functions on ChH K such that each ln and −un is lower semicontinuous, un < g < ln , un g and ln g. Let C be a real number such that −C ≤ un < ln ≤ C for every n ∈ N. Fix n ∈ N. We claim that un∗,ChH K = inf{h ∈ H : h ≥ un on ChH K} < ln on ChH K. Indeed, let x be a point in ChH K. We use [4, Lemma 1.1] and ﬁnd a probability measure µ ∈ Mx (H) such that µ(un ) = un∗,ChH K (x) and spt µ ⊂ ChH K. Then un∗,ChH K (x) = µ(un ) < µ(g) = µ(fˆ) ≤ inf{µ(h) : h ≥ f on ChH K, h ∈ B1b (K) ∩ A(H)} = inf{h(x) : h ≥ f on ChH K, h ∈ B1b (K) ∩ A(H)} = fˆ(x) < ln (x) . As x ∈ ChH K is arbitrary, un∗,ChH K < ln on ChH K. For every x ∈ ChH K we ﬁnd a continuous H–aﬃne function hx so that un ≤ hx on ChH K and hx (x) < ln (x). It follows from the continuity of hx and the semicontinuity of ln that the inequality hx < ln holds on some neighbourhood Ux of x. Compactness of ChH K yields the existence of ﬁnitely many points x1 , . . . , xk ∈ ChH K such that ChH K ⊂ ki=1 Uxi . Then the function kn := hx1 ∧ · · · ∧ hxk ∧ C

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is an H–concave continuous function on K which satisﬁes un ≤ kn ≤ ln on ChH K and kn ≤ C on K. As −C ≤ un ≤ kn on ChH K, it follows from Proposition 3.2 and from the construction that the sequence {kn } is bounded. Since kn → g on ChH K, the proof of the claim is ﬁnished. Claim 3.5. Let µ, ν be positive measures supported by ChH K such that µ ≺ ν. Then µ(g) = ν(g). Proof. Let {kn } be a bounded sequence of H–concave continuous functions converging pointwise to g on ChH K. Then µ(kn ) ≥ ν(kn ), n ∈ N, and from the Lebesque dominated convergence theorem it follows that µ(g) ≥ ν(g). On the other hand, Claim 3.4 obviously admits its counterpart. Thus we may ﬁnd a bounded sequence of H–convex continuous functions which pointwise converges to g on ChH K. Hence µ(g) ≤ ν(g) and the claim is proved. Claim 3.6. Let x be a point of K and µ1 , µ2 ∈ M1 (ChH K) be a couple of measures representing x. Then µ1 (g) = µ2 (g). Proof. Let νi , i = 1, 2, be maximal measures with µi ≺ νi , i = 1, 2. Then each νi represents x and ν1 (g) = ν2 (g) due to our assumption. As the support of both measures ν1 and ν2 is contained in ChH K, Claim 3.5 implies that µi (g) = νi (g) for i = 1, 2. Combining these equalities together we get that µ1 (g) = µ2 (g). Claim 3.7. There exists an H–aﬃne Baire–one function h on K such that h = f on ChH K. Proof. We deﬁne the function h : K → R as h(x) := µ(g) ,

µ ∈ M1 (ChH K) ∩ Mx (H) ,

x∈K.

According to Claim 3.6, the function h is well deﬁned and h = g on ChH K. Let L and r : L → K be as in Proposition 3.3. If g˜ : L → R is deﬁned as g˜(µ) := µ(g), µ ∈ L, then g˜ = h ◦ r. et {gn } be a bounded sequence of continuous functions on ChH K such that gn → g. Then each function g˜n : µ → µ(gn ), µ ∈ L, is continuous on L and g˜n → g˜. Thus g˜ is a g −1 (U )) is an Baire–one function on L. Let U be an open subset of R. Then h−1 (U ) = r(˜ Fσ –set in K because g˜−1 (U ) is an Fσ –set in L and r maps Fσ –sets in L to Fσ –sets in K. Thus h is a Baire–one function on K. It follows from the deﬁnition that µ(h) = h(x) for any x ∈ K and µ ∈ Mx (H) with the support in ChH K. Proposition 3.3 implies that h is H–aﬃne and the proof of the claim is ﬁnished. To conclude the proof of the theorem it remains to verify the last assertion concerning

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the unicity of the extension. But this easily follows from the minimum principle contained in Proposition 3.2. Theorem 3.8. Let f be a bounded function on ChH K. Then the following two conditions are equivalent: (i) there is a Baire–one function h ∈ B1bb (H) (respectively h ∈ B1bb (Ac (H))) on X such that f = h on ChH K; (ii) fˆ = fˇ on ChH K, fˆ is a Baire–one function on ChH K and µ(fˆ) = 0 for every boundary measure µ ∈ H⊥ (respectively µ ∈ (Ac (H))⊥ ). Moreover, if this extension of f exists, it is uniquely determined. Proof. Let {ln } and {un } be bounded sequences of semicontinuous functions on ChH K as in the proof of Theorem 3.1. We ﬁnd a real number C satisfying −C ≤ un < ln ≤ C and extend functions un and ln on the whole space K by setting un := −C and ln := C on K \ ChH K. Then every un and −ln is an upper semicontinuous function and un < ln . We are going to use the following version of the Hahn–Banach separation theorem: Let H be a function space on a compact space K and ϕ1 , −ϕ2 be upper semicontinuous function on K with ϕ1 < ϕ2 . Then there exists a function h ∈ H such that ϕ1 < h < ϕ2 if and only if µ1 (ϕ1 ) < µ2 (ϕ2 ) for every couple of measures µ1 , µ2 ∈ M1 (K) satisfying µ1 − µ2 ∈ H⊥ . (For the proof see [11, Lemma 3.3].) Let µ1 , µ2 ∈ M1 (K) be measures satisfying µ1 (h) = µ2 (h) for each h ∈ H. For i = 1, 2 we ﬁnd a maximal measue µ ˆi with µi K\ChH K ≺ µ ˆi and set λi := µi ChH K +ˆ µi . Then λ1 (h) = λ2 (h) for any h ∈ H, µ1 (un ) ≤ λ1 (un ) and λ2 (ln ) ≤ µ2 (ln ). We ﬁnd maximal measures ν1 , ν2 ∈ M1 (K) so that λi ≺ νi , i = 1, 2. As g satisﬁes the assumptions (ii) of Theorem 3.1, λi (g) = νi (g) due to Claim 3.5 for i = 1, 2. Since ν1 (g) = ν2 (g) due to our assumption, we get µ1 (un ) ≤ λ1 (un ) < λ1 (g) = ν1 (g) = ν2 (g) = λ2 (g) < λ2 (ln ) ≤ µ2 (ln ) . Hence we may apply the italicized result and ﬁnd a function hn ∈ H such that un < hn < ln . Then {hn } is a bounded sequence of functions from H which converges pointwise to g on ChH K. If x is an arbitrary point of K and µ is a maximal measure in Mx (H), hn (x) = µ(hn ) for every n ∈ N. It follows from the Lebesgue dominated convergence theorem that {hn (x)} is a convergent sequence. Hence, by setting h(x) := lim hn (x) , n→∞

x∈K,

we obtain the desired function h ∈ B1bb (H). If µ(g) = 0 for every boundary measure µ ∈ (Ac (H))⊥ , we apply the Hahn–Banach separation theorem for the function space Ac (H) instead of H. The unicity of the extension again follows from Proposition 3.2.

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Remarks 3.9. (a) For a simplicial function space H, any function f ∈ B1b (K)∩A(H) is in fact a pointwise limit of a bounded sequence of functions from Ac (H), i.e., B1b (K)∩A(H) = B1bb (Ac (H)). This assertion was proved in [10, Theorem 6.3]. (b) If f is a Baire–one aﬃne function on a compact convex set X, then f is a pointwise limit of a bounded sequence of aﬃne continuous functions. The proof of this assertion can be found in [13, Th´eor`eme 80]. If we write A(X) for the space of aﬃne functions on X, we have the following equalities A(Ac (X)) ∩ B1 (X) = A(X) ∩ B1 (X) = B1bb (Ac (X)) = B1b (Ac (X)) = B1 (Ac (X)) . The ﬁrst equality is the Choquet barycentric theorem [6] (see also [1, Theorem I.2.6]). The inclusion A(X) ∩ B1 (X) ⊂ B1bb (X) follows from the aforementioned [13, Th´eor`eme 80] and the remaining inclusions are trivial. With these facts in mind, we can rewrite Theorems 3.1 and 3.8 for the “convex case” in the form laid down in Corollary 3.10. Corollary 3.10. Let X be a compact convex set and f be a bounded function on ext X. Then the following conditions are equivalent: (i) there is a Baire–one aﬃne function h on X such that f = h on ext X; (ii) fˆ = fˇ on ext X, fˆ is a Baire–one function on ext X and µ(fˆ) = ν(fˆ) or every couple µ, ν of maximal probability measures on X with the same barycenter. Moreover, if this extension of f exists, it is uniquely determined. The following example shows that the assumption of the “topological” quality of the function fˆ in condition (ii) of Theorem 3.1 is necessary. Proposition 3.11. Let H be a simplicial function space on a metrizable compact space K. If f is a bounded Baire–one function on ChH K, then fˆ(x) = fˇ(x) = δx (f ) for every x ∈ K and µ(fˆ) = 0 for every boundary measure µ ∈ (Ac (H))⊥ . In particular, if ChH K is not an Fσ –set, then there exists a bounded Baire–one function f on ChH K such that fˆ = fˇ on ChH K, µ(fˆ) = 0 for every boundary measure µ ∈ (Ac (H))⊥ and f cannot be extended to an H–aﬃne Baire–one function. Proof. Obviously, fˇ(x) ≤ δx (f ) ≤ fˆ(x) for every x ∈ K. To verify the equality fˆ(x) = fˇ(x), ﬁx x ∈ K. Since K is supposed to be metrizable, δx (ChH K) = 1 and thus we can ﬁnd a sequence {Kn } of compact sets Kn ⊂ ChH K such that δx (Kn ) → 1. Let {ln } be a bounded sequence of lower semicontinuous functions on ChH K satisfying ln f and C be an upper bound of the sequence {ln }. If we deﬁne ˜ln := ln on Kn , C on K \ Kn , we obtain a sequence {˜ln } of H–concave lower semicontinuous functions on K which converges to f almost everywhere with respect to δx . Then {T ˜ln } is a bounded sequence of lower semicontinuous (and hence also Baire–one) H–aﬃne functions such that f ≤ T ˜ln

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on ChH K. Moreover, from the Lebesque dominated convergence theorem it follows that T ˜ln (x) → δx (f ). Hence fˆ(x) = δx (f ). Similarly we verify that fˇ(x) = δx (f ) for any x ∈ K. As was mentioned in the introduction, T f is a Borel function. Since H is a simplicial function space, there is no nonzero boundary measure µ ∈ (Ac (H))⊥ , and thus the condition µ(fˆ) = 0 for every boundary measure µ ∈ (Ac (H))⊥ is vacuously satisﬁed. If ChH K is not an Fσ –set, according to [14, Theorem] there exists a bounded Baire– one function f on K such that T f is not of the ﬁrst Baire class. Hence the function f ChH K cannot be extended to an H–aﬃne Baire–one function deﬁned on the whole space K.

Acknowledgments ˇ 201/03/0935, GACR ˇ 201/03/D120 and Research supported in part by the grants GACR in part by the Research Project MSM 1132 00007 from the Czech Ministry of Education.

References [1] E.M. Alfsen: Compact convex sets and boundary integrals, Springer–Verlag, New York-Heidelberg, 1971. [2] E.M. Alfsen: “Boundary values for homomorphisms of compact convex sets”, Acta Math., Vol. 120, (1968), pp. 149–159. [3] E.M. Alfsen: “On the Dirichlet problem on the Choquet boundary”, Math. Scand., Vol. 19, (1965), pp. 113–121. [4] N. Boboc and A. Cornea: “Convex cones of lower semicontinuous functions on compact spaces”, Rev. Roumaine Math. Pures Appl., Vol. 12, (1967), pp. 471–525. [5] G. Choquet: Lectures on analysis I - III., W.A. Benjamin Inc., New York– Amsterdam, 1969. [6] G. Choquet: “Remarque `a propos de la d´emonstration de l’unicit´e de P.A. Meyer”, S´eminaire Brelot–Choquet–Deny (Th´eorie de Potentiel), Vol. 8, (1961/62) 6 ann´ee. [7] E.G. Eﬀros: “Structure in simplexes II.”, J. Funct. Anal., Vol. 1, (1967), pp. 361–391. [8] E. Hewitt and K. Stromberg: Real and abstract analysis, Springer–Verlag, New York– Berlin, 1969. [9] A. Lazar: “Aﬃne products of simplexes”, Math. Scand., Vol. 22, (1968), pp. 165–175. [10] J. Lukeˇs, J. Mal´ y, I. Netuka, M. Smrˇcka and J. Spurn´ y: “On approximation of aﬃne Baire-one functions” Israel Jour. Math., Vol. 134, (2003), pp. 255–289. [11] J. Lukeˇs, T. Mocek, M. Smrˇcka and J. Spurn´ y: “Choquet like sets in function spaces”, Bull. Sci. Math., Vol. 127, (2003), pp. 397–437. [12] J. Lukeˇs, J. Mal´ y and L. Zaj´ıˇcek: Fine topology methods in real analysis and potential theory, Lecture Notes in Math., Vol. 1189, Springer–Verlag, 1986. [13] M. Rogalski: “Op´erateurs de Lion, projecteurs bor´eliens et simplexes analytiques”, J. Funct. Anal., Vol. 2, (1968), pp. 458–488.

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CEJM 2(2) 2004 272–293

Generalized Mukai conjecture for special Fano varieties Marco Andreatta1∗ , Elena Chierici1† , Gianluca Occhetta1‡ Dipartimento di Matematica via Sommarive 14 I-38050 Povo (TN)

Received 24 October 2003; accepted 26 March 2004 Abstract: Let X be a Fano variety of dimension n, pseudoindex iX and Picard number ρX . A generalization of a conjecture of Mukai says that ρX (iX − 1) ≤ n. We prove that the conjecture if X admits an unsplit covering family of holds for a variety X of pseudoindex iX ≥ n+3 3 rational curves; we also prove that this condition is satisfied if ρX > 1 and either X has a fiber type extremal contraction or has not small extremal contractions. Finally we prove that the conjecture holds if X has dimension five. c Central European Science Journals. All rights reserved. Keywords: Fano varieties, Rational curves MSC (2000): 14J45, 14E30

1

Introduction

Let X be a Fano variety, that is a smooth complex projective variety whose anticanonical bundle −KX is ample. We denote with rX the index of X and with iX the pseudoindex of X, deﬁned respectively as rX = max{m ∈ N | − KX = mL for some line bundle L}, iX = min{m ∈ N | − KX · C = m for some rational curve C ⊂ X}. In 1988, Mukai [11] proposed the following conjecture: ∗ † ‡

E-mail: [email protected] E-mail: [email protected] E-mail: [email protected]

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273

Conjecture 1.1 (A). Let X be a Fano variety of dimension n. Then ρX (rX − 1) ≤ n. A more general conjecture (since iX ≥ rX ), which we will consider here, is the following: Conjecture 1.2 (B). Let X be a Fano variety of dimension n. Then ρX (iX − 1) ≤ n, with equality if and only if X (PiX −1 )ρX . then ρX = 1; in that paper he implicIn 1990 Wi´sniewski [13] proved that if iX > n+2 2 itly noticed that the statement of Conjecture 1.2 is more natural. In 2002 Bonavero, Casagrande, Debarre and Druel [2] explicitely posed Conjecture 1.2 and proved it in the following situations: (a) X has dimension 4, (b) X is a toric variety of pseudoindex iX ≥ n+3 or of dimension ≤ 7, (c) X is a homogeneous Fano variety. In this paper we 3 prove the following two theorems: ; then Conjecture Theorem 1.3. Let X be a Fano variety of dimension n and iX ≥ n+3 3 1.2 holds if X admits an unsplit covering family of rational curves. This condition is satisﬁed if ρX > 1 and either X has a ﬁber type extremal contraction or has not small extremal contractions. Theorem 1.4. If X is a Fano variety of dimension ﬁve then conjecture 1.2 holds for X. We use the language of the Minimal Model Program, or Mori theory; therefore for us an extremal contraction is a map with connected ﬁbers from X onto a normal projective variety; such a map contracts all curves in an extremal face of the Kleiman-Mori cone N E(X) ⊂ N1 (X). Recall that, since X is Fano, N E(X) is contained in the half space deﬁned by {z ∈ N1 (X) | KX ·z < 0} and so, by the Cone theorem, N E(X) is a polyhedral closed cone. We use the typical tools for this kind of problems, in particular the existence of “many” rational curves on X, a fundamental property of Fano varieties shown by Mori [10]. We work with families of rational curves, i.e. components of the scheme Ratcurvesn (X) which parametrizes birational morphisms P1 → X up to automorphisms of P1 , and families of rational 1-cycles, i.e. components of Chow(X), which we call Chow families; we will denote families of rational curves by capital letters and Chow families by calligraphic letters. To a family of rational curves V one can associate a Chow family V, taking the closure of the image of V in Chow(X) via the natural morphism Ratcurvesn (X) → Chow(X); if V is an unsplit family, i.e. if V is a proper scheme, then the two notions essentially agree and we can identify V with V. Fano varieties are rationally connected, i.e. through every pair of points x, y ∈ X there exists a rational curve; this was proved in [4] and in [9]. In this paper, as in [1], we

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use the notion of rational connectedness with respect to some chosen Chow families of rational curves V 1 , . . . , V k : roughly speaking, X is rc(V 1 , . . . , V k ) connected if through every pair of points x, y ∈ X there passes a connected 1-cycle whose components belong to the families V 1 , . . . , V k . To the rc(V 1 , . . . , V k ) relation one can associate a proper ﬁbration, called rationally connected ﬁbration, deﬁned on an open set of X, whose ﬁbers are equivalence classes for the relation; this was proved again in [4] and [9]. Using this fact we prove that if X is rationally connected with respect to k unsplit families V 1 , . . . , V k then ρX ≤ k; we show that if iX ≥ n+3 and X admits an unsplit covering 3 family then X is rationally connected with respect to k ≤ 3 unsplit families, and equality holds if and only if X = (PiX −1 )3 . Next we prove that if iX ≥ n+3 and ρX > 1 then X admits an unsplit covering family if 3 either X has a ﬁber type extremal contraction or has not small extremal contractions. The proof of theorem 1.4 is more diﬃcult: we prove that X is rationally connected with respect to a suitable number of proper families, but one of them could be a non unsplit Chow family, so to get the result we have to bound the number of its possible splittings.

2

Families of rational curves

We recall some of our basic deﬁnitions; our notation is basically consistent with the one in [8] to which we refer the reader. Let X be a normal projective variety and let Hom(P1 , X) be the scheme parametrizing morphisms f : P1 → X; we consider Hombir (P1 , X) ⊂ Hom(P1 , X), the open subscheme corresponding to those morphisms which are birational onto their image, and its normalization Homnbir (P1 , X); the group Aut(P1 ) acts on Homnbir (P1 , X) and the quotient exists. Definition 2.1. The space Ratcurvesn (X) is the quotient of Homnbir (P1 , X) by Aut(P1 ), and the space Univ(X) is the quotient of the product action of Aut(P1 ) on the space Homnbir (P1 , X) × P1 . We have the following commutative diagram: Homnbir (P1 , X) × P1 ❄

Homnbir (P1 , X)

U

✲ Univ(X)

p ❄ u ✲ Ratcurvesn (X)

where u and U are principal Aut(P1 )-bundles and p is a P1 -bundle. Definition 2.2. We deﬁne a family of rational curves to be an irreducible component V ⊂ Ratcurvesn (X). Given a rational curve f : P1 → X we will call a family of deformations of f any irreducible component V ⊂ Ratcurvesn (X) containing u(f ).

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Given a family of rational curves, we have the following basic diagram: p−1 (V ) =: U p

i✲

X

❄

V where i is the map induced by the evaluation ev : Homnbir (P1 , X) × P1 → X and p is a P1 -bundle. We deﬁne Locus(V ) to be the image of U in X; we say that V is a covering family if i is dominant, i.e. if Locus(V ) = X. We will denote by deg V the anticanonical degree of the family V , i.e. the integer −KX · C for any curve C ∈ V . If we ﬁx a point x ∈ X, everything can be repeated starting from the scheme Hom( P1 , X ; 0 → x) which parametrizes morphisms f : P1 → X sending 0 ∈ P1 to x. Again we obtain a commutative diagram Homnbir (P1 , X; 0 → x) × P1 ❄

Homnbir (P1 , X; 0 → x)

U

✲ Univ(X, x)

p

❄

(1)

u ✲ Ratcurvesn (X, x)

and, given a family V ⊆ Ratcurvesn (X), we can consider the subscheme V ∩ Ratcurvesn (X, x) parametrizing curves in V passing through x. We usually denote by Vx a component of this subscheme. Definition 2.3. Let V be a family of rational curves on X. Then (a) V is unsplit if it is proper; (b) V is locally unsplit if for the general x ∈ Locus(V ) every component Vx is proper; (c) V is generically unsplit if there is at most a ﬁnite number of curves of V passing through two general points of Locus(V ). Remark 2.4. Note that (a) ⇒ (b) ⇒ (c). Proposition 2.5. [8, IV.2.6] Let X be a smooth projective variety and let V be a family of rational curves. Assume either that V is generically unsplit and x is a general point in Locus(V ) or that V is unsplit and x is any point in Locus(V ). Then (a) dim X + deg V ≤ dim Locus(V ) + dim Locus(Vx ) + 1 = dim V ; (b) deg V ≤ dim Locus(Vx ) + 1. Definition 2.6. We deﬁne a Chow family of rational curves to be an irreducible component V ⊂ Chow(X) parametrizing rational connected 1-cycles. Given a Chow family of rational curves, we have a diagram as before, coming from the

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universal family over Chow(X). U p

❄

i✲

X (2)

V In the diagram i is the map induced by the evaluation and the ﬁbers of p are connected and have rational components. Both i and p are proper (see for instance [8, II.2.2]). By [8, IV.4.10] the family V deﬁnes a proper prerelation in the sense of [8, IV.4.6] (note that schemes and morphisms appearing in that deﬁnition are those of the normal form [8, IV.4.4.5]). Definition 2.7. If V is a family of rational curves we can consider the closure of the image of V in Chow(X), and call it the Chow family associated to V . Remark 2.8. If V is proper, i.e. if the family is unsplit, then V corresponds to the normalization of the associated Chow family V; in particular V itself deﬁnes a proper prerelation.

3

Chains of rational curves

Let X be a normal proper variety, V 1 , . . . , V k Chow families of rational curves on X and Y a subset of X. Definition 3.1. We denote by Locus(V 1 , . . . , V k ) the set of points x ∈ X such that there exist cycles C1 , . . . , Ck with the following properties: • Ci belongs to the family V i ; • Ci ∩ Ci+1 = ∅; • x ∈ C1 ∪ · · · ∪ Ck , i.e. Locus(V 1 , . . . , V k ) is the set of points which belong to a connected chain of k cycles belonging respectively to the families V 1 , . . . , V k . Note that if V is a Chow family then Locus(V) is the image of U in X through i in diagram 2, so, since V, p and i are proper, Locus(V) is a closed subset of X. Definition 3.2. We denote by Locus(V 1 , . . . , V k )Y the set of points x ∈ X such that there exist cycles C1 , . . . , Ck with the following properties: • Ci belongs to the family V i ; • Ci ∩ Ci+1 = ∅; • C1 ∩ Y = ∅ and x ∈ Ck , i.e. Locus(V 1 , . . . , V k )Y is the set of points that can be joined to Y by a connected chain of k cycles belonging respectively to the families V 1 , . . . , V k . Note that Locus(V 1 , . . . , V k )Y ⊂ Locus(V k ).

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Remark 3.3. If Y is a closed subset, then Locus(V 1 , . . . , V k )Y is closed. We prove the statement by induction, since we have Locus(V 1 , . . . , V k )Y = Locus(V k )Locus(V 1 ,...,V k−1 )Y . With the notation of diagram 2 let VY = p(i−1 (Y ∩ Locus(V))) be the subset of V parametrizing cycles of V meeting Y ; Locus(V)Y is just i(p−1 (VY )), so it is closed by the properness of i and p. Definition 3.4. We denote by ChLocusm (V 1 , . . . , V k )Y the set of points x ∈ X such that there exist cycles C1 , . . . , Cm with the following properties: • Ci belongs to a family V j ; • Ci ∩ Ci+1 = ∅; • C1 ∩ Y = ∅ and x ∈ Cm , i.e. ChLocusm (V 1 , . . . , V k )Y is the set of points that can be joined to Y by a connected chain of at most m cycles belonging to the families V 1 , . . . , V k . Remark 3.5. Note that ChLocusm (V 1 , . . . , V k )Y =

Locus(V i(1) , . . . , V i(m) )Y ;

1≤i(j)≤k

in particular, if Y is a closed subset then ChLocusm (V 1 , . . . , V k )Y is closed. Definition 3.6. We deﬁne a relation of rational connectedness with respect to V 1 , . . . , V k on X in the following way: x and y are in rc(V 1 , . . . , V k ) relation if there exists a chain of rational curves in V 1 , . . . , V k which joins x and y, i.e. if y ∈ ChLocusm (V 1 , . . . , V k )x for some m. Remark 3.7. The rc(V 1 , . . . , V k ) relation is nothing but the set theoretic relation U1 , . . . , Uk associated to the proper proalgebraic relation Chain(U1 , . . . , Uk ) in the language of [8, IV.4.8]. To the rc(V 1 , . . . , V k ) relation we can associate a ﬁbration, at least on an open subset. Theorem 3.8. [8, IV.4.16] There exist an open subvariety X 0 ⊂ X and a proper morphism with connected ﬁbers π : X 0 → Z 0 such that (a) the rc(V 1 , . . . , V k ) relation restricts to an equivalence relation on X 0 ; (b) the ﬁbers of π are equivalence classes for the rc(V 1 , . . . , V k ) relation; (c) for every z ∈ Z 0 any two points in π −1 (z) can be connected by a chain of at most 2dim X−dim Z − 1 cycles in V 1 , . . . , V k .

4

Bounding the Picard number of X

Lemma 4.1. Let Y ⊂ X be a closed subset, V a Chow family of rational curves. Then every curve contained in Locus(V)Y is numerically equivalent to a linear combination

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with rational coeﬃcients of a curve contained in Y and irreducible components of cycles parametrized by V which intersect Y . Proof. Let VY = p(i−1 (Y ∩ Locus(V))), let UY = p−1 (VY ) and consider the restriction of diagram 2 UY p

i✲

X

❄

VY

Let C be a curve in Locus(V)Y which is not an irreducible component of a cycle parametrized by V. Then i−1 (C) contains an irreducible curve C which is not contained in a ﬁber of p and dominates C via i. Let B = p(C ) and let S be the surface p−1 (B). Note that there is a curve CY in S which dominates B and such that i(CY ) is contained in Y : this is due to the fact that the image via i of every ﬁber of p|S meets Y . By [8, II.4.19] every curve in S is algebraically equivalent to a linear combination with rational coeﬃcients of CY and of the irreducible components of ﬁbers of p|S (in [8, II.4.19] take X = S, Y = B and Z = CY ). Thus any curve in i(S), and in particular C, is algebraically, hence numerically, equivalent in i(UY ) = Locus(V)Y (and hence in X) to a linear combination with rational coeﬃcients of i∗ (CY ) and of irreducible components of cycles parametrized by VY . Corollary 4.2. Let Y ⊂ X be a closed subset, V 1 , . . . , V k Chow families of rational curves, m a positive integer. Then every curve contained in ChLocusm (V 1 , . . . , V k )Y is numerically equivalent to a linear combination with rational coeﬃcients of a curve contained in Y and irreducible components of cycles in V 1 , . . . , V k .

Proof. By 3.5, ChLocusm (V 1 , . . . , V k )Y = 1≤i(j)≤k Locus(V i(1) , . . . , V i(m) )Y , so every irreducible component of ChLocusm (V 1 , . . . , V k )Y is contained in i(1) i(m) Locus(V , . . . , V )Y for some m-uple (i(1), . . . , i(m)). Then we note that the corollary is true for Locus(V i(1) , . . . , V i(m) )Y , applying m times lemma 4.1 with Y0 = Y and Yj = Locus(V i(1) , . . . , V i(j) )Y . Proposition 4.3. Let V 1 , . . . , V k be Chow families of rational curves on X and let π : X 0 → Z 0 be the rc(V 1 , . . . , V k ) ﬁbration. Let Y ⊂ X be a closed subset which dominates Z 0 via π; then every curve in X is numerically equivalent to a linear combination with rational coeﬃcients of a curve contained in Y and irreducible components of cycles in V 1 , . . . , V k . Proof. By theorem 3.8 and the assumption, every couple of points in a general ﬁber of π can be connected by a chain of cycles belonging to V 1 , . . . , V k of length at most M = 2dim X−dim Z − 1. In particular it follows that ChLocusM (V 1 , . . . , V k )Y is dense in

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X and, being closed by remark 3.5, it coincides with X. Then the claim follows from corollary 4.2. Corollary 4.4. Suppose that X is rationally connected with respect to some Chow families V 1 , . . . , V k ; then every curve in X is numerically equivalent to a linear combination with rational coeﬃcients of irreducible components of cycles in V 1 , . . . , V k . In particular if X is rationally connected with respect to k unsplit families, then ρX ≤ k. Proof. We apply proposition 4.3 with π : X → {∗} the contraction of X to a point and Y any point in X. The second part follows from the fact that any cycle parametrized by an unsplit family is irreducible.

5

Unsplit families

The results in the previous section can be enriched if we consider unsplit families of rational curves instead of Chow families. Lemma 5.1. [12, Lemma 1] Let Y ⊂ X be a closed subset, V an unsplit family of rational curves. Then Locus(V )Y is closed and every curve contained in Locus(V )Y is numerically equivalent to a linear combination with rational coeﬃcients λCY + µCV , where CY is a curve in Y , CV belongs to the family V and λ ≥ 0. Note that the improvement with respect to lemma 4.1 is the claim λ ≥ 0. Corollary 5.2. Let R = R+ [Γ] be an extremal ray of X, VΓ a family of deformations of a minimal extremal curve, x a point in Locus(VΓ ) and V an unsplit family of rational curves, independent from VΓ . Then every curve contained in Locus(VΓ , V )x is numerically equivalent to a linear combination with rational coeﬃcients λCV + µCΓ , where CV is a curve in V , CΓ belongs to the family VΓ and λ, µ ≥ 0. Proof. By lemma 5.1, if C is a curve in Locus(VΓ , V )x = Locus(V )Locus(VΓ )x , then C ≡ λCΓ + µCV , with λ ≥ 0 so we have only to prove that µ ≥ 0. If µ < 0, then we can write CΓ ≡ αCV + βC with α, β ≥ 0, but since CΓ is extremal, this implies that both [C] and [CV ] belong to R, a contradiction. One of the advantages of using unsplit families is given by the existence of good estimates for the dimension of Locus(V 1 , . . . , V k )x :

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Theorem 5.3. [2, Th´eor`eme 5.2] Let V 1 , . . . , V k be unsplit families of rational curves on X. If the corresponding classes in N1 (X) are independent, then either Locus(V 1 , . . . , V k )x is empty or it has dimension greater or equal to deg V i − k. Using the same techniques as in the proof of theorem 5.3 we obtain the following: Lemma 5.4. Let Y ⊂ X be a closed subset and V an unsplit family. Assume that curves contained in Y are numerically independent from curves in V , and that Y ∩Locus(V ) = ∅. Then for a general y ∈ Y ∩ Locus(V ) (a) dim Locus(V )Y ≥ dim(Y ∩ Locus(V )) + dim Locus(Vy ); (b) dim Locus(V )Y ≥ dim Y + deg V − 1. Moreover, if V 1 , . . . , V k are numerically independent unsplit families such that curves contained in Y are numerically independent from curves in V 1 , . . . , V k then either Locus(V 1 , . . . , V k )Y = ∅ or (c) dim Locus(V 1 , . . . , V k )Y ≥ dim Y + deg V i − k. Proof. We refer to diagram 2. Since V is unsplit, for a point y in Y ∩ Locus(V ) we have dim i−1 (y) = dim Vy = dim Locus(Vy ) − 1. So, setting VY = y ∈ Y ∩ Locus(V ),

p(i−1 (Y )) and UY

=

p−1 (VY ),

we have for general

dim UY = dim(Y ∩ Locus(V )) + dim Locus(Vy ) ≥ ≥ dim Y + dim Locus(V ) − n + dim Locus(Vy ) ≥ ≥ dim Y + deg V − 1. Since Locus(V )Y = i(UY ), (a) and (b) will follow if we prove that i : UY → X is generically ﬁnite. To show this we take a point x ∈ i(UY )\Y and we suppose that i−1 (x) ∩ UY contains a curve C which is not contained in any ﬁber of p; let B be the curve p(C ) ⊂ VY and let ν : B → B be the normalization of B . By base change we obtain the following diagram SB pB

j ✲ X

❄

B Let CY be a curve in SB which dominates B and whose image via j is contained in Y ; such a curve exists since the image via j of every ﬁber of pB meets Y . Now two cases are possible: either j(CY ) is a point, and therefore we have a one-parameter family of curves passing through two ﬁxed points, contradicting the fact that V is unsplit (see for instance [8, IV.2.3]) or j(CY ) is a curve in Y ∩ Locus(Vy ), so a curve in Y is numerically proportional to a curve parametrized by V , against the assumptions.

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To show (c) it is enough to recall that, as already observed in remark 3.3, we have Locus(V 1 , . . . , V k )Y = Locus(V k )Locus(V 1 ,...,V k−1 )Y . Remark 5.5. If in the previous theorem V 1 is not a covering family and Locus(V 1 , . . . , V k )x is nonempty, then dim Locus(V 1 , . . . , V k )x ≥

deg V i − k + 1.

In fact Locus(V 1 , . . . , V k )x = Locus(V 2 , . . . , V k )Locus(Vx1 ) , and we can apply part (c) of lemma 5.4, recalling that dim Locus(Vx1 ) = deg V 1 − 1 implies that V 1 is covering (see proposition 2.5).

6

Rational curves on Fano varieties

The geometry of Fano varieties is strongly related to the properties of families of rational curves of low degree. The ﬁrst result in this direction is a fundamental theorem, due to Mori: Theorem 6.1. Through every point of a Fano variety X there exists a rational curve of anticanonical degree ≤ dim X + 1. Remark 6.2. The families {V i ⊂ Ratcurvesn (X)} containing rational curves with degree ≤ n + 1 are only a ﬁnite number, so for at least one index i we have that Locus(V i ) = X. Among these families we choose one with minimal anticanonical degree, and call it a minimal dominating family. Note that every such family is locally unsplit. A relative version of Mori’s theorem is the following Theorem 6.3. [9, Theorem 2.1] Let X be a Fano manifold. Suppose that there exist a nonempty open subset X 0 of X, a smooth quasiprojective variety of positive dimension Z 0 and a proper surjective morphism π : X 0 → Z 0 . Let z be a general point on Z 0 . Then there exists a rational curve C on X satisfying (a) C ∩ π −1 (z) = ∅; (b) C is not contained in π −1 (z); (c) −KX · C ≤ n + 1. Remark 6.4. The families {V i ⊂ Ratcurvesn (X)} containing the horizontal curves with degree ≤ n+1 are only a ﬁnite number, so for at least one index i we have that Locus(V i ) dominates Z 0 . Among these families we choose one with minimal anticanonical degree, and call it a minimal horizontal dominating family for π. A typical situation where these morphisms arise is the construction of rationally connected ﬁbrations associated to families of rational curves, as we have explained in section 2, or more generally to a ﬁnite number of proper connected prerelations, as done in [8, IV.4.16].

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✲ Z be the rationally connected Lemma 6.5. Let X be a Fano variety, and let π : X ﬁbration associated to m proper connected prerelations on X; suppose that dim Z > 0 and let V be a minimal horizontal dominating family for π. Then (a) curves parametrized by V are numerically independent from curves contracted by π; (b) V is locally unsplit; (c) if x is a general point in Locus(V ) and F is the ﬁber containing x, then

dim(F ∩ Locus(Vx )) = 0. Proof. (a) Since X is normal and Z is proper, the indeterminacy locus E of π in X has codimension ≥ 2 [6, 1.39]. Pull back an ample divisor from Z and observe that it is zero on curves contracted by π. On the other hand it intersects nontrivially curves which are not contracted by π and are not contained in E, like curves of V , since V is dominant. (b) If for the general x ∈ Locus(V ) a curve in Vx degenerates into a reducible cycle, then at least one component of this cycle is horizontal, otherwise curves in V would be numerically equivalent to curves in the ﬁbers. But this contradicts the minimality of V among horizontal dominating families. (c) From lemma 4.1 we know that any curve in Locus(Vx ) is numerically proportional to V , while proposition 4.3 applied to F implies that all curves in F can be written as linear combinations of curves contracted by π. ✲ Z be the rationally conCorollary 6.6. Let X be a Fano variety, and let π : X nected ﬁbration associated to m proper connected prerelations on X; let V be a minimal horizontal dominating family for π. Then

deg V ≤ dim Z + 1. Proof. It follows from lemma 6.5 and the fact that dim Locus(Vx ) ≥ deg V − 1.

7

Special Fano varieties of high pseudoindex

In this section we will prove Theorem 1.3, dividing it in two steps. Theorem 7.1. Let X be a Fano variety of dimension n and pseudoindex ; if there exists a family V of rational curves which is unsplit and covering iX ≥ n+3 3 then Conjecture B is true for X. Note that if X has Picard number ρX = 1 then the inequality in conjecture B (which is equivalent to iX ≤ n+1) follows immediately from theorem 6.1, while the characterization of Pn as the only case for which equality holds is contained in [5] and [7]. Proof. Consider the rcV ﬁbration π : X 0 → Z 0 : if dim Z 0 = 0 then ρX = 1 by corollary 4.4 and we conclude. Otherwise take a minimal horizontal dominating family V ; from lemma 6.5 we know that Vx is unsplit for general x ∈ Locus(V ). Then applying lemma

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283

5.4 (b) with Y = Locus(Vx ) we obtain n ≥ dim Locus(V )Locus(Vx ) ≥

≥ dim Locus(Vx ) + deg V − 1 ≥

≥ deg V + deg V − 2 so deg V ≤ 2iX − 1 and therefore V is unsplit. Take the rc(V, V ) ﬁbration π : X → Z : if dim Z = 0 then from corollary 4.4 we have ρX = 2 and and conjecture B follows since we can assume that iX < n+2 (the 2 n+2 case iX > 2 has been dealt with in [13], while the characterization of the equality is contained in [12]) , otherwise take a minimal dominating family V with respect to π . For general x ∈ Locus(V ), denote by F the ﬁber of π containing x: then F is an equivalence class with respect to the rc(V, V ) relation, so F ⊇ Locus(V, V )y for some y; then theorem 5.3 implies dim F ≥ deg V + deg V − 2 ≥ 2iX − 2. By lemma 6.5 we have dim(Locus(Vx ) ∩ F ) = 0, so n ≥ dim F + dim Locus(Vx ) ≥ 2iX − 2 + deg V − 1, that is deg V ≤ n + 3 − 2iX ≤ iX . This is impossible unless deg V = deg V = deg V = iX and dim Locus(Vx ) = dim Locus(Vx ) = dim Locus(Vx ) = iX − 1. Proposition 2.5 implies that all these families ¨ are covering and from [ThEorEme 5.2] [2] it follows that X is rc(V, V , V ) connected, so we can apply [12, Theorem 1] to obtain that X (PiX −1 )3 . Theorem 7.2. Let X be a Fano variety of dimension n, pseudoindex iX ≥ n+3 and 3 Picard number ρX > 1. If X has a ﬁber type extremal contraction or has not small contractions then there exists a covering unsplit family V of rational curves. Proof. First of all suppose that there exists a ﬁber type contraction ϕ : X → W ; let Vϕ be a minimal horizontal dominating family for ϕ; since every ﬁber type extremal contraction can be seen as the rcV-ﬁbration with respect to some covering Chow family V, we can apply corollary 6.6 and obtain that deg Vϕ ≤ dim W + 1. Let F be a general ﬁber of ϕ; we have that dim F ≤ dim X − deg Vϕ + 1 ≤ 2iX − 2. By adjunction we have KF = (KX )F , so F is a Fano variety; in particular there exists a minimal dominating family VF of degree ≤ dim F + 1 ≤ 2iX − 1. This means that through a general point of X there passes a curve of degree ≤ 2iX − 1, and since the families of rational curves with bounded degree are a ﬁnite

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number, one of them must be covering; the bound on the degree implies that this family is also unsplit. Suppose now that all the extremal contractions of X are divisorial and, by contradiction, that there does not exist any unsplit covering family of rational curves. Let V be a minimal dominating family of rational curves; since we are assuming that V is not unsplit we have deg V ≥ 2iX . Consider the Chow family V associated to V : since deg V ≤ n + 1 < 3iX , reducible cycles in V split into exactly two irreducible components. To each one of them we associate the corresponding irreducible component of Ratcurvesn (X), which is an unsplit family. We denote by B the ﬁnite set of pairs of families (W i , W i+1 ) satisfying: • [W i ] is numerically independent from [W i+1 ]; • [W i ] + [W i+1 ] = [V ]; • W i and W i+1 contain irreducible components of cycles of V. Consider now the rcV ﬁbration π : X 0 → Z 0 . Claim.

dim Z 0 = 0.

Suppose by contradiction that Z 0 has positive dimension, and take V a minimal horizontal dominating family for π; we know from lemma 6.5 (c) that for a general ﬁber F we have dim Locus(Vx ) + dim F ≤ n, which implies deg V ≤ n + 1 − dim F ≤ n − 2iX + 2 < iX , a contradiction which proves the claim. As a corollary we obtain that N1 (X) is generated as a vector space by the numerical classes of the irreducible components of cycles in V (proposition 4.3) so, since ρX > 1 the set B is not empty. Note also that if [V ] is extremal in N E(X), then all the irreducible components of cycles in V are numerically proportional to [V ] and ρX = 1, so we can assume that [V ] is not extremal. Take now R1 = R+ [C1 ] to be a divisorial extremal ray of X, let E1 be its exceptional locus and V 1 an unsplit family of deformations of a minimal extremal rational curve C1 . First of all we claim that E1 ·V = 0; otherwise for a general x ∈ X the set Locus(V 1 )Locus(Vx ) would be nonempty, so by lemma 5.4 and proposition 2.5 dim Locus(V 1 )Locus(Vx ) ≥ dim Locus(Vx ) + deg V 1 − 1 ≥ 3iX − 2 > dim X. In particular, since N1 (X) is generated as a vector space by the numerical classes of the irreducible components of cycles in V we ﬁnd a pair (W 1 , W 2 ) ∈ B such that E1 · W 1 < 0 and E1 · W 2 > 0. Suppose that [W 1 ] = [λV 1 ] and let x be a point in Locus(W 1 ) ∩ Locus(W 2 ). Let C be a curve whose degree is minimum among curves in R1 passing through x; and consider the associated proper family V 1 x ; by corollary 5.2, the class of every curve in

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Locus(V 1 , W 1 )x can be written as a linear combination with positive coeﬃcients of [V 1 ] and [W 1 ], so, for x ∈ Locus(W 1 ) ∩ Locus(W 2 ),

dim(Locus(Wx2 ) ∩ Locus(V 1 , W 1 )x ) = 0; on the other hand, if [W 1 ] = [λV 1 ], by remark 5.5 we have

dim Locus(V 1 , W 1 )x ≥ 2iX − 1, and therefore

dim(Locus(Wx2 ) ∩ Locus(V 1 , W 1 )x ) ≥ 3iX − 2 − n > 0. We thus get a contradiction, unless [W 1 ] = [λV 1 ]. Note that this argument also shows that for all i = 1, 2 we have E1 · W i = 0. Since X is Fano and E1 is eﬀective there exists an extremal ray R2 on which E1 is positive (this is due to the fact that every eﬀective curve on a Fano manifold can be written as a linear combination with positive coeﬃcients of extremal curves: see [3, Lemma 2]); let E2 be the exceptional locus of R2 . We repeat the same argument and we ﬁnd a pair (W 3 , W 4 ) such that [V 2 ] = [µW 3 ] and E2 · W 4 > 0. If the plane Π1 spanned in N1 (X) by the classes [V ] and [V 1 ] is diﬀerent from the plane Π2 spanned by [V ] and [V 2 ], then [V 1 ], [V 2 ] and [W 4 ] are independent, and Locus(W 4 , V 2 , V 1 )x is nonempty for every x ∈ Locus(W 4 ). By remark 5.5 we get dim Locus(W 4 , V 2 , V 1 )x ≥ 3iX − 2 > n, a contradiction. So we suppose that Π1 = Π2 := Π and we choose a basis of N1 (X) formed by [V 1 ], [V ] and by classes [W i ] not contained in Π. Since the divisors E1 and E2 are zero on all the elements of the basis but [V 1 ], they are proportional in N 1 (X); but E1 · V 1 < 0 and E2 · V 1 > 0, so E1 = −kE2 with k > 0. One can now compute the intersection number of E1 and E2 with any curve which meets E1 ∪ E2 without being contained in it, and this leads to a contradiction.

8

Fano fivefolds with a covering unsplit family

This section and the following one are devoted to the proof of Theorem 1.4. Let X be a Fano variety of dimension 5 and let V ⊆ Ratcurvesn (X) be a minimal dominating family; by remark 6.2 we have that deg V ≤ 6 and Vx is unsplit for a general x ∈ X. If deg V = 6 then X = Locus(Vx ) and ρX = 1 by lemma 4.1, therefore we can assume deg V ≤ 5. First of all we note that if iX ≥ 3, then V is unsplit; moreover in this case we can apply theorem 7.1 and obtain the result, so from now on we assume that iX = 2 (and we thus have to prove that ρX ≤ 5).

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We divide the proof into two main cases: in this section we will deal with the case in which V is unsplit, while in the next one we will assume that V is not unsplit. Consider the rcV ﬁbration π : X 0 → Z 0 : if dim Z 0 = 0 then ρX = 1 by corollary 4.4 and we conclude; otherwise take a minimal horizontal dominating family V . Case 1.

V is not unsplit.

Note that in this situation deg V ≥ 4, so dim Locus(Vx ) ≥ 3; in particular, since V is horizontal and dominates Z 0 , we have also dim Z 0 ≥ 3. If dim Z 0 = 3 take a general point x ∈ Locus(V ), so that Vx is unsplit. Note that Y = Locus(Vx ) dominates Z 0 , so we can apply proposition 4.3 to get ρX = 2. If dim Z 0 = 4 consider the rc(V, V ) ﬁbration π : X → Z . Claim.

dim Z = 0.

Assume that this is not the case and denote by F a general ﬁber of π . Then there exists a minimal horizontal dominating family V satisfying 0 = dim(F ∩ Locus(Vx )) ≥ dim F + dim Locus(Vx ) − 5 ≥ 4 + dim Locus(Vx ) − 5 ≥ deg V − 2 for every x ∈ F ∩ Locus(V ). Thus deg V = 2 and dim Locus(Vx ) = 1, so by proposition 2.5 V is covering. Since V is horizontal also with respect to the ﬁbration π this contradicts the minimality of V , thus the claim is proved. From corollary 6.6 it follows that deg V ≤ 5, so every reducible cycle in V splits into exactly two irreducible components; moreover the family of deformations of each component is unsplit and non covering because of the minimality of V . Consider the pairs (W i , W i+1 ) of unsplit families satisfying • [W i ] + [W i+1 ] = [V ], • W i and W i+1 contain irreducible components of a cycle in V , and let B be the set of these pairs. If the numerical class of every pair in B lies in the plane Π ⊆ N1 (X) spanned by [V ] and [V ] then, by corollary 4.4 we have that ρX = 2 and we are done. Assume therefore by contradiction that there exists a pair (W 1 , W 2 ) ∈ B whose classes don’t lie in Π, call Π the plane spanned by [W 1 ] and [W 2 ] and set BΠ,Π = {(W i , W i+1 ) ∈ B | [W i ], [W i+1 ] ∈< Π, Π > and [W i ], [W i+1 ] = [λV ]}. For every (W i , W i+1 ) ∈ BΠ,Π , for every cycle Ci + Ci+1 ∈ W i + W i+1 and for every point by remark 5.5, we have x ∈ Ci we consider Locus(W i , V, W i+1 )x : i i+1 i+1 dim Locus(W , V, W )x ≥ 4; since W is not covering every irreducible component i i+1 of Locus(W , V, W )x is an eﬀective divisor on X, which is contained in Locus(W i+1 ). Since W i+1 does not dominate Z 0 , the intersection of any of these divisors with V is zero. We claim that the intersection of any of these divisors with V is also zero.

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In fact, if D = Locus(W i , V, W i+1 )x is such that D.V > 0, then every curve in V intersects Locus(W i+1 ). Since V is covering we have Locus(V )Locus(Vx ) ⊇ Locus(Vx ), so Locus(V, W i+1 )Locus(Vx ) = ∅; we apply lemma 5.4 (c) and we obtain that dim Locus(V, W i+1 )Locus(Vx ) = 5, which implies that W i+1 is covering, a contradiction. Obviously we can repeat the same argument with Locus(W i+1 , V, W i )x for every x ∈ Ci+1 , and we obtain eﬀective divisors which are contained in Locus(W i ) and whose intersection with V and V is zero. Call T the union of all these divisors. Now take a point y ∈ X \ T ; since X is rc(V, V ) connected, y can be joined to T by a chain of curves in V and cycles in V . In particular there exists a cycle Γ either in V or in V which intersects T but is not contained in it, and since every component of T has intersection zero with V and V , it must be of the form C3 + C4 , with (W 3 , W 4 ) ∈ B and [W 3 ], [W 4 ] ∈< Π, Π >. So, up to exchange W 3 and W 4 , there exists a component D of T such that D · W 3 > 0; then Locus(W 3 )D is nonempty and, by lemma 5.4 (b), dim Locus(W 3 )D ≥ dim D + deg W 3 − 1 ≥ 5 and W 3 is covering, a contradiction. Case 2.

V is unsplit.

Consider the rc(V, V ) ﬁbration π : X → Z ; if Z is a point then ρX = 2 and we conclude, otherwise take a minimal horizontal dominating family V . If V is not unsplit then deg V ≥ 4, so dim Locus(Vx ) ≥ 3; moreover, since dim Z ≤ 3, Locus(Vx ) dominates Z . Take a general point x ∈ Locus(V ), so that Vx is unsplit and apply proposition 4.3 with V, V and Y = Locus(Vx )to obtain ρX = 3. If V is unsplit we can take the rc(V, V , V ) ﬁbration π : X → Z : either Z is a point or every minimal horizontal dominating family is unsplit. We consider the new ﬁbration and we repeat the same argument. Finally we ﬁnd at most ﬁve independent unsplit families on X such that X is rationally connected with respect to them, so ρX ≤ 5 by corollary 4.4. If there are exactly ﬁve independent families, then they must be covering and of degree 2 and from [12] we conclude that X (P1 )5 .

9

Fano fivefolds without a covering unsplit family

We assume now that every minimal dominating family V of X is not unsplit, which implies that deg V ≥ 4.

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By the discussion at the beginning of the previous section we can also assume that iX = 2 and that deg V ≤ 5, so every reducible cycle in the associated Chow family V splits into exactly two irreducible components; moreover any family of deformations of each component is unsplit and non covering because of the minimality of V . Consider the pairs (W i , W i+1 ) of unsplit families satisfying • [W i ] + [W i+1 ] = [V ], • W i and W i+1 contain irreducible components of a cycle in V, and let B be the set of these pairs. Claim.

If deg V = 5 then ρX = 1.

Assume by contradiction that deg V = 5 and ρX ≥ 2. Suppose that all the irreducible components of cycles in V are numerically proportional to V , and consider the rcV ﬁbration π : X 0 → Z 0 . Now, either Z 0 is a point and in our assumptions ρX = 1 by corollary 4.4, or there exists a minimal horizontal dominating family V ; then for a general x ∈ Locus(V ), if F is the ﬁber through x, we know from lemma 6.5 that dim Locus(Vx ) + dim F ≤ 5, and since dim F ≥ deg V − 1 = 4 we have deg V − 1 ≤ dim Locus(Vx ) ≤ 1, forcing deg V = 2 and dim Locus(Vx ) = 1; hence by proposition 2.5 V is covering, against the assumptions. So there exists a pair (W 1 , W 2 ) ∈ B such that [W 1 ] = [αV ]. Let D be an irreducible component of Locus(Vx ) for a general x ∈ X; since V is locally unsplit we have N1 (D) =< [V ] >. By proposition 2.5, dim D ≥ deg V − 1 ≥ 4; as we are assuming ρX ≥ 2 it cannot be D = X, so D is an eﬀective divisor. If D · V = 0 then D would be negative on at least a family W i and so it would contain curves in W i , contradicting the fact that N1 (D) =< [V ] >. If else D · V > 0, then either D · W 1 > 0 or D · W 2 > 0; but in this case either Locus(Wx1 ) ∩ D or Locus(Wx2 ) ∩ D would be nonempty. Since W i is not covering we have dim Locus(Wxi ) ≥ 2, therefore dim(Locus(Wx1 ) ∩ D) ≥ 1, against the fact that N1 (Locus(Wxi )) =< [W i ] >. So the claim is proved and we can assume from now on that deg V = 4. Consider the rcV ﬁbration π : X 0 → Z 0 . Case 1

dim Z0 > 0.

In this case we actually prove that ρX = 2. Choose V to be a minimal horizontal dominating family for π; again we know that dim Locus(Vx ) + dim F ≤ 5, but in this case dim F ≥ 3 so deg V − 1 ≤ dim Locus(Vx ) ≤ 2. On the other hand dim Locus(Vx ) ≥ deg V ≥ 2, since otherwise V would be covering

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and of degree 2 by 2.5, contradicting the minimality of V . It follows that dim F = 3, dim Locus(Vx ) = 2 and deg V = 2, so V is unsplit and dim Locus(V ) = 4. Moreover, since Locus(Vx ) meets the general ﬁber of π, then X is rc(V, V ) connected. Let Π be the plane spanned by [V ] and [V ] and let BΠ = {(W i , W i+1 ) ∈ B | [W i ] and [W i+1 ] ∈ Π}. If BΠ = B then we have ρX = 2 by corollary 4.4. Suppose that this is not the case and let D be an irreducible component of Locus(V ). Since D does not contain the general ﬁber F of π and the general F coincides with Locus(Vx ) for some x, there exists a curve of V meeting D but not entirely contained in it; therefore D · V > 0. Let VD be the closed subfamily of V such that Locus(VD ) = D ; by lemma 5.4 (a) dim Locus(VD )D ∩F ≥ 4 i.e. Locus(VD )D ∩F = D . Since N1 (D ∩ F ) =< [V ] >, lemma 4.1 implies that N1 (D ) =< [V ], [V ] >. Let (W 1 , W 2 ) be a pair in B \ BΠ ; since D · V > 0 either D · W 1 > 0 or D · W 2 > 0, so we can assume Locus(W 1 )D = ∅; but this implies by lemma 5.4 that dim Locus(W 1 )D ≥ 5, and therefore that W 1 is covering, a contradiction. Case 2

dim Z0 = 0 i.e. X is rcV connected.

In this case by corollary 4.4 N1 (X) is generated as a vector space by the numerical classes of the irreducible components of cycles in V. We want to show that ρX ≤ 3, so by contradiction we assume that there exist three pairs (W 1 , W 2 ), (W 3 , W 4 ) and (W 5 , W 6 ) in B whose classes generate a four dimensional vector space inside N1 (X). Let Π ⊂ N1 (X) be the plane generated by [W 1 ] and [W 2 ], and let BΠ = {(W i , W i+1 ) ∈ B | [W i ] and [W i+1 ] ∈ Π}. For every pair (W i , W i+1 ) ∈ BΠ let {Dki } be the components of Locus(W i ) which intersect Locus(W i+1 ) and let {Dji+1 } be the components of Locus(W i+1 ) which intersect Locus(W i ). Let us note that, by proposition 2.5, every component of Locus(W i ) has dimension greater than three, so, since the families W i are not covering, we have dim Dki = 3 or 4. Case 2a

For every i and every k there exists j such that Dki = Dji+1 and viceversa.

If dim Dki = 3 then by proposition 2.5 dim Locus(Wxi ) = 3 and Dki is a component of Locus(Wxi ) for some x, so N1 (Dki ) =< [W i ] >; but since Dki = Dji+1 for some j, we have also N1 (Dki ) =< [W i+1 ] >, a contradiction. So we can assume that Dki is a divisor for every k; moreover Dki is a component of Locus(W i )Locus(Wxi+1 ) , and so N1 (Dki ) =< [W i ], [W i+1 ] >. Let us consider the intersection number of one of these divisors, say D11 =: D, with the family V ; if D · V > 0 then, up to exchange W 3 and W 4 , we have D · W 3 > 0.

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By lemma 5.4, since Locus(W 3 )D is nonempty, we have dim Locus(W 3 )D = 5, a contradiction since W 3 is not covering. Therefore D · V = 0, hence D · W i < 0 for some i; since N1 (D) is generated by the classes of W 1 and W 2 , the class of W i must belong to the plane Π. In particular for every pair (W i , W i+1 ) ∈ BΠ we have that (D·W i )(D·W i+1 ) < 0, yielding that D = Locus(W i ) = Locus(W i+1 ). Let now x be a point outside D and let z be a point of D; since X is rcV connected there exists a chain of cycles in V which connects x and z; let Γ be the ﬁrst irreducible component of one of these chains which meets D. Since D · V = 0 then Γ cannot belong to V or to a family which is proportional to V . Moreover, since Γ ⊂ D then Γ does not belong to a family whose class is contained in the plane Π. Therefore Γ belongs to a family W i whose class is not in Π; we can thus apply lemma 5.4 and obtain dim Locus(W i )D = 5, a contradiction, since W i is not covering. We have proved that case 2a cannot occur. We can therefore assume, up to rename the pairs in BΠ , that there exist meeting components D1 and D2 of Locus(W 1 ) and Locus(W 2 ) such that D1 = D2 . Case 2b

dim D1 = dim D2 = 4.

We claim that we cannot have D1 · V = D2 · V = 0. In fact, if D1 · V = 0, then for at least a family W i we have D1 · W i < 0. If i = 1, 2 then D1 = Locus(W i ) = Locus(W i )Locus(Wx1 ) and N1 (D1 ) =< [W 1 ], [W i ] >, hence, by lemma 5.4, dim Locus(W 2 )D1 = 5, a contradiction since W 2 is not covering. If D1 · W 2 < 0, then D2 ⊆ D1 , against our assumptions, so we have D1 · W 1 < 0 (and, in the same way D2 · W 2 < 0). It follows that D1 = Locus(W 1 ) and D2 = Locus(W 2 ); moreover the locus of every family of a pair belonging to BΠ is contained either in D1 or in D2 . Let T = D1 ∪ D2 , let z ∈ T and let x be a general point of X. Since X is rcV connected there exists a chain of cycles in V connecting x and z; let Γ be the ﬁrst irreducible component which meets T . The curve Γ cannot be numerically proportional to V , since D1 · V = D2 · V = 0, and its class cannot lie in the plane Π, so Γ belongs to an unsplit family W i which is independent from W 1 and W 2 ; so either D1 ·W i > 0 or D2 ·W i > 0, which implies that either D1 ·W i+1 or D2 · W i+1 is negative, a contradiction which proves the claim. Therefore we can assume that D1 · V > 0; up to exchange W 3 and W 4 we can also assume that D1 · W 3 > 0. Let x be a point on a curve in W 4 : then H = Locus(W 4 , W 3 , W 1 )x is nonempty and has dimension four by remark 5.5. If H · V > 0 then up to exchange W 5 and W 6 we can assume that H · W 5 > 0, and so by

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lemma 5.4 dim Locus(W 5 )H = 5, a contradiction. If H · V = 0, for some pair (W i , W i+1 ) we have H. · W i < 0 and H · W i+1 > 0. It cannot be i = 1, since in this case H = Locus(W 1 ) = D1 , but we are assuming that D1 · V > 0; therefore H · W i < 0 for some i such that W i is independent from W 1 . Let WH1 be the closed subfamily of W 1 whose locus is H; then H = Locus(WH1 )Locus(Wxi ) and so N1 (H) =< [W 1 ], [W i ] >. By construction, H ∩ Locus(W 3 ) is nonempty, so either i = 3, H contains Locus(W 3 ) and dim Locus(W 4 )H = 5, a contradiction, or i = 3 and dim Locus(W 3 )H = 5, again a contradiction. So case 2b cannot occur either. Case 2c

dim D1 = 3.

If D1 has dimension 3, then D1 is a component of Locus(Wx1 ) by proposition 2.5; therefore dim Locus(W 2 )Locus(Wx1 ) ≥ 4 by lemma 5.4. Let D2 be a component of Locus(W 2 )Locus(Wx1 ) ; since W 2 is not covering, D2 is a divisor in X and, by lemma 4.1, N1 (D2 ) =< [W 1 ], [W 2 ] >. Suppose that D2 · V > 0; then up to exchange W 3 and W 4 we have D2 · W 3 > 0, hence Locus(W 3 )D2 is nonempty and, by lemma 5.4, dim Locus(W 3 )D2 = 5, a contradiction. So we have D2 · V = 0; in this case D2 must be negative on one of the W i , but, since N1 (D2 ) =< [W 1 ], [W 2 ] >, [W i ] must belong to Π. In particular for every pair (W i , W i+1 ) ∈ BΠ we have that (D2 · W i )(D2 · W i+1 ) < 0. Moreover, if D2 · W i < 0, then D2 = Locus(W i ); in fact, if dim Locus(W i ) = 3 then we can apply lemma 5.4 (a) and get dim Locus(W 2 )Locus(Wxi ) = 5, a contradiction. Let T be the union of Locus(W i , W i+1 ) for (W i , W i+1 ) ∈ BΠ , let z ∈ T and let x be a point outside T ; since X is rcV connected we can join x to z with a chain of cycles in V; let Γ be the ﬁrst irreducible curve in the chain which meets T First of all we note that Γ cannot meet D2 ; in fact, since D2 · V = 0, Γ would be a curve in a family W i whose class does not lie in the plane Π, so that, by lemma 5.4 dim Locus(W i )D2 = 5, a contradiction. Therefore Γ meets a component Di = D2 of the locus of a family W i of a pair in BΠ such that Locus(W i+1 ) = D2 and such that D2 ∩ Di = ∅. If Di has dimension four, then we go back to case 2b, so we can assume that dim Di = 3, i.e. without loss of generality that Di = D1 . By construction, Γ cannot belong to a family W i whose class is contained in the plane Π and is not proportional to V ; on the other hand, if Γ belongs to a family W i whose class is not contained in Π, then, by lemma 5.4 dim Locus(W i , W i+1 )D1 = 5, a contradiction. It follows that either Γ belongs to an unsplit family αV whose numerical class is proportional to V or Γ belongs to V . In the ﬁrst case Locus(αV )D1 is a divisor D such that N1 (D ) =< [W 1 ], [αV ] >; if D ·V > 0, then we can assume that Locus(W 3 )D is nonempty and so dim Locus(W 3 )D = 5, a contradiction. Therefore D · V = 0, but, since D meets D1 and D ⊃ D1 then D · W 1 > 0 and D · W 2 < 0, so D = D2 and the curve is contained in T , a contradiction. Finally, if Γ belongs to V , we use the following

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Lemma 9.1. Let C be an irreducible curve in V . Then either C ⊂ Locus(Vx ) for some x such that Vx is unsplit or C ⊂ Locus(W i ) for some unsplit family W i such that [V ] = [W i ] + [W i+1 ]. Proof. If there exists a point x ∈ C such that Vx is unsplit, then we are in the ﬁrst case. Otherwise, for every x ∈ C there passes a reducible cycle Cxi + Cxj ∈ V. Since the families such that [W i ] + [W i+1 ] = [V ] are only a ﬁnite number, it follows that C ⊂ Locus(W i ) for some i. We thus have two possibilities for Γ: either Γ ⊂ Locus(Vx ), with Vx unsplit, so Locus(Vx )∩ D1 = ∅ and therefore dim Locus(Vx )∩D1 ≥ 1, a contradiction because N1 (Locus(Vx )) =< [V ] > and N1 (D1 ) =< [W 1 ] >, or Γ ⊂ Locus(W i ) with [W i ] ∈ Π; in this case Locus(W i , W i+1 )D1 is nonempty and by lemma 5.4 dim Locus(W i , W i+1 )D1 = 5, a contradiction.

Acknowledgments We would like to thank Cinzia Casagrande, Laurent Bonavero and Stephane Druel for some helpful remarks.

References [1] M. Andreatta and J.A. Wi´sniewski: “On manifolds whose tangent bundle contains an ample subbundle“, Invent. Math., Vol. 146, (2001), pp. 209–217. [2] L. Bonavero, C. Casagrande, O. Debarre and S. Druel: “Sur une conjecture de Mukai“, Comment. Math. Helv., Vol. 78, (2003), pp. 601–626. [3] L. Bonavero, F. Campana and J.A. Wi´sniewski: “Vari´et´es complexes dont l’´eclat’ee en un point est de Fano“, C.R. Math. Acad. Sci. Paris, Vol. 334, (2002), pp. 463–468. ´ [4] F. Campana: “Connexit´e rationnelle des vari´et´es de Fano“, Ann. Sci. Ecole Norm. Sup., Vol. 25, (1992), pp. 539–545. [5] K. Cho, Y. Miyaoka and N.I. Shepherd-Barron: “Characterizations of projective space and applications to complex symplectic manifolds”, In: Higher dimensional birational geometry (Kyoto, 1997), Adv. Stud. Pure Math., Vol. 35, Math. Soc. Japan, Tokyo, 2002, pp. 1–88. [6] O. Debarre: Higher-Dimensional Algebraic Geometry, Universitext, Springer-Verlag, New York, 2001. [7] S. Kebekus: “Characterizing the projective space after Cho, Miyaoka and ShepherdBarron”, In: Complex geometry (G¨ ottingen, 2000), Springer, Berlin, 2002, pp. 147– 155. [8] J. Koll´ar:Rational Curves on Algebraic Varieties, Ergebnisse der Math., Vol. 32, Springer-Verlag, 1996. [9] J. Koll´ar, Y. Miyaoka and S. Mori: “Rational connectedness and boundedness of Fano manifolds“, J. Diﬀ. Geom., Vol. 36, (1992), pp. 765–779.

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[10] S. Mori: “Projective manifolds with ample tangent bundle“, Ann. Math., Vol. 110, (1979), pp. 595–606. [11] S. Mukai: “Open problems“, In: Birational geometry of algebraic varieties, Taniguchi Foundation, Katata, 1988. [12] G. Occhetta: A characterization of products of projective spaces, preprint, February 2003, http://www.science.unitn.it/∼occhetta. [13] J.A. Wi´sniewski: “On a conjecture of Mukai“, Manuscripta Math., Vol. 68, (1990), pp. 135–141.

CEJM 2(2) 2004 294–331

Cycles of polynomial mappings in two variables over rings of integers in quadratic fields T. Pezda1∗ Department of Mathematics, University of Wroclaw, Pl.Grunwaldzki 2/4, 50-384 Wroclaw, Poland

Received 30 October 2003; accepted 9 March 2004 Abstract: We find all possible cycle-lengths for polynomial mappings in two variables over rings of integers in quadratic extensions of rationals. c Central European Science Journals. All rights reserved. Keywords: polynomial cycles, quadratic integers MSC (2000): 11R09, 11R11, 11E05

1

Introduction

For a commutative ring R with unity and Φ = (Φ1 , ..., ΦN ), where Φi ∈ R[X1 , ..., XN ] we deﬁne a cycle for Φ as a k-tuple x¯0 , x¯1 , ..., x¯k−1 of diﬀerent elements of RN such that Φ(¯ x0 ) = x¯1 , Φ(¯ x1 ) = x¯2 , ..., Φ(¯ xk−1 ) = x¯0 . The number k is called the length of this cycle. The set of all possible cycle lengths for polynomial mappings in N variables having coeﬃcients from R will be denoted by CYCL(R, N ). Remark 1.1. Clearly k ∈ CYCL(R, N ) implies l ∈ CYCL(R, N ) for every divisor l of k ( it suﬃces to take a suitable iteration). Owing to the last remark we see that a ﬁnite set CYCL(R, N ) is uniquely determined by CYCLMAX (R, N )— the set of maximal, in the sense of divisibility, elements ∗

E-mail: [email protected]

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of CYCL(R, N ). For example if CYCL(R, N ) = {30, 24, 15, 12, 10, 8, 6, 5, 4, 3, 2, 1} then CYCLMAX (R, N ) = {30, 24}. If K is a ﬁnite extension √ of Q then ZK will denote the set of all its integers. If [K : Q] ≤ 2 then√K = Q( d) for a uniquely determined square-free rational integer d. When writing Q( d) we assume that d is of this form. J.Boduch [Bo] and G.Baron [Ba] obtained the following result: √ Proposition 1.2. Let K = Q( d). Then CYCLMAX (ZK , 1) equals {2} for d = −3, −1, 2, 5; {4} for d = −1, 2; {6} for d = −3 and {4, 3} for d = 5. In [Pe1] it was shown that CYCL(Z, 2) = {24, 18, 16, 12, 9, 8, 6, 4, 3, 2, 1} and in this paper we will obtain an analogue of this result for rings of integers in quadratic ﬁelds and prove the following result: √ Theorem 1.3. Let K = Q( d). Then CYCLMAX (ZK , 2) equals (i) {24, 18, 16} for d ≡ 1 (mod 8); (ii) {32, 24, 18} for d ≡ 2 (mod 4); (iii) {48, 36, 32} for d ≡ 3 (mod 4); (iv) {72, 64, 56, 54, 48, 42, 40, 30} for d ≡ 13 (mod 24); (v) {192, 168, 144, 120, 108} for d ≡ 21 (mod 72); (vi) {192, 168, 144, 120, 108, 45} for d ≡ 69 (mod 72); (vii) {480, 360, 300, 192, 168, 156, 144, 132, 108, 105, 70} for d ≡ 29, 149, 221, 389, 581, 701, 749, 821 (mod 840); (viii) {480, 360, 300, 192, 168, 156, 144, 132, 110, 108, 105, 70, 65} for d ≡ 101, 269, 341, 461, 509, 629 (mod 840); (ix) {480, 360, 300, 210, 192, 168, 156, 144, 140, 132, 108} for d ≡ 365, 485, 1085, 1565, 2165, 2765, 2885, 4085 (mod 4200); (x) {480, 420, 360, 300, 192, 168, 156, 144, 132, 108} for d ≡ 245, 1205, 2045, 2405, 3005, 3245, 3605, 3845 (mod 4200); (xi) {480, 360, 300, 210, 192, 168, 156, 144, 140, 132, 130, 110, 108} for d ≡ 965, 1685, 2285, 3365, 3485, 3965 (mod 4200); (xii) {480, 420, 360, 300, 270, 192, 168, 156, 144, 132, 108} for d ≡ 53, 77, 197, 317, 413, 533, 557, 653 (mod 840); (xiii) {480, 420, 360, 300, 192, 168, 156, 144, 132, 130, 110, 108} for d ≡ 5, 605, 845, 1445, 1805, 2645 (mod 4200); (xiv) {480, 450, 420, 390, 360, 330, 300, 270, 192, 168, 156, 144, 132, 108} for d ≡ 173, 293, 437, 677, 773, 797 (mod 840). Note that in this list every square-free integer d appears, hence all rings of integers of quadratic ﬁelds are covered by the theorem.

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Auxiliary results and some notations

2.1 The main auxiliary theorem Proposition 2.1 (Pe2). Let R be a Dedekind ring. Then for N ≥ 2 we have p, N ), CYCL(Rp, N ) = CYCL(R CYCL(R, N ) = p

p

where the intersection is taken over all non-zero prime ideals p of R, Rp denotes the p is the completion of Rp with respect to the obvious corresponding localization and R valuation. This holds, in particular, for the rings ZK of integers in ﬁnite extensions K of Q. One of the aims of this paper is to show that the above result may be used to determine cycle lengths in an eﬀective manner.

2.2 Cycles in some local domains Owing to proposition 2.1 it is useful to recall some results concerning cycles in discrete valuation domains. In this subsection let R be a discrete valuation domain of characteristic zero, P its unique maximal ideal. We assume that the quotient ﬁeld R/P is ﬁnite and has pf elements, p prime. Let π be a generator of the principal ideal P and let v be the norm of R, normalized so that v(π) = p1 . By w we denote the corresponding exponent, deﬁned for x = 0 and w(0) = ∞. by w(x) = − loglogv(x) p We put w(p) = e, so e is the ramiﬁcation index of R. We extend w to RN putting

w((x1 , ..., xN )) = min{w(xi ), i = 1, ..., N }. The congruence symbol x¯ ≡ y¯ (mod P d ) will be used for x¯, y¯ ∈ RN to indicate that their corresponding components are congruent (mod P d ), or equivalently w(¯ x − y¯) ≥ d. N N N N The image of x¯ ∈ R under the canonical mapping R → R /P R = (R/P )N will be denoted by x¯ + P RN . We apply a similar convention to matrices. By ¯0 we shall denote the element (0, 0, ..., 0) ∈ RN . For a polynomial mapping Φ = (Φ1 , ..., ΦN ) we shall denote by Φ the matrix ∂Φi . ∂xj i,j=1,...,N The letter I will denote the unit matrix. Let Mm×n (R) be the set of all m × n matrices with coeﬃcients from R. A cycle x¯0 , ..., x¯k−1 will be called a (*)-cycle if for all i, j one has

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w(¯ xi − x¯j ) ≥ 1. CYCL∗(R, N ) will denote the set of all possible lengths of (*)-cycles in RN . Similarily to remark 1.1 we have Remark 2.2. If k ∈ CYCL ∗ (R, N ) then we have l ∈ CYCL ∗ (R, N ) for every divisor l of k. If x¯0 , ..., x¯k−1 is a cycle then we shall sometimes extend it periodically by putting x¯k = x¯0 , x¯k+1 = x¯1 , x¯k+2 = x¯2 and so on. Remark 2.3. If we write for instance Φ(x, y) = (π + x − y + 2x2 + ..., x + ...) then it means that Φ(0, 0) = (π, 0), the ﬁrst component of Φ has 1 as a coeﬃcient of x, −1 as a coeﬃcient of y and 2 as a coeﬃcient of x2 and some other non-speciﬁed coeﬃcients of other monomials of the form xi y j with i + j > 0, whereas the second component of Φ has 1 as a coeﬃcient of x and some other non-speciﬁed coeﬃcients of other monomials of the form xi y j with i + j > 0. Whatever non-speciﬁed coeﬃcients appear in proofs they do not aﬀect subsequent calculations and therefore there is no point to write them explicitly. Hence we use dots. Lemma 2.4. Let Φ : R2 → R2 be a polynomial mapping over R satisfying w(Φ(¯0)) ≥ 1, in other words the free terms of Φ are divisible by π. Put Φ(x, y) = (x1 +αx+βy+k1 x2 +k2 xy+k3 y 2 +..., y1 +γx+δy+K1 x2 +K2 xy+K3 y 2 + ¯ + k¯1 x2 + k¯2 xy + k¯3 y 2 , γ¯ x + δy ¯ + K¯1 x2 + K¯2 xy + K¯3 y 2 ) αx + βy ...). Also put Ψ1 (x, y) = (¯ ¯ + k¯2 xy, γ¯ x + δy ¯ + K¯2 xy), where t¯ ≡ t (mod P ) for t = and Ψ2 (x, y) = (¯ αx + βy α, β, γ, δ, k1 , k2 , k3 , K1 , K2 , K3 . Then (i) For every natural n the coeﬃcients of x2 , xy, y 2 in both components of Φn are congruent (mod P ) to the corresponding components of Ψn1 . (ii) If additionally charR/P = 2 ( in other words p = 2) then for every n ≥ 1 the coeﬃcients of xy in both components of Φn are congruent (mod P ) to the corresponding components of Ψn2 . Proof. By induction on n. Note that the asumption π|x1 , y1 is essential. In the proof of (ii) we clearly use 2 ∈ P . The following lemma contains some properties of polynomial mappings, which we shall use. Lemma 2.5. (i) Let Φ : RN → RN be a polynomial mapping with coeﬃcients from R. Put Φ(¯0) = x¯ and x¯ ≡ ¯0 (mod P d ). Then for every r ≥ 0 we have (Φr ) (¯0) ≡ (Φ (¯0))r (mod P d ). x) = d and and let r ≥ 0. If AM x¯ ≡ ¯0 (ii) Let A ∈ MN ×N (R) and x¯ ∈ RN , w(¯ (mod P d+r ) for some natural M then AN r x¯ ≡ ¯0 (mod P d+r ). (iii) If A, B ∈ MN ×N (R); x¯ ∈ RN ; w(¯ x) = r; A ≡ B (mod P d ) then A¯ x ≡ B x¯ d+r (mod P ).

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If A ∈ MN ×N (R); x¯, y¯ ∈ RN ; A ≡ 0 (mod P d ); x¯ ≡ y¯ (mod P r ) then A¯ x ≡ A¯ y d+r (mod P ). (iv) If A ∈ MN ×N (R) is invertible and x¯ ∈ RN then w(A¯ x) = w(¯ x). Proof. (i) For x¯1 , x¯2 ∈ RN the congruence x¯1 ≡ x¯2 (mod P d ) implies Φ(¯ x1 ) ≡ Φ(¯ x2 ) d d (mod P ) and Φ (¯ x1 ) ≡ Φ (¯ x2 ) (mod P ). Hence ¯0 ≡ Φ(¯0) ≡ Φ2 (¯0) ≡ ... ≡ Φr−1 (¯0) (mod P d ) and therefore from the chain rule we get (Φr ) (¯0) ≡ Φ (Φr−1 (¯0)) ◦ ... ◦ Φ (¯0) ≡ (Φ (¯0))r (mod P d ). (ii) This is lemma 6.2 from [Pe2]. (iii) Clear. (iv) For all B ∈ MN ×N (R) we have w(B x¯) ≥ w(¯ x) and it remains to observe that −1 x¯ = A A¯ x. 2 Lemma 2.6. (i) If k ∈ CYCL ∗ (R,  2) and k ≥ 2 then in R there is a (*)-cycle π x¯0 , x¯1 , ..., x¯k−1 with x¯0 = ¯0, x¯1 =  . 0 (ii) Let ¯0 = x¯0 , x¯1 , ..., x¯m−1 be a (*)-cycle in RN . Then w(¯ xr ) ≤ w(¯ xrs ) for all natural

r, s. (iii) Let x¯0 , x¯1 , ..., x¯m−1 be a (*)-cycle in RN . Then w(¯ xi+r − x¯i ) = w(¯ xj+r − x¯j ) for all i, j, r ≥ 0. (iv) Let ¯0 = x¯0 , x¯1 , ..., x¯m−1 be a (*)-cycle in RN for a mapping Φ. Put Φ (¯0) = A. Then for every k ≥ 0 we have x¯k ≡ (Ak−1 + Ak−2 + ... + A + I)¯ x1

(mod P 2w(¯x1 ) ).

x1 ) = r > 0. (v) Let ¯0 = x¯0 , x¯1 , ..., x¯m−1 be a (*)-cycle in RN of length m > 1. Put w(¯ Assume that m2 is the smallest j with w(¯ xj ) > w(¯ x1 ). Then the following elements N of (R/P ) : ¯0 + P RN , π −r x¯1 + P RN , π −r x¯2 + P RN , ..., π −r x¯m2 −1 + P RN are pairwise diﬀerent. (vi) Let ¯0 = x¯0 , x¯1 , ..., x¯m−1 be a (*)-cycle in R2 of length m > 1 for a mapping Φ. Put w(¯ x1 ) = r > 0. If π −r x¯1 + P R2 , π −r x¯2 + P R2 are independent over R/P  linearly    π 0 then in R2 there is a (*)-cycle of the form ¯0 = y¯0 ,   = y¯1 ,   = y¯2 , ..., y¯m−1 . 0 π (vii) Let ¯0 = x¯0 , x¯1 , ..., x¯m−1 be a (*)-cycle in R2 of length m > 1 for a mapping Φ such that Φ (¯0) (mod P ) has an eigenvalue λ + P for a suitable λ ∈ R. Then in R2 there is a (*)-cycle of the  form ¯0= y¯0 , y¯1 , ..., y¯m−1 for a mapping Ψ with w(¯ xi ) = w(¯ yi ) for λ ∗ i ≥ 0 and Ψ (¯0) ≡   (mod P ). Moreover we can assume that Ψ is of the 0∗ form Ψ = B −1 ◦ Φ ◦ B for a suitable invertible B ∈ M2×2 (R).

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(viii) Assume in addition that R is complete. Let ¯0 = x¯0 , x¯1 , ..., x¯m−1 be a (*)-cycle in R2 of length m > 1 for a mapping Φ such that Φ (¯0) (mod P ) has two diﬀerent eigenvalues µ1 + P, µ2 + P with µ1 , µ2 ∈ R. Then in R2 there is a (*)-cycle of the form¯0 = y¯0 , y¯1 , ..., y¯m−1 for a mapping Ψ with w(¯ xi ) = w(¯ yi ) for i ≥ 0 and  λ1 0  Ψ (¯0) =   for suitable λ1 , λ2 ∈ R such that λ1 ≡ µ1 0 λ2

(mod P ) and λ2 ≡ µ2

(mod P ). Proof. (i) Let x¯0 , x¯1 , ..., x¯k−1 be a (*)-cycle of length k in R2 for a mapping Φ. Then the k-tuple x¯0 − x¯0 , x¯1 − x¯0 , x¯2 − x¯0 , ..., x¯k−1 − x¯0 is a (*)-cycle for a polynomial mapping Ψ ¯ = Φ(X ¯ + x¯0 ) − x¯0 . deﬁned by Ψ(X) −r x1 ) = r > Thus we can assume that x¯0 = ¯0. Put w(¯ 0. Therefore w(π x¯1 ) = 0 and 1 there is an invertible matrix B ∈ M2×2 (R) such that B( ) = π −r x¯1 . Let the mapping 0 −(r−1) −1 r−1 ¯ As w(Φ(¯0)) = r > r − 1 we easily ¯ =π Ψ be deﬁned by Ψ(X) B ◦ Φ ◦ B(π X). see that Ψ has all its coeﬃcients in R. Moreover for all i ≥ 0 we have Ψ(B −1 π −(r−1) x¯i ) = −1 −(r−1) B −1 π −(r−1) x¯i+1 .  From x¯0 = ¯0 and  the observation that B π

π B −1 π −(r−1) x¯1 =   we obtain the assertion. 0 (ii) This is lemma 4.3 from [Pe2]. (iii) This is lemma 4.1(iii) from [Pe2]. (iv) This is lemma 4.6 from [Pe2]. (v) By (ii) the elements π −r x¯1 + P RN , π −r x¯2 + P RN , ..., π −r x¯m2 −1 + P RN are well deﬁned elements of (R/P )N . They are non-zero by the very deﬁnition of m2 . If for some xj − x¯i ) > r and by (iii) 0 < i < j < m2 we have π −r x¯i + P RN = π −r x¯j + P RN then w(¯ w(¯ xj−i ) = w(¯ xj−i − x¯0 ) = w(¯ xj − x¯i ) > r = w(¯ x1 ), which in view of j − i < m2 clearly contradicts the deﬁnition of m2 .     0 1 (vi) Let B be a 2×2 matrix such that B( ) = π −r x¯1 and B( ) = π −r x¯2 . Note 1 0 that because of our assumption and since R is a discrete valuation ring, B is invertible. ¯ and y¯i = B −1 π −(r−1) x¯i . Then y¯0 = ¯0, y¯1 = ¯ = π −(r−1) B −1 ◦ Φ ◦ B(π r−1 X), Put Ψ(X)     π 0  , y¯2 =   and y¯0 , y¯1 , ..., y¯m−1 is a (*)-cycle for Ψ. 0 π (vii) For every invertible matrix B ∈ M2×2 (R) and all x¯ ∈ RN we have by lemma 2.5(iv) w(B −1 x¯) = w(¯ x). −1 Putting y¯i = B x¯i we obtain a (*)-cycle ¯0 = y¯0 , y¯1 , ..., y¯m−1 for a mapping Ψ = −1 B ◦ Φ ◦ B with w(¯ yi ) = w(¯ xi ).

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Let z¯ ∈ R2 be such that Φ (¯0)¯ z ≡ λ¯ z (mod P ) and w(¯ z ) =  0. 1 Now let B ∈ M2×2 (R) be invertible and such that B( ) = z¯. For Ψ deﬁned as 0   λ ∗ above we obtain Ψ (¯0) = B −1 Φ (¯0)B ≡   (mod P ). 0∗    µ1 ∗  (viii) By (vii) and its proof it suﬃces to ﬁnd for every matrix A ≡   0 µ2    λ1 0  (mod P ) an invertible B ∈ M2×2 (R) such that B −1 AB =   with suitable λ1 , λ2 . 0 λ2      µ1 µ   1 −t  Write then A =  . Then for C =   we have C −1 AC = πa µ2 0 1     2  ∗ µ + t(µ2 − µ1 ) − t πa   µ1 ∗   and C −1 AC ≡   (mod P ). As µ2 − µ1 ∈ P , using  ∗ ∗ 0 ∗   ∗ 0 Hensel’s lemma for suitable t ∈ R we get C −1 AC =  . ∗∗    µ1 0  Thus we can assume that A =  . Taking πa µ2   1 0  B=  we obtain the assertion. πa(µ1 − µ2 )−1 1 Lemma 2.7. Let ¯0 = x¯0 , x¯1 , ..., x¯pα −1 be a (*)-cycle in R2 of length pα for a mapping Φ. i Assume that α ≥ 1 and put (Φp ) (¯0) = Ai ∈ M2×2 (R). (i) If p ≥ 3 then w(¯ x1 ) < w(¯ xp ) < w(¯ xp2 ) < ... < w(¯ xpα−1 ). (ii) If p = 2 then w(¯ x2 ) < w(¯ x4 ) < ... < w(¯ x2α−1 ). ¯ 2 If additionally Φ (0) = A0 ≡ B (mod P ) for suitable B ∈ M2×2 (R) then also w(¯ x1 ) < w(¯ x2 ). (iii) For all i > 0 and 0 ≤ β < α we have xpβ ≡ ¯0 (Ai − I)¯

(mod P w(¯xpβ )+1 ).

(iv) Assume that p = 2. If additionally Φ (¯0) = A0 ≡ B 2 B ∈ M2×2 (R) then for all 0 ≤ β < α we have xpβ ≡ ¯0 (A0 − I)¯

(mod P w(¯xpβ )+1 ).

(mod P ) for suitable

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(v) The matrix Ai (mod P ) has an eigenvalue 1 + P for all i ≥ 0. In particular Φ (¯0) (mod P ) has an eigenvalue 1 + P . (vi) Assume that α ≥ 2 and w(¯ xpα−1 ) ≥ e + 1. Then ((Aα−1 − I)p−1 + pI)¯ xpα−1 ≡ ¯0 (mod P w(¯xpα−1 )+e+1 ) and (Aα−1 − I)p x¯pα−1 ≡ ¯0 (mod P w(¯xpα−1 )+e+1 ). Proof. (i) As for every 0 ≤ β < α the tuple ¯0 = x¯0 , x¯pβ , x¯2pβ , ..., x¯(pα−β −1)pβ is a (*)-cycle β for Φp , it suﬃces to prove w(¯ x1 ) < w(¯ xp ). α α α have Ap −1 + Ap −2 + ... + A + I = (A − I)p −1 +  every square matrix  For  A we  α α  p   p pα −2 + ... +  (A − I) + pα I.  (A − I) α 1 p −2   α p  Thus in view of p|  for 0 < i < pα and 2w(¯ x1 ) ≥ w(¯ x1 ) + 1 by lemma 2.6(iv) i α we have ¯0 = x¯pα ≡ (A0 − I)p −1 x¯1 (mod P w(¯x1 )+1 ). Using lemma 2.5(ii) we then obtain (A0 − I)2 x¯1 ≡ ¯0 (mod P w(¯x1 )+1 ) and in view of p − 1 ≥ 2 we obtain (A0 − I)p−1 x¯1 ≡ ¯0 (mod P w(¯x1 )+1 ). Now lemma 2.6(iv) gives x¯p ≡ (Ap−1 + Ap−2 + ... + A0 + I)¯ x1 ≡ 0 0     p  p  x1 ≡ (A0 − I)p−1 x¯1 ≡ ¯0 ((A0 − I)p−1 +  (A0 − I)p−2 + ... +  (A0 − I) + pI)¯ 1 p−2

(mod P w(¯x1 )+1 ) and w(¯ xp ) > w(¯ x1 ) follows. (ii) As for every 0 < β < α the tuple ¯0 = x¯0 , x¯pβ , x¯2pβ , ..., x¯(pα−β −1)pβ is a (*)-cycle for β β−1 β−1 pβ Φ and by lemma 2.5(i) (Φp ) (¯0) ≡ ((Φp ) (¯0))p ≡ ((Φp ) (¯0))2 (mod P ), it suﬃces to prove the second part of the statement. So let Φ (¯0) ≡ B 2 (mod P ) for a suitable B ∈ M2×2 (R). α In the same way as in the proof of (i) we have (A0 −I)2 −1 x¯1 ≡ ¯0 (mod P w(¯x1 )+1 ). In α view of A0 ≡ B 2 (mod P ), using lemma 2.5(iii), we obtain (B 2 − I)2 −1 x¯1 ≡ ((B − I)2 + α α+1 2(B − I))2 −1 x¯1 ≡ (B − I)2 −2 x¯1 ≡ ¯0 (mod P w(¯x1 )+1 ). By lemma 2.5(ii) we then have (B − I)2 x¯1 ≡ ¯0 (mod P w(¯x1 )+1 ). Thus using lemma 2.6(iv) and lemma 2.5(iii) we obtain x1 ≡ (B 2 + I)¯ x1 ≡ ((B − I)2 + 2B)¯ x1 ≡ (B − I)2 x¯1 ≡ ¯0 (mod P w(¯x1 )+1 ) x¯2 ≡ (A0 + I)¯ and w(¯ x2 ) > w(¯ x1 ) follows. (iii) Considering for every 0 ≤ β < α the (*)-cycle ¯0 = x¯0 , x¯pβ , x¯2pβ , ..., x¯(pα−β −1)pβ of length pα−β for Φpβ we get, in the same way as in the α−β proof of (i), (Aβ − I)p −1 x¯pβ ≡ ¯0 (mod P w(¯xpβ )+1 ). β β By lemma 2.5(i) Aβ −I ≡ Ap0 −I ≡ (A0 −I)p (mod P ). Hence using lemma 2.5(iii) β α−β we obtain (A0 − I)p (p −1) x¯pβ ≡ ¯0 (mod P w(¯xpβ )+1 ). By lemma 2.5(ii) we therefore i have (A0 − I)2 x¯pβ ≡ ¯0 (mod P w(¯xpβ )+1 ) and in view of pi ≥ 2 also (A0 − I)p x¯pβ ≡ ¯0 (mod P w(¯xpβ )+1 ). In view of lemma 2.5(iii) and, obtained by lemma 2.5(i), Ai − I ≡ i i Ap0 − I ≡ (A0 − I)p (mod P ) we therefore obtain the statement.

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(iv) In the proof of (iii) we obtained (A0 − I)2 x¯pβ ≡ ¯0 (mod P w(¯xpβ )+1 ). As A0 ≡ B 2 (mod P ) we then have (B − I)4 ≡ (B 2 − I)2 ≡ (A0 − I)2 (mod P ) and then by lemma 2.5(iii) (B − I)4 x¯pβ ≡ ¯0 (mod P w(¯xpβ )+1 ). By lemma 2.5(ii) we therefore have (B − I)2 x¯pβ ≡ ¯0 (mod P w(¯xpβ )+1 ). Now we get the statement by lemma 2.5(iii) and (B − I)2 ≡ A0 − I (mod P ). (v) The statement concerning i > 0 clearly follows from (iii). As from lemma 2.5(i) (A1 − I) (mod P ) = (Ap0 − I) (mod P ) = (A0 − I)p (mod P ) the matrix (A0 − I) (mod P ) is not invertible. (vi) By lemma 2.6(iv) and w(¯ xpα ) ≥ e+1, applied to a (*)-cycle ¯0 = x¯0 , x¯pα−1 , x¯2pα−1 , ..., x¯(p−1)pα−1 , we have   p p−1 ¯0 = x¯pα ≡ (Ap−1 α−1 ≡ ((Aα−1 − I) + ... + A + I)¯ x +  (Aα−1 − I)p−2 + ... + α−1 p α−1 1    p  xpα−1 (mod P w(¯xpα−1 )+e+1 ).  (Aα−1 − I) + pI)¯ p−2 As α − 1 ≥ 1 we get by (iii) (Aα−1 − I)¯ xpα−1 ≡ 0¯ (mod P w(¯xpα−1 )+1 ). Thus we get the ﬁrst congruence of the statement by w(p) = e. The second congruence follows now from the ﬁrst by applying to both sides the matrix Aα−1 − I. N ) and Proposition 2.8. (i) For every natural N we have CYCL(R, N ) = CYCL(R, N ), where R is the completion of R with respect to v. CYCL ∗ (R, N ) = CYCL ∗ (R, (ii) For all natural k, N we have k ∈ CYCL(R, N ) if and only if k can be written in the form ab with a ≤ pf N and b ∈ CYCL ∗ (R, N ). (iii) If a1 , ..., ar are such that a1 + ... + ar ≤ N then for every k|[pf a1 − 1, ..., pf ar − 1] we have k ∈ CYCL ∗ (R, N ). (iv) Let ¯0 = x¯0 , x¯1 , ..., x¯m−1 be a (*)-cycle in RN for a mapping Φ. Put Φ (¯0) = A. Let (−1)N X a0 (X − 1)b0 F1 (X)b1 · ... · Fr (X)br be the characteristic polynomial of the matrix B = A (mod P ). We assume that Fi are pairwise diﬀerent, monic, and irreducible over R/P , Fi = X, X − 1; r ≥ 0, a0 , b0 ≥ 0; b1 , ..., br > 0. Put deg Fi = ai ( in particular a1 + ... + ar ≤ a0 + b0 + a1 b1 + ... + ar br = N ). Then m = k · pα with k|[pf a1 − 1, ..., pf ar − 1]. In particular if m = k · pα ∈ CYCL ∗ (R, N ) and p does not divide k then k|[pf a1 − 1, ..., pf ar − 1] for suitable a1 , ..., ar such that a1 + ... + ar ≤ N . (v) Suppose that N > 1. Let m = k · pα ∈ CYCL ∗ (R, N ) and p does not divide k. Suppose that k does not divide [pf a1 − 1, ..., pf ar − 1] whenever a1 + ... + ar < N . In particular k > 1. x1 ) ≥ 2 for a mapping Then in RN there is a (*)-cycle ¯0 = x¯0 , x¯1 , ..., x¯pα −1 with w(¯ ¯ Φ satisfying Φ (0) ≡ I (mod P ). (vi) Suppose that N > 1. Let ¯0 = x¯0 , x¯1 , ..., x¯m−1 be a (*)-cycle in RN of length m > 1. Put m = k · pα with p not dividing k. Assume that k does not divide

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[pf a1 − 1, ..., pf ar − 1] whenever a1 + ... + ar < N . In particular k > 1. Then w(¯ xl ) > w(¯ x1 ) for suitable l = [pf b1 − 1, ..., pf bs − 1] with b1 + ... + bs = N . In particular, if ¯0 = x¯0 , x¯1 , ..., x¯m−1 is a (*)-cycle in R2 of length m > 1, m = k · pα with p not dividing k, and k not dividing pf − 1, then w(¯ xp2f −1 ) > w(¯ x1 ). (vii) Let ζn be a primitive root of unity of order n. Assume that ζn is a root of a polynomial X 2 + aX + b ∈ R[X]. Then n ∈ CYCL ∗ (R, 2). In particular 1, 2, 3, 4, 6 ∈ CYCL ∗ (R, 2). Proof. (i) This is proposition 4.1 from [Pe2]. (ii) This is lemma 4.4 from [Pe2]. (iii) This is proposition 3.1(ii) from [Pe3]. (iv) The ﬁrst part is proposition 3.1(i) from [Pe3]. The remaining follows now from the fact that if y¯0 , y¯1 , ..., y¯m−1 is a (*)-cycle for a mapping Ψ then ¯0 = y¯0 − y¯0 , y¯1 − y¯0 , ..., y¯m−1 − y¯0 is a (*)-cycle for a mapping Ψ1 deﬁned by Ψ1 (Y¯ ) = Ψ(Y¯ + y¯0 ) − y¯0 . (v) By the proof of (iv) we can assume that in RN there is a (*)-cycle ¯0 = y¯0 , y¯1 , ..., y¯m−1 of length m for a mapping Ψ. Put Ψ (¯0) = A. Let (−1)N X a0 (X −1)b0 F1 (X)b1 ·...·Fr (X)br be the characteristic polynomial of the matrix B = A (mod P ). We assume that Fi are pairwise diﬀerent, monic, and irreducible over R/P , Fi = X, X − 1; r ≥ 0, a0 , b0 ≥ 0; b1 , ..., br > 0. Put deg Fi = ai . From (iv) we have k|[pf a1 − 1, ..., pf ar − 1]. As k > 1 we then have [pf a1 − 1, ..., pf ar − 1] > 1. By the assumptions we have a1 +...+ar = N and from a0 +b0 +a1 b1 +...+ar br = N we fa fa f ar obtain a0 = b0 = 0 and b1 = b2 = ... = br = 1. As Fi (X)|X p i −1 − 1|X [p 1 −1,...,p −1] − 1 fa f ar we have F (X) = (−1)N F1 (X)...Fr (X)|X [p 1 −1,...,p −1] − 1. By the well known theorem from linear algebra we obtain F (B) = 0 and fa f ar [pf a1 −1,...,pf ar −1] B − I = 0, which is equivalent to A[p 1 −1,...,p −1] − I ≡ 0 (mod P ). fa f ar As b0 = 0 we have that B − I is invertible and from B [p 1 −1,...,p −1] − I = fa f ar (B − I)(B [p 1 −1,...,p −1]−1 + ... + B 2 + B + I) we obtain fa f ar B [p 1 −1,...,p −1]−1 + ... + B 2 + B + I = 0, or, equivalently, fa f ar A[p 1 −1,...,p −1]−1 + ... + A2 + A + I ≡ 0 (mod P ). fa f ar Put Ψ[p 1 −1,...,p −1] = Φ. By lemma 2.5(i) we then have Φ (¯0) ≡ I (mod P ) and fa f ar by lemma 2.5(iii) and lemma 2.6(iv) y¯[pf a1 −1,...,pf ar −1] ≡ (A[p 1 −1,...,p −1]−1 + ... + A2 + A + I)¯ y1 ≡ ¯0 (mod P w(¯y1 )+1 ). Put x¯j = y¯j[pf a1 −1,...,pf ar −1] . Then we easily see that ¯0 = x¯0 , ..., x¯pα −1 is a (*)-cycle for Φ such that Φ (¯0) ≡ I (mod P ) and w(¯ x1 ) = w(¯ y[pf a1 −1,...,pf ar −1] ) ≥ w(¯ y1 ) + 1 ≥ 2. (vi) The proof of (v) gives the ﬁrst part. The second follows from the ﬁrst, lemma 2.6(ii) and pf − 1|p2f − 1.    −a −1  (vii) Let K be the ﬁeld of fractions of the ring R. Take a matrix A =  ∈ b 0 M2×2 (R). Then An has the eigenvalue ζnn = 1 ∈ K. Hence there is a non-zero x¯ =

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

 x1    ∈ K 2 such that x¯ is an eigenvector of An corresponding to 1. Clearly we can x2 assume x1 , x2 ∈ P . If x¯, A¯ x, ..., An−1 x¯ are pairwise distinct then x¯, A¯ x, ..., An−1 x¯ is a (*)-cycle of length n in R2 for the mapping A, otherwise from An x¯ = x¯ and simple properties of periodic points we obtain that j = min{r : Ar x¯ = x¯} is smaller than n and divides n. Thus A has an eigenvalue ξ, lying in some algebraic closure of K, and being a primitive root of unity of order j. So the only eigenvalues of A are ξ and ζn , which are diﬀerent. As j|n we then obtain that An = I.   y1  Let y¯ =   with y1 , y2 ∈ P be such that x¯, y¯ treated as elements of K 2 are linearly y2 independent over K. If y¯, A¯ y , ..., An−1 y¯ are pairwise diﬀerent then in the same way as above we are done. If not then in the similar way the number l deﬁned by l = min{r : Ar y¯ = y¯} is smaller than n and divides n. Moreover A has an eigenvalue ξ1 which is a primitive root of unity of order l. As ζn , ξ and ξ1 would be eigenvalues of A we have l = j. As x¯, y¯ are linearly independent over K, from Aj x¯ = x¯ and Aj y¯ = y¯ we obtain Aj = I, and clearly such a matrix cannot have as an eigenvalue a primitive root of unity of order n. The statement concerning the numbers 1, 2, 3, 4, 6 follows from the consideration of the polynomials X 2 + 1, X 2 − X + 1 ∈ R[X] and remark 2.2. 2 When writing for instance ”let R be such that p = 2, e = 2, f = 1, π ≡ 2 (mod P 4 )” we mean that R is as in subsection 2.2 and moreover complete, R/P is of characteristic 2, #R/P = pf = 2, w(p) = 2, and the uniformizing parameter π is such that π 2 ≡ 2 (mod P 4 ). All the notations from subsection 2.2 still hold.

2.3 Embeddings of ZK into discrete valuation rings √ Let K = Q( d),d = 1 squarefree. Then it is well known that the decomposition of the prime ideal pZ of integers is as follows: for a prime p > 2 one has pZK = p for ( dp ) = −1; pZK = p1 p2 for ( dp ) = 1 and pZK = p2 for p|d; for p = 2 one has 2ZK = p for d ≡ 5 (mod 8); 2ZK = p1 p2 for d ≡ 1 (mod 8) and ﬁnally 2ZK = p2 for d ≡ 2, 3 (mod 4). √ In the case K = Q( 1) = Q all the decompositions are clear. If p lies above pZ then #ZK /p = pf , (Z K )p is a discrete valuation domain, p(ZK )p is f its unique maximal ideal and (Z K )p/p(ZK )p has p elements. The ramiﬁcation index e of (Z K )p is the same as the ramiﬁcation index of p over pZ. The number e is the same as the exponent appearing by p in the decomposition of pZ as a multiplication of prime

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ideals from ZK . If p lies above pZ and f = e = 1 then (Z K )p is isomorphic to Zp - the ordinary p-adic ring. In any case Zp is embeddable into (ZK )p. √ Remark 2.9. Note, that for K = Q( d), and R = (Z K )p we have ef ≤ 2.

3

Cycles for R with p = 2

3.1 Cycles for p = 2, e = 2, f = 1 In this case R/P has 2 elements. Hence by proposition 2.8(iv) (*)-cycles in R2 may have lengths only of the form 2α , 3 · 2α . Lemma 3.1. Let R be such that p = 2, e = 2, f = 1. y1 ) ≥ 2 for (i) There are no (*)-cycles in R2 of the form ¯0 = y¯0 , y¯1 , ..., y¯7 with w(¯ ¯ 2 a mapping Φ such that Φ (0) ≡ B (mod P ) for a suitable B ∈ M2×2 (R) ( in particular if Φ (¯0) ≡ I 2 ≡ I (mod P )). (ii) There are no (*)-cycles of length 24 in R2 . (iii) There are no (*)-cycles of length 16 in R2 . y1 ) ≥ 2 be a (*)-cycle in R2 for a mapping Φ Proof. (i) Let ¯0 = y¯0 , y¯1 , ..., y¯7 with w(¯ i such that Φ (¯0) ≡ B 2 (mod P ). Put (Φ2 ) (¯0) = Ai . By lemma 2.7(ii) we then have y2 ) < w(¯ y4 ). In particular w(¯ y4 ) ≥ w(¯ y2 ) ≥ 3. Hence, by lemma 2.5(i) 2 ≤ w(¯ y1 ) < w(¯ 2 3 2 2 2 A2 ≡ A1 (mod P ), A1 ≡ A0 (mod P ), A0 ≡ B (mod P ). By lemma 2.7(vi) ¯0 ≡ ((A2 − I) + 2I)¯ y4 (mod P w(¯y4 )+3 ) and (A2 − I)2 y¯4 ≡ ¯0 (mod P w(¯y4 )+3 ). By lemma 2.7(iv) we get (A0 − I)¯ y4 ≡ ¯0 (mod P w(¯y4 )+1 ). Hence, by lemma 2.5(i),(iii) and w(2) ≥ 2, we get ¯0 ≡ (A2 − I)2 y¯4 ≡ (A40 − I)2 y¯4 ≡ y4 ≡ (B 2 − ((A0 − I)4 + 4(A0 − I)3 + 6(A0 − I)2 + 4(A0 − I))2 y¯4 ≡ (A0 − I)7 (A0 − I)¯ I)7 (A0 − I)¯ y4 ≡ (B − I)14 (A0 − I)¯ y4 (mod P w(¯y4 )+2 ). Thus, using (A0 − I)¯ y4 ≡ ¯0 (mod P w(¯y4 )+1 ), by lemma 2.5(ii),(iii) and A1 ≡ A20 (mod P 2 ) we get ¯0 ≡ (B − I)2 (A0 − I)¯ y4 ≡ (B 2 − I)(A0 − I)¯ y4 ≡ (A0 − I)(A0 − I)¯ y4 ≡ 2 w(¯ y4 )+2 w(¯ y4 )+2 ¯ (A0 − I)¯ y4 ≡ (A1 − I)¯ y4 (mod P ), and (A1 − I)¯ y4 ≡ 0 (mod P ) follows. 2 3 Thus by A2 ≡ A1 (mod P ), lemma 2.5(iii), w(4) = 2w(2) ≥ 4 and A1 ≡ A20 (mod P 2 ) we obtain ¯0 ≡ (A2 − I)2 y¯4 ≡ (A21 − I)2 y¯4 ≡ ((A1 − I)4 + 4(A1 − I)3 + 4(A1 − I)2 )¯ y4 ≡ (A1 −I)3 (A1 −I)¯ y4 ≡ (A20 −I)3 (A1 −I)¯ y4 ≡ (A0 −I)6 (A1 −I)¯ y4 (mod P w(¯y4 )+3 ). In view of (A1 − I)¯ y4 ≡ ¯0 (mod P w(¯y4 )+2 ), lemma 2.5(ii), A1 ≡ A20 (mod P ) and A2 ≡ A21 (mod P 3 ) we therefore get ¯0 ≡ (A0 − I)2 (A1 − I)¯ y4 ≡ (A20 − I)(A1 − I)¯ y4 ≡ 2 w(¯ y4 )+3 y4 ≡ (A1 − I)¯ y4 ≡ (A2 − I)¯ y4 (mod P ). Thus (A2 − I)¯ y4 ≡ ¯0 (A1 − I)(A1 − I)¯ (mod P w(¯y4 )+3 ) and already obtained ¯0 ≡ ((A2 −I)+2I)¯ y4 (mod P w(¯y4 )+3 ) gives 2¯ y4 ≡ ¯0 (mod P w(¯y4 )+3 ), a contradiction since w(2¯ y4 ) = w(2) + w(¯ y4 ) = 2 + w(¯ y4 ) < 3 + w(¯ y4 ). (ii) The statement follows from (i) and proposition 2.8(v). (iii) Let y¯0 , y¯1 , ..., y¯15 be a (*)-cycle of length 16 in R2 for a mapping Φ. Put Φ (¯0) = A.

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  π By lemma 2.6(i) we can assume y¯0 = ¯0, y¯1 =  . If w(¯ y2 ) ≥ 2 then ¯0 = y¯0 , y¯2 , y¯4 , ..., y¯14 0 would be a (*)-cycle of length 8 with w(¯ y2 ) ≥ 2 for the mapping Φ2 fulﬁlling, by lemma 2.5(i), (Φ2 ) (¯0) ≡ (Φ (¯0))2 (mod P ), which is impossible by (i). Hence w(¯ y2 ) = 1. By −1 2 −1 lemma 2.6(v) we obtain, in its notation, m2 ≥ 3. Thus π y¯1 + P R , π y¯2 + P R2 are non-zero and diﬀerent elements of (R/P )2 , and therefore linearly independent  over  the 0 two-element ﬁeld R/P . By lemma 2.6(vi) we can additionally assume y¯2 =  . By π     π 0 y1 = (I + A)  (mod P 2 ) and by lemma 2.6(iv) we have   = y¯2 ≡ (I + A)¯ 0 π lemma 2.7(iii) ((Φ2 ) (¯0) − I)¯ y1 ≡ (A2 − I)¯ y1 ≡ ¯0 (mod P 2 ). Using two last congruences   1 0 by a straightforward calculation we obtain A ≡   (mod P ). In particular by 11   π y1 ≡   (mod P 2 ). lemma 2.6(iv) y¯3 ≡ (I + A + A2 )¯ π Write then Φ(x, y) = (π+(1+πa)x+πby+mxy+..., (1+πc)x+(1+πd)y+M xy+...). By lemma 2.4 the coeﬃcient of xy in the ﬁrst component of Φ2 is congruent to 0 (mod P ), whereas the coeﬃcient component of  Φ2 iscongruent to m (mod P ).  of xy in the second   1 + πu πv  0 Put (Φ2 ) (¯0) =  . In view of y¯2 =   and an earlier remark we πs 1 + πt π obtain    πv   1 + πu πv   1 + πu y2 ) ◦ (Φ2 ) (¯0) ≡  (Φ4 ) (¯0) = (Φ2 ) (¯   πs + πm 1 + πt πs 1 + πt     1 0  0 0 ≡ +π  (mod P 2 ). 01 m0 2 ¯ ¯ ≡ I (mod P ) we obtain by lemma 2.4 As (Φ )(0) ≡ 0¯ (mod P ) and (Φ2 ) (0)

that the coeﬃcients of xy in both components of Φ4 are congruent to 0 (mod P ). This together with w(¯ y4 ) ≥ 2 and w(2) = 2 gives (Φ4 ) (¯ y ) ≡ (Φ4 ) (¯0) (mod P 3 ) and  2 4    1 0  1 0 (Φ8 ) (¯0) = (Φ4 ) (¯ y4 )◦(Φ4 ) (¯0) ≡ ((Φ4 ) (¯0))2 ≡   ≡  ≡ I (mod P 3 ). πm 1 2πm 1 By lemma 2.6(iv), and w(¯ y8 ) ≥ 3( which follows from lemma 2.7(ii)) ¯0 = y¯16 ≡ ((Φ8 ) (¯0) + I)¯ y8 ≡ 2¯ y8 (mod P w(¯y8 )+3 ). This gives a contradiction in view of w(2¯ y8 ) =

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2 + w(¯ y8 ) < 3 + w(¯ y8 ).

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Lemma 3.2. Let R be such that p = 2, e = 2, f = 1. Then there are (*)-cycles of length 8 in R2 . Proof. As 2 ≡ π 2 (mod P 3 ) we can write 2 = π 2 + π 3 5, with a certain 5 ∈ R. 1st case, 5 ≡ 1 (mod P ). Take a mapping Φ deﬁned byΦ(x, y) = (π − x + (π +  xi  π 2 c)xy, x + y + xy). If we put for i = 0, 1, 2, ..., 8 that x¯i =   = Φi (¯0) then by a yi       π 0 π straightforward calculation we get x¯0 = ¯0, x¯1 =  , x¯2 =  , x¯3 =  , x¯4 = 0 π π         2 2  (π + π c)π  ¯ π 0  π    = 0, x¯5 ≡   (mod P 2 ), x¯6 ≡   (mod P 2 ), x¯7 ≡   2 2 π + 2π 0 π π+π   0   (mod P 3 ) and x¯8 ≡   (mod P 7 ). 5 π (1 + 5 + cπ) We ﬁx an element c ∈ R such that π 2 |1 + 5 + cπ. Then x¯8 ≡ 0¯ (mod P 7 ). Now take Ψ(x, y) = Φ(x, y) + (t1 y(x − x2 )(y − y3 )(x − x4 )(y − y5 )(x − x6 ), t2 y(x − x2 )(y − y3 )(x − x4 )(y − y5 )(x − x6 )). One easily sees that Ψi (¯0) = Φi (¯0) for i ≤ 7. As x¯8 ≡ ¯0 (mod P 7 ) and w(y7 (x7 − x2 )(y7 − y3 )(x7 − x4 )(y7 − y5 )(x7 − x6 )) = 7 then for suitable t1 , t2 ∈ R we have Ψ8 (¯0) = ¯0. As x¯4 = ¯0 we get the (*)-cycle ¯0, x¯1 , ..., x¯7 of lenght 8 for the mapping Ψ. 2nd case, 5 ≡ 0 (mod P ). Take Φ(x, y) = (π + (1 + π)x + π(1  + πb)y,  x+ π y). With the same notations like in the 1st case we get x¯0 = ¯0, x¯1 =  , x¯2 ≡ 0         2 π   π   0  π   (mod P 3 ), x¯4 ≡   (mod P 3 ), x¯5 ≡     (mod P 3 ), x¯3 ≡  π π + π2 π2 π2     2  π  π and (mod P 3 ), x¯6 ≡   (mod P 3 ), x¯7 ≡   (mod P 3 ) π + π2 π   0   x¯8 ≡   (mod P 7 ). 5 π (5 + π(1 + b)) Now ﬁx b ∈ R such that π 2 |5 + π(1 + b). Therefore x¯8 ≡ ¯0 (mod P 7 ). As w(y7 (x7 − x2 )(y7 − y3 )(x7 − x4 )(y7 − y5 )(x7 − x6 )) = 7 in a similar manner to the

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1st case we arrive at the statement. By lemmas 3.1, 3.2, proposition 2.8(vii) and the beginning of this section in order to determine CYCL ∗ (R, 2) ( and by proposition 2.8(ii) also CYCL(R, 2)) we shall only examine the existence of (*)-cycles of length 12 in R2 . As π 2 ≡ 2 (mod P 3 ) we then have two possibilities to consider, namely 2 ≡ π 2 +π 3 (mod P 4 ) and 2 ≡ π 2 (mod P 4 ). Lemma 3.3. Let R be such that p = 2, e = 2, f = 1 and 2 ≡ π 2 + π 3 there is a (*)-cycle of length 12 in R2 .

(mod P 4 ). Then

Proof. We have 2 cases. 1st case, 2 ≡ π 2 + π 3 (mod P 5 ). In this case 1 + (1 + π)2 = 2 + 2π + π 2 ≡ 2 + 2π + 2 − π 3 ≡ 4 + π 3 + π 4 − π 3 ≡ 0 (mod P 5 ). So (1 + π)2 ≡ −1 (mod P 5 ). Thus √ the congruence x2 ≡ −1 (mod P 5 ) is solvable and Hensel’s lemma implies i = −1 ∈ R. As the polynomial X 2 − iX − 1 ∈ R[X] has a root being a primitive root of unity of order 12 we are done by proposition 2.8(vii). 2nd case, 2 ≡ π 2 + π 3 + π 4 (mod P 5 ). In this case (1 + π)2 − 3 = 1 + 2π + π 2 − 3 ≡ √ π 2 + π(π 2 + π 3 + π 4 ) − π 2 − π 3 − π 4 ≡ 0 (mod P 5 ). So, by Hensel’s lemma, 3 ∈ R. √ As the polynomial X 2 + 3X + 1 ∈ R[X] has a root being a primitive root of unity of order 12 we are done by proposition 2.8(vii). Lemma 3.4. Let R be such that p = 2, e = 2, f = 1 and 2 ≡ π 2 are no (*)-cycles of length 12 in R2 .

(mod P 4 ). Then there

  π Proof. Assume the contrary. By lemma 2.6(i) we then can assume that ¯0 = x¯0 ,   = 0 x¯1 , ..., x¯11 is a (*)-cycle of length 12 in R2 for a mapping Φ. By proposition 2.8(vi) we have w(¯ x3 ) > w(¯ x1 ). If w(¯ x2 ) > w(¯ x1 ) then by lemma 2.6(iii) we get w(¯ x1 ) = w(¯ x1 − x¯0 ) = w(¯ x3 − x¯2 ) ≥ x2 )} > w(¯ x1 ), a contradiction. min{w(¯ x3 ), w(¯ Thus from lemma 2.6(ii) we have w(¯ x2 ) = w(¯ x1 ) and by lemma 2.6(v) π −1 x¯1 + P R2 , π −1 x¯2 + P R2 are non-zero and diﬀerent elements of (R/P )2 , hence linearlyinde 0 pendent over R/P . So, by lemma 2.6(vi) we can additionally assume that x¯2 =  . π x1 (mod P 2), x¯3  ≡ As x¯3 ≡ ¯0 (mod P 2 ) by lemma 2.6(iv) we get x¯2 ≡ (I + Φ (¯0))¯ 1 1 x1 (mod P 2 ). Thus by a simple calculation Φ (¯0) ≡  (I + Φ (¯0) + (Φ (¯0))2 )¯  10 3 ¯ ¯ 3 (mod P ) and in particular (Φ ) (0) ≡ (Φ (0)) ≡ I (mod P ). Write then Φ(x, y) = (π + (1 + πa)x + (1 + πb)y + gxy + ..., (1 + πc)x + πdy + Gxy + ...). As ¯0 = x¯0 , x¯3 , x¯6 , x¯9 forms a (*)-cycle of length 4 for Φ3 , with (Φ3 ) (¯0) ≡ I (mod P )

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we obtain by lemma 2.7(ii) w(¯ x6 ) > w(¯ x3 ) ≥ 2 and w(¯ x6 ) ≥ 3 follows. 3 ¯ 3 ¯ ¯ As Φ (0) ≡ 0 (mod P ) and (Φ ) (0) ≡ I (mod P ) we obtain by lemma 2.4 that the coeﬃcients of x2 , xy, y 2 in both components of Φ6 are congruent to 0 (mod P ). Thus from w(¯ x6 ) ≥ 3 and Taylor’s expansion we obtain ¯0 = x¯12 ≡ (I + (Φ6 ) (¯0))¯ x6 ≡ ¯0 (mod P w(¯x6 )+4 ).

(1)

¯ = Φ (¯ By a straightforward calculation (Φ3 ) (0) x2 ) ◦ Φ (¯ x1 ) ◦ Φ (¯0)≡     1 + πa + πg 1 + πb   1 + πa 1 + πb + πg   1 + πa 1 + πb      1 + πc + πG πd 1 + πc πd + πG 1 + πc πd (mod P 2 ) and  1 (Φ3 ) (¯0) ≡  0







0 a + d + g + G b + c + d + g  +π  1 b+c+d+g a+b+c+G

(mod P 2 ).

(2)

Lemma 2.4 gives that the coeﬃcients of xy in both components of Φ3 are congruent to 0 (mod P ). Thus in view of w(2) = 2 and w(¯ x3 ) ≥ 2 we get (Φ3 ) (¯ x3 ) ≡ (Φ3 ) (¯0) (mod P 3 ) and therefore 3 ¯ 2 (Φ6 ) (¯0) = (Φ3 ) (¯ x3 ) ◦ (Φ3 ) (¯0) ≡ ((Φ  ) (0)) ≡ a + d + g + G b + c + d + g  2 (I + π  ) ≡ b+c+d+g a+b+c+G   2 2 2 2 2 2 2 2 a + b + c + G b + c + d + g  I + π2   b2 + c2 + d2 + g 2 a2 + d2 + g 2 + G2

(mod P 3 ) and as for all y ∈ R we have

y2 ≡ y

(mod P ) we obtain by a suitable counterpart of lemma 2.5(iii) that  a + b + c + G b + c + d + g  (Φ6 ) (¯0) = I + π 2   (mod P 3 ) and from 2 ≡ π 2 b+c+d+g a+d+g+G

(mod P 3 ) we get 



1 + a + b + c + G b + c + d + g  (I + (Φ6 ) (¯0)) ≡ π 2   b+c+d+g 1+a+d+g+G

(mod P 3 ).

(3)

x6 ≡ ¯0 (mod P w(¯x6 )+3 ) and (3) together with >From (1) it follows that (I + (Φ6 ) (¯0))¯ lemma  2.5(iii) gives  1 + a + b + c + G b + c + d + g    x¯6 ≡ ¯0 b+c+d+g 1+a+d+g+G

(mod P w(¯x6 )+1 ).

By lemma 2.5(iv) we therefore obtain that a matrix

310



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

1 + b + c + G + a b + c + d + g    is not invertible, or, equivalently, its determib+c+d+g 1+a+d+g+G nant lies in P . We have that this determinant lies in P if and only if 1 + (a + b + c + G)(1 + b + c + d + g) ≡ 0 (mod P ) and a + b + c + G ≡ 1 (mod P ), b + c + d + g ≡ 0 (mod P ) follows. In particular 1 ≡ (a + b + c + G) + (b + c + d + g) ≡ (a + d + g + G) (mod P ).   1 + π 0  Hence by (2) we obtain (Φ3 ) (¯0) ≡   0 1+π

(mod P 2 ).

2.6(iv), w(2) = 2 and w(¯ x3 ) ≥ 2 we obtain x¯6 ≡ (I + (Φ3 ) (¯0))¯ x3 ≡  Thus by lemma  2 + π 0  x3 (mod P w(¯x3 )+2 ). In particular w(¯ x6 ) = 1 + w(¯ x3 ).   x¯3 ≡ π¯ 0 2+π 6 Let πβ1 , πβ2 be the of xy in ﬁrst  coeﬃcients   and  second components of Φ respectively.

α β   z1  x3 ) ≥ 2 we get w(z1 ), w(z2 ) ≥ Put (Φ3 ) (¯0) =  . Write x¯3 =  . As w(¯ z2 γ δ 2. Hence from w(2) = 2, w(¯ x3 ) ≥ 2 and (Φ3 ) (¯0) ≡ I (mod P ) we have (Φ6 ) (¯0) = (Φ3 ) (¯ x3 ) ◦ (Φ3 ) (¯0) ≡       2  α + πβ1 z2 β + πβ1 z1   α β   α β   β1 z2 β1 z1    ≡  (mod P 4 ).  +π γ + πβ2 z2 δ + πβ2 z1 β2 z2 β2 z1 γ δ γ δ     2  α β  1 + π 0  α β  As  ≡  ≡  (mod P 2 ) and w(2) = 2 we easily obtain  γ δ γ δ 0 1+π   2 0  1 + π + 2π    (mod P 4 ). 0 1 + π 2 + 2π       2 0   β1 z2 β1 z1   2π 0   2 + π + 2π Thus I + (Φ6 ) (¯0) ≡  +π ≡ + 2 β2 z2 β2 z1 0 2π 0 2 + π + 2π    β1 z2 β1 z1  π  (mod P 4 ) ( we used the assumption 2 ≡ π 2 (mod P 4 ) and therefore β2 z2 β2 z1 ¯ ≡ 0 (mod P 3 ). 2 + 2π + π 2 ≡ 4 + 2π ≡ 2π (mod P 4 )). In particular I + (Φ6 ) (0) ¯ ≡ 0 (mod P 3 ), w(¯ Hence from I + (Φ6 ) (0) x6 ) = w(¯ x3 ) + 1, x¯6 ≡ π¯ x3 (mod P w(¯x3 )+2 ), w(¯ x3 ) ≥ 2, lemma 2.5(iii) and (1) we obtain









 2π 0   z1  ¯0 ≡ (I + (Φ6 ) (¯0))¯ x6 ≡ (I + (Φ6 ) (¯0))π¯ x3 ≡ (I + (Φ6 ) (¯0))π   ≡ ( + z2 0 2π

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

 











311





 z1   2β1 z1 z2   z1   β1 z2 β1 z1   z1  π )π   = 2π 2   + π 2   ≡ 2π 2   β2 z2 β2 z1 z2 z2 2β2 z1 z2 z2 This gives  acontradiction since  z1  x3 ). x3 ) < 5 + w(¯ w(2π 2  ) = 4 + w(¯ z2 >From considerations of this subsection we get

(mod P w(¯x3 )+5 ).

Proposition 3.5. Let R be such that p = 2, e = 2, f = 1. (i) If 2 ≡ π 2 +π 3 (mod P 4 ) then CYCL∗(R, 2) = {1, 2, 3, 4, 6, 8, 12} and by proposition 2.8(ii) CYCLMAX (R, 2) = {48, 36, 32}. (ii) If 2 ≡ π 2 (mod P 4 ) then CYCL∗(R, 2) = {1, 2, 3, 4, 6, 8} and by proposition 2.8(ii) CYCLMAX (R, 2) = {32, 24, 18}.

3.2 Cycles for p = 2, e = 1, f = 2 In this case R/P has 4 elements. By proposition 2.8(iv) we have that (*)-cycles in R2 may have their lengths only of the form 2α , 3 · 2α , 5 · 2α and 15 · 2α . Lemma 3.6. Let R be such that p = 2, e = 1, f = 2. y1 ) ≥ 2 for (i) There are no (*)-cycles in R2 of the form ¯0 = y¯0 , y¯1 , y¯2 , y¯3 with w(¯ ¯ 2 a mapping Φ such that Φ (0) ≡ B (mod P ) for a suitable B ∈ M2×2 (R) ( in particular if Φ (¯0) ≡ I 2 ≡ I (mod P )). (ii) There are no (*)-cycles of length 20, 60 in R2 . (iii) There are no (*)-cycles of length 8, 24 in R2 . Proof. (i) Let ¯0 = y¯0 , y¯1 , y¯2 , y¯3 with w(¯ y1 ) ≥ 2 be a (*)-cycle in R2 for a mapping Φ i such that Φ (¯0) ≡ B 2 (mod P ). Put (Φ2 ) (¯0) = Ai . By lemma 2.7(ii) we then have 2 ≤ w(¯ y1 ) < w(¯ y2 ). Hence, by lemma 2.5(i) A1 ≡ A20 (mod P 2 ), A0 ≡ B 2 (mod P ). By lemma 2.7(vi) we have ¯0 ≡ ((A1 − I) + 2I)¯ y2 (mod P w(¯y2 )+2 ) and (A1 − I)2 y¯2 ≡ ¯0 (mod P w(¯y2 )+2 ). By lemma 2.7(iv) we get (A0 − I)¯ y2 ≡ ¯0 (mod P w(¯y2 )+1 ). Hence, by lemma 2.5(iii), A1 ≡ A20 (mod P 2 ), A0 ≡ B 2 (mod P ) and w(4) = y2 ≡ 2w(2) = 2 we get ¯0 ≡ (A1 −I)2 y¯2 ≡ (A20 −I)2 y¯2 ≡ ((A0 −I)4 +4(A0 −I)3 +4(A0 −I)2 )¯ 3 2 3 6 w(¯ y2 )+2 (A0 − I) (A0 − I)¯ y2 ≡ (B − I) (A0 − I)¯ y2 ≡ (B − I) (A0 − I)¯ y2 (mod P ). w(¯ y2 )+1 2 ¯ Thus, using (A0 − I)¯ y2 ≡ 0 (mod P ), by lemma 2.5(ii), A0 ≡ B (mod P ) 2 2 ¯ and A1 ≡ A0 (mod P ) we get 0 ≡ (B − I)2 (A0 − I)¯ y2 ≡ (B 2 − I)(A0 − I)¯ y2 ≡ 2 w(¯ y2 )+2 (A0 − I)(A0 − I)¯ y2 ≡ (A0 − I)¯ y2 ≡ (A1 − I)¯ y2 (mod P ), and (A1 − I)¯ y2 ≡ ¯0 (mod P w(¯y2 )+2 ) follows. Now we get a contradiction with ¯0 ≡ ((A1 − I) + 2I)¯ y2 w(¯ y2 )+2 (mod P ) since w(2¯ y2 ) = w(2) + w(¯ y2 ) = 1 + w(¯ y2 ) < 2 + w(¯ y2 ). 1 (ii) As 5 does not divide 4 − 1 the assertion concerning 20 follows from (i) and proposition 2.8(v). The assertion concerning 60 follows now from remark 2.2. (iii) By remark 2.2 it suﬃces to deal with 8. Let y¯0 , y¯1 , ..., y¯7 be a (*)-cycle of length 8 in

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  π R2 for a mapping Φ. Put Φ (¯0) = A. By lemma 2.6(i) we can assume y¯0 = ¯0, y¯1 =  . 0 y2 ) ≥ 2 If w(¯ y2 ) ≥ 2 then ¯0 = y¯0 , y¯2 , y¯4 , y¯6 would be a (*)-cycle of length 4 with w(¯ 2 2 ¯ ¯ 2 for the mapping Φ fulﬁlling, by lemma 2.5(i), (Φ ) (0) ≡ (Φ (0)) (mod P ), which is impossible by (i). Hence w(¯ y2 ) = 1. By lemma 2.7(ii) we obtain w(¯ y4 ) ≥ 2.





 cπ  If π −1 y¯1 + P R2 , π −1 y¯2 + P R2 are linearly dependent over R/P then y¯2 ≡   0 y2 ) = 1), c ∈ R not equivalent to0 (mod P ). As by lemma (mod P 2 ) for some ( as w(¯  c − 1 ∗ 2.6(iv) y¯2 ≡ (I + A)¯ y1 (mod P 2 ) we obtain A ≡   (mod P ). 0 ∗ y1 (mod P 2 ) and Hence, using lemma 2.6(iv) we get ¯0 = y¯8 ≡ (I + A + ... + A7 )¯

1 + (c − 1) + (c − 1)2 + ... + (c − 1)7 ≡ 0 (mod P ) follows. This in turns gives c ≡ 0 (mod P ), which we already know to be impossible. independent over R/P and by lemma Hence π −1 y¯1 + P R2 , π −1 y¯2 + P R2 are linearly   0 2.6(vi) we can additionally assume that y¯2 =  . π





1 ∗ (mod P 2 ) we obtain A ≡   (mod P ). 1∗ 2 ¯ ¯ From lemma 2.5(i)and y1 ≡ (A2 − I)¯ y1 (mod P 2 )   lemma 2.7(iii) 0 ≡ ((Φ ) (0) − I)¯ 1 0 and therefore A ≡   (mod P ). 11   π y1 (mod P 2 ) and therefore y¯3 ≡   By lemma 2.6(iv) we have y¯3 ≡ (I + A + A2 )¯ π y1 As by lemma 2.6(iv) y¯2 ≡ (I + A)¯

(mod P 2 ) follows. Write Φ(x, y) = (π + (1 + πα)x + πβy + dxy + ..., (1 + πγ)x + (1 + πδ)y + Dxy + ...). Then    1 + πα + πd πβ + πd  (Φ4 ) (¯0) = Φ (¯ y3 ) ◦ Φ (¯ y2 ) ◦ Φ (¯ y1 ) ◦ Φ (¯0) ≡   1 + πγ + πD 1 + πδ + πD      1 + πα + πd πβ   1 + πα πβ + πd   1 + πα πβ     ≡ 1 + πγ + πD 1 + πδ 1 + πγ 1 + πδ + πD 1 + πγ 1 + πδ

T. Pezda / Central European Journal of Mathematics 2(2) 2004 294–331

  1  πd

313

 0  1

(mod P 2 ).

¯ >From w(¯ y4 ) ≥2, lemma  2.5(iii) and lemma 2.6(iv) we therefore obtain 0 = y¯8 ≡  2 0 (I + (Φ8 ) (¯0))¯ y4 ≡   y¯4 (mod P w(¯y4 )+2 ). πd 2    2 0 Thus we obtain a contradiction since w( y4 ) < 2 + w(¯ y4 ).  y¯4 ) = 1 + w(¯ πd 2 Lemma 3.7. Let R be such that p = 2, e = 1, f = 2. (i) There is a (*)-cycle of length 30 in R2 . (ii) There is a (*)-cycle of length 12 in R2 . Proof. Let ξ ∈ R/P , ξ = 0, 1. (i) The minimal polynomial over Z of the primitive root of unity of order 30 is X 8 + X 7 − X 5 − X 4 − X 3 + X + 1 Treating the last polynomial (mod P ) and writing it as a product of irreducibles we get X 8 + X 7 − X 5 − X 4 − X 3 + X + 1 = (X 2 + X + ξ)(X 2 + X + ξ + 1)(X 2 + ξX + ξ)(X 2 + (ξ + 1)X + ξ + 1). The polynomials at the right-hand side of the last equality are pairwise coprime. Hence by Hensel’s lemma there is a polynomial X 2 + αX + β with α, β ∈ R whose roots are primitive roots of unity of order 30. We therefore get the statement by proposition 2.8(vii). (ii) The minimal polynomial over Z for the primitive root of unity of order 6 is 2 X −X +1. Treating it as a polynomial over R/P we get X 2 −X +1 = (X −ξ)(X −(ξ+1)). Hence from Hensel’s lemma there exists a primitive root c of unity of order 6 in R and we may apply proposition 2.8(vii) to the polynomial X 2 − c ∈ R[X], whose roots are primitive roots of unity of order 12. >From the considerations in this subsection we therefore obtained Proposition 3.8. Let R be such that p = 2, e = 1, f = 2. Then CYCL ∗ (R, 2) = {1, 2, 3, 4, 5, 6, 10, 12, 15, 30}. Hence by proposition 2.8(ii) CYCLMAX (R, 2) = {480, 450, 420, 390, 360, 330, 300, 270, 192, 168, 156, 144, 132, 108}.

3.3 Cycles for p = 2, e = 1, f = 1 Let R be such that p = 2, e = 1, f = 1. Then R is isomorphic to the 2-adic ring Z2 . Proposition 2.8(vii) implies that we have (*)-cycles of lengths 1, 2, 3, 4, 6 in R2 . Let R1 be such that p = 2, e = 1, f = 2 and R2 be such that p = 2, e = 2, f = 1, 2 ≡ π 2 (mod P 4 ) ( clearly there are such R1 , R2 ). Then R is embeddable into both R1 and R2 . As via the mentioned embeddings every (*)-cycle in R2 becomes a (*)-cycle in R12 and

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R22 then by propositions 3.5 and 3.8 we obtain that the lengths of (*)-cycles in R2 are contained in the set {30, 15, 12, 10, 6, 5, 4, 3, 2, 1} ∩ {8, 6, 4, 3, 2, 1} = {1, 2, 3, 4, 6}. Taking into account proposition 2.8(ii) we therefore get Proposition 3.9. Let R be such that p = 2, e = 1, f = 1. Then in R2 there are only (*)-cycles of lengths 1, 2, 3, 4, 6. Hence CYCLMAX (R, 2) = {24, 18, 16}.

3.4 Some remarks If K/Q is of degree ≤ 2 and p is a prime ideal of ZK lying over 2Z then the completion (Z K )p of the localization (ZK )p of ZK with respect to p is one of the discrete valuation rings considered in propositions 3.5,3.8 and 3.9. As ZK is embeddable into (Z K )p one can easily see that CYCL(ZK , 2) ⊂ CYCL((Z K )p, 2) ⊂ {30n, 12n : 1 ≤ n ≤ 16; and their divisors }. Put D = {30n, 12n : 1 ≤ n ≤ 16; and their divisors }. Thus proposition 2.1 gives CYCL(ZK , 2) = CYCL(ZK , 2) ∩ D = p(CYCL((ZK )p, 2) ∩ D), where the intersection is taken over all non-zero prime ideals p of ZK . For d square-free and prime p let Γ(d, p) = p(CYCL((Z K )p, 2) ∩ D), where K = √ non-zero prime ideals p of ZK lying over pZ. Q( d), and the intersection is taken over all √ Thus CYCL(ZK , 2) = p Γ(d, p) for K = Q( d). Lemma 3.10. (i) Let R be such that p ≥ 11 and e, f arbitrary. Then CYCL(R, 2) ⊃ D = {30n, 12n √ : 1 ≤ n ≤ 16; and their divisors }. (ii) Let K = Q( d). Then CYCL(ZK , 2) = Γ(d, 2) ∩ Γ(d, 3) ∩ Γ(d, 5) ∩ Γ(d, 7). (iii) Let R be such that p ≥ 3, e = 1, f = 2 then CYCL(R, 2) ⊃ D = {30n, 12n : 1 ≤ n ≤ 16; and their divisors }. Proof. (i) As for n ≤ 16 we have 12n = 3n · 4, 3n ≤ 48 < 112 ≤ p2 ≤ p2f and 30n = 5n · 6, 5n ≤ 80 < 112 ≤ p2 ≤ p2f by remark 2.2 and proposition 2.8(ii),(vii) we get the statement. (ii) Follows directly from (i) and preceding remarks. (iii) As for n ≤ 16 we have 12n = 3n · 4, 3n ≤ 48 < 34 ≤ p4 = p2f and 30n = 5n · 6, 5n ≤ 80 < 34 ≤ p4 = p2f by remark 2.2 and proposition 2.8(ii),(vii) we get the statement.

3.5 Evaluation of Γ(d, 2) Lemma 3.11. Let d be a square-free number. Then (i) Γ(d, 2) = {24, 18, 16 and their divisors } for d ≡ 1 (ii) Γ(d, 2) = {32, 24, 18 and their divisors } for d ≡ 2 (iii) Γ(d, 2) = {48, 36, 32 and their divisors } for d ≡ 3

(mod 8); (mod 4); (mod 4);

T. Pezda / Central European Journal of Mathematics 2(2) 2004 294–331

315

(iv) Γ(d, 2) = D = {30n, 12n : 1 ≤ n ≤ 16; and their divisors } for d ≡ 5 (mod 8). Proof. (i) We use subsection 2.3 and obtain that 2ZK = p1 p2 for d = 1 and 2ZK = p1 for d = 1, where p1 , p2 are diﬀerent prime ideals of ZK such that (Z K )p1 , (ZK )p2 are isomorphic to R from subsection 3.3 √ and we use proposition 3.9. 2 (ii) In this case 2ZK = p and d may serve the role of a uniformizing parameter π 4 in (Z K )p. As π divides 2 − d in (ZK )p we see that (ZK )p is isomorphic to some R such that p = 2, e = 2, f = 1 and 2 ≡ π 2 (mod P 4 ), and we use proposition 3.5(ii). √ (iii) In this case 2ZK = p2 and 1 + d may serve the role of a uniformizing parameter √ 2 √ √ √ π in (Z d) + (1 + d)3 − 2 = 4d + (d + 5) d is divisible by (1 + d)4 K )p. As (1 + in (Z K )p we see that (ZK )p is isomorphic to some R such that p = 2, e = 2, f = 1 and 2 3 4 2 ≡ π + π (mod P ), and we use proposition 3.5(i). (iv) In this case 2ZK = p is prime and (Z K )p is isomorphic to some R such that p = 2, e = 1, f = 2, and we use proposition 3.8.

4

Cycles for R with p = 3

If R is such that p = 3, e = 1, f = 2 then by lemma 3.10(iii) CYCL(R, 2) ⊃ D. For ef ≤ 2 the remaining possibility is f = 1, e ≤ 2. If R is such that p = 3, f = 1, e ≤ 2 then, by proposition 2.8(iv), (*)-cycles in R2 may have lengths only of the form 3α , 2·3α , 4·3α , 8·3α .

4.1 Cycles for R such that p = 3, e = 2, f = 1 Let R be such that p = 3, e = 2, f = 1. By proposition 2.8(vii) we have that {1, 2, 3, 4, 6} ⊂ CYCL ∗ (R, 2). Lemma 4.1. Let R be such that p = 3, e = 2, f = 1. Then 8, 12, 24 ∈ CYCL ∗ (R, 2). Proof. In view of remark 2.2 it suﬃces to deal with 24. We have π 2 ≡ ±3 (mod P 3 ) so 3 = π 2 (5 + πb) for some b ∈ R and 5 ∈ {+1, −1}. √ √ If 5 = 1 then by Hensel’s lemma 3 ∈ R. If 5 = −1 then similarly −3 ∈ R. Note √ √ that for every a ∈ R we have 1 + πa ∈√R,√hence in√particular −2 ∈ R. √ 2 3+1 2 3−1 ◦ ◦ Note that ζ24 = cos 15 + i sin 15 = 2 ( 2 ) + i 2 ( 2 ) is a primitive root of unity √ √ √ √ of order 24. We put −3 = 3i, −2 = 2i. √ √ √ √ −2X−1 and X 2 + −3+1 −2X− We see that ζ24 is a root of the polynomials X 2 − 3−1 2 2 √ 1 (−1 + −3). At least one of them has all its coeﬃcients in R and we are done by propo2 sition 2.8(vii). Lemma 4.2. Let R be such that p = 3, e = 2, f = 1. (i) In R2 there are no (*)-cycles ¯0 = x¯0 , x¯1 , ..., x¯8 for any mapping Φ such that Φ (¯0) (mod P ) has two diﬀerent eigenvalues. (ii) Let ¯0, x¯1 , ..., x¯8 be a (*)-cycle in R2 of length 9 for a mapping Φ, and let Φ be of the

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form Φ(x, y) = (x1 + (1 + πa)x + by + kx2 + gxy + hy 2 + ..., y1 + πcx + (1 + πd)y + Kx2 + Gxy + Hy 2 + ...). Then b ≡ 0 (mod P )( in particular Φ (¯0) ≡ I (mod P )). Moreover K ≡ 0 (mod P ) or G ≡ k (mod P ). (iii) There are no (*)-cycles of length 18, 36, 72 in R2 . (iv) There are no (*)-cycles of length 27 in R2 . Proof. (i) Assume the contrary. Owing to lemma 2.6(viii) and lemma 2.7(v) we can 2 is a (*)-cycle ¯0 = x¯0 , x¯1 , ..., x¯8 for a mapping Φ such that A = assume that  in R there   1 + πa 0  Φ (¯0) =   for d ≡ 1 (mod P ). 0 d      x1   x3  x3 ) ≥ 2. Put x¯1 =   and x¯3 =  . Then by lemma 2.7(i) we have t = w(¯ y1 y3 Put Φ3 = Ψ.





0  0 x1 ≡  As by lemma 2.6(iv) ¯0 = x¯9 ≡ (I + A + ... + A8 )¯  x¯1 8 0 1 + d + ... + d (mod P w(¯x1 )+1 ), in view of 1 + d + ... + d8 ≡ 0 (mod P ), we obtain w(y1 ) > w(x1 ). In particular w(y1 ) ≥ 2. In the same manner considering a (*)-cycle ¯0 = x¯0 , x¯3 , x¯6 for  the mapping Ψ,we 0  0 obtain ¯0 = x¯9 ≡ (I + Ψ (¯0) + (Ψ (¯0))2 )¯ x3 ≡ (I + A3 + A9 )¯ x3 ≡   x¯3 0 1 + d3 + d6 (mod P w(¯x3 )+1 ) and, in view of 1+d3 +d6 ≡ 0 (mod P ), we obtain ﬁnally w(y3 ) > w(x3 ). In particular from the very deﬁnition of w(¯ x3 ) we obtain t = w(x3 ). 2 Write Φ(x, y) = (x1 +(1+πa)x+kx +lxy+my 2 +..., y1 +dy+Kx2 +Lxy+M y 2 +...) and ˜ 2 + ˜lxy+ my ˜ Kx ˜ 2 + Lxy+ ˜ ˜ y 2 +...). Ψ(x, y) = (x3 +(1+π˜ a)x+π˜by+ kx ˜ 2 +..., y3 +π˜ cx+ dy+ M By lemma 2.5(i) one easily sees that d˜ ≡ d3 (mod P ). Using lemma 2.4 we obtain k˜ ≡ 0 (mod P ) and from Ψ (¯0) = Φ (¯ x2 ) ◦ Φ (¯ x1 ) ◦ Φ (¯0) we obtain, using w(y1 ) ≥ 2, by a direct calculation that a˜ ≡ 0 (mod P ). ˜ a In view of k, ˜ ≡ 0 (mod P ), w(y3 ) > t = w(x3 ) ≥ 2 we obtain by a straightforward calculation that the ﬁrst coordinate of Ψ3 (¯0) minus the second coordinate of Ψ3 (¯0) multiplied by (d˜ + 2)˜bπ is equivalent to ˜ ˜bπy3 c˜˜bπ 2 x3 (1 − (d˜ + 2)2 ) + 3x3 − (d˜ + 2)(d˜2 + d)

(mod P t+3 ).

As d˜ ≡ d3 (mod P ), d ≡ 0, −1 (mod P ), w(y3 ) ≥ t + 1, w(x3 ) = t, 1 − (d˜ + 2)2 , (d˜ + ˜ ∈ P and Ψ3 (¯0) = Φ9 (¯0) = ¯0 we therefore obtain 3x3 ≡ 0 (mod P t+3 ). Now 2)(d˜2 + d) from w(3x3 ) = w(3) + w(x1 ) = 2 + t < 3 + t we obtain a contradiction. (ii) From lemma 2.7(i) w(¯ x3 ) > w(¯ x1 ), and in particular t = w(¯ x3 ) ≥ 2. 3 Write Φ = Ψ. Then we easily see that Ψ can be written in the form Ψ(x, y) = ˜ 2 + g˜xy + hy ˜ 2 + ..., y3 + π˜ ˜ + Kx ˜ 2 + Gxy ˜ + Hy ˜ 2 + ...). (x3 + (1 + π˜ a)x + π˜by + kx cx + (1 + π d)y

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˜≡ By a straightforward calculation we obtain k˜ ≡ 0 (mod P ), g˜ ≡ −Kb2 (mod P ), h 2 2 ˜ ≡ 0 (mod P ), G ˜ ≡ 0 (mod P ), H ˜ ≡ −Kb (mod P ) b (G − k + Kb) (mod P ), K 2 2 2 2 and π˜ a ≡ −kby1 − bKx1 + bGy1 (mod P ), π˜b ≡ kb y1 + b πc − b Kx1 + b2 Gy1 + bkx1 + bGx1 − bHy1 (mod P 2 ), π˜ c ≡ −Kby1 (mod P 2 ), π d˜ ≡ Kb2 y1 + bKx1 (mod P 2 ). As Ψ3 (¯0) = ¯0 by a straightforward calculation we get ˜ 2 + 2˜ ˜ 2 + π 2 x3 (˜ ˜ 2 y3 ≡ 0 2kx g x3 y3 + 3x3 + 2hy a2 + ˜b˜ c) + ˜b˜ aπ 2 y3 + ˜bdπ 3 3 (mod P t+3 ); ˜ 2 x3 + c˜a ˜ 32 + 2Gx ˜ 3 y3 + 3y3 + 2Kx ˜ 23 + y3 π 2 (d˜2 + ˜b˜ 2Hy c) + c˜dπ ˜ π 2 x3 ≡ 0 (mod P t+3 ).

(4)

˜ ≡K ˜ ≡G ˜ ≡H ˜ ≡0 Notice that if b ≡ 0 (mod P ) then a ˜ ≡ ˜b ≡ c˜ ≡ d˜ ≡ k˜ ≡ g˜ ≡ h (mod P ) and (4) gives 3x3 ≡ 0 (mod P t+3 ), 3y3 ≡ 0  (mod P t+3 ), a contradiction  x3  because of w(3) = 2 < 3 and the very deﬁnition of w( ). y3 So we obtained b ≡ 0 (mod P ). If K ≡ 0 (mod P ) and G ≡ k (mod P ) then from the calculation provided earlier ˜ ≡ 0 (mod P ). Similarly in the proof we have π˜ a ≡ −bKx1 +by1 (G−k) (mod P 2 ) and a ˜ ˜ ˜ c˜ ≡ d ≡ g˜ ≡ h ≡ H ≡ 0 (mod P ). By (4) we then get 3x3 ≡ 0 (mod P t+3 ), 3y3 ≡ 0 (mod P t+3 ), and in the same way as before a contradiction follows. (iii) Assume the contrary. According to remark 2.2 it suﬃces to deal with 18. By lemma 2.6(i) we can therefore assume that in R2 there is a (*)-cycle ¯0 = x¯0 , x¯1 , ..., x¯17 of length 18 for a mapping Φ. Put Φ (¯0) = A. As by lemma 2.6(iv) ¯0 = x¯18 ≡ (I + (Φ9 ) (¯0))¯ x9 ≡ (I + A9 )¯ x9 (mod P w(¯x9 )+1 ) we obtain by lemma 2.5(iv) that (I + A9 ) (mod P ) = (I + A)9 (mod P ) is not invertible and A (mod P ) has the eigenvalue −1 + P . ¯  So by  without any  lemma 2.6(vii) we can assume  loss of generality that Φ (0) ≡  −1  0

∗  ∗

 −1 + πa b  (mod P ). Write Φ (¯0) =  . Now by lemma 2.5(i) (Φ2 ) (¯0) ≡ πc m    1 −b + bm  (Φ (¯0))2 ≡   (mod P ). 0 m2 Now observe that ¯0, x¯2 , x¯4 , ..., x¯16 is a (*)-cycle of length 9 for Φ2 , hence by (i) m2 ≡ 1

(mod P ). If m ≡ 1 (mod P ) then (Φ2 ) (¯0) ≡ I (mod P ) contradicting (ii)( b ≡ 0 (mod P )). Hence m ≡ −1 (mod P ). We may thus write Φ(x, y) = (x1 + (−1 + πa)x + by + kx2 + gxy + hy 2 + ..., y1 + πcx + (−1 + πd)y + Kx2 + Gxy + Hy 2 + ...). Using lemma 2.4 we then get by an easy calculation that the coeﬃcient of x2 in the second coordinate of Φ2 is congruent to 0 (mod P ). The coeﬃcient of x2 in the

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ﬁrst coordinate of Φ2 is congruent to bK (mod P ). The coeﬃcient of xy in the second coordinate of Φ2 is congruent to −2bK ≡ bK (mod P ). This clearly contradicts (ii). (iv) Assume the contrary. By lemma 2.6(i),(vii) and lemma 2.7(v) we can therefore assume that there a (*)-cycle ¯0 = x¯0 , x¯1 , ..., x¯26 in R2 of length 27 for a mapping Φ such  is 1 that Φ (¯0) ≡  0

∗  ∗

(mod P ). But then ¯0 = x¯0 , x¯3 , x¯6 , ..., x¯24 would be a (*)-cycle

of length 9 for a mapping Ψ = Φ3 such that Ψ (¯0) ≡ (Φ (¯0))3 (mod P ) has either two diﬀerent eigenvalues or is congruent to I (mod P ). These two possibilities contradict either (i) or (ii). Lemma 4.3. Let R be such that p = 3, e = 2, f = 1, π 2 ≡ −3 (mod P 3 ). Then there are (*)-cycles of length 9 in R2 . 

 2

 3π + π  Proof. Take Φ(x, y) = ((1+π)x+y, π+πx+y+mx2 +qxy). Then Φ3 (¯0) =   = ∗   K(m, q)  ¯0 and by a direct calculation Φ9 (¯0) = π 4   , where K(m, q) = 2m3 + 2qm2 + L(m, q) (q 2 +2q)m+q 3 −1+πW1 (m, q); L(m, q) = 2m3 +(2q+1)m2 +2q 2 m−m−2q−1+πW2 (m, q) for some polynomials W1 , W2 with coeﬃcients from R.     Clearly

K(1, −1), L(1, −1) ≡ 0

(mod P )

and

∂K ∂K ∂q

 ∂m 

∂L ∂L ∂m ∂q

 1  (1, −1) ≡  2

2  2

(mod P ), which is an invertible matrix. Hence, by two-dimensional Hensel’s lemma and completeness of R, for some m, q ∈ R we get K(m, q) = L(m, q) = 0 and then Φ9 (¯0) = ¯0. As Φ3 (¯0) = ¯0 from simple properties of periodic points we therefore obtain that ¯0, Φ(¯0), Φ2 (¯0), ..., Φ8 (¯0) is a (*)-cycle of length 9 for Φ. Lemma 4.4. Let R be such that p = 3, e = 2, f = 1, π 2 ≡ 3 no (*)-cycles of length 9 in R2 .

(mod P 3 ). Then there are

Proof. Assume a contrary. By lemma 2.6(i),(vii), lemma 2.7(v) and lemma 4.2(i) we 2 ¯ ¯ can  assume that inR there is a (*)-cycle 0, x¯1 , ..., x¯8 for a mapping Φ such that Φ (0) =  1 + πA B  x1 ) = 1.   and w(¯ πC 1 + πD Put Φ(x, y) = (U π + (1 + πA)x + By + Jx2 + Gxy + Hy 2 + ..., V π + πCx + (1 + πD)y +

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

319



U  M x2 + Qxy + Ky 2 + ...) with w( ) = 0. V A straightforward calculation gives 



∗ x¯3 = Φ3 (¯0) ≡   (mod P 3 ) b1

(5)

where b1 = (2M U 2 + M U V B + 2QV U + M V 2 B 2 + 2QBV 2 + CV B + 2KV 2 )π 2 . By lemma 2.7(i) we have s = w(¯ x3 ) ≥ 2. If we now put Φ3 = Ψ then by lemma  3 1 B  2.5(i) Ψ (¯0) ≡   ≡ I (mod P ). Hence we can write Ψ in the form Ψ(x, y) = 0 1 2 2 s 2 2 uπ s + (1 +πa)x + πby + jx + gxy + hy + ..., vπ + πcx + (1 + πd)y + mx + qxy + ky + ...),

u where w( ) = 0. v So by (5) π s−2 v ≡ 2M U 2 + M U V B + 2QV U + M V 2 B 2 + 2QBV 2 +CV B + 2KV 2

(mod P ).

(6)

Moreover by a direct calculation we get a ≡ 2M BU + 2JBV + QV B (mod P ); b ≡ 2M B 2 U + JBU + QBU + 2KV B + B 2 QV + B 2 C + 4B 2 JV (mod P ); c ≡ 2M BV (mod P ); d ≡ M BU + M V B 2 (mod P ); g ≡ −M B 2 (mod P ); h ≡ −JB 2 + M B 3 + QB 2 (mod P ); k ≡ −M B 2 (mod P ) and m ≡ j ≡ q ≡ 0 (mod P ). 2 3 P ), we By a direct   calculation, using 3 ≡ π (mod P ) and j ≡ m ≡ q ≡ 0 (mod (1 + a2 + bc + 2g(π s−2 v))u + π s+2 (ab + db + 2h(π s−2 v))v  (mod P s+3 ). s+2 s+2 2 s−2 π (ac + dc)u + π (bc + d + 1 + 2k(π v))v >From the just enlisted values of a, b, c, d, g, h, k, m, j, q and substitution into the last formula the right-hand side of (6) instead of π s−2 v and dividing by π s+2 we then get [(M B − J + Q)M B 2 V (BV − U ) + V 2 B 2 (Q − J)2 + 1]u + [B(M B − J + Q)((M B − J + Q)(−B 2 V 2 − BU V ) + M BU (U − V B))]v ≡ 0 (mod P ) and [−M B 2 V 2 (M B −J +Q)]u+[(M B −J +Q)M B 2 V (U +V B)−2M B 3 V 2 M B +1]v ≡ 0 (mod P ). Treating the two last as a system of two linear equations in u, v over the  congruences  π get Ψ (¯0) ≡ 

s+2

3

u ﬁeld R/P having, as   ≡ ¯0 v

(mod P ), a non-zero solution, we obtain

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



 a2 b 2  det  ≡0 c2 d2

(mod P )

(7)

where a2 = (M B − J + Q)M B 2 V (BV − U ) + V 2 B 2 (Q − J)2 + 1; b2 = B(M B − J + Q)((M B − J + Q)(−B 2 V 2 − BU V ) + M BU (U − V B)); c2 = −M B 2 V 2 (M B − J + Q) and d2 = (M B − J + Q)M B 2 V (U + V B) − 2M B 3 V 2 M B + 1. >From (7) we immediately have that B, V ≡ 0 (mod P ). Thus B 2 ≡ V 2 ≡ 1 (mod P ). Similarly M ≡ 0 (mod P ). So by (7) we get    a3 b 3  det  (8)  ≡ 0 (mod P ) c3 d3 where a3 = (M B − J + Q)M (B − U V ) + (Q − J)2 + 1; b3 = B(M B − J + Q)((M B − J + Q)(−1 − BU V ) + M B(U 2 − U V B)); c3 = −M (M B − J + Q) and d3 = (M B − J + Q)M (U V + B) − 1. Put T = M B − J + Q, T M = L. Then in particular Q − J = T − M B and by (8) we have T ≡ 0 (mod P ). Hence, as M ≡ 0 (mod P ), we get L ≡ 0 (mod P ). It follows from (8) that the determinant   2 2  T M (B − U V ) + (T − M B) + 1 BT (T (−1 − BU V ) + M B(U − U V B))  det   lies −M T T M (U V + B) − 1

in P and, as T 2 , (M B)2 ≡ 1 (mod P ) and T M = L, we get   2  L(B − U V ) + LB B(−1 − BU V + LB(U − U V B))  det  ≡0 −L L(U V + B) − 1 

(mod P ).

(9)



 2LB −B  If U ≡ 0 (mod P ) then (9) gives 0 ≡ det   ≡ 2L2 B 2 ≡ 2 (mod P ), a −L LB − 1 contradiction. Thus we must have U ≡ 0 (mod P ) and putting W = U V ( note that W ≡ 0 (mod P ) asU, V ≡ 0 (mod P )) we get by (9), as W, L, B ≡  0 (mod P ), that  L(B − W ) + LB B(−1 − BW + LB(1 − BW ))  0 ≡ det   ≡ −1 −L L(W + B) − 1 (mod P ), a contradiction. So by considerations in this subsection we obtained

Proposition 4.5. Let R be such that p = 3, e = 2, f = 1. Then (i) If 3 ≡ π 2 (mod P 3 ) then CYCL ∗ (R, 2) = {1, 2, 3, 4, 6, 8, 12, 24}. Hence, by proposition 2.8(ii), CYCLMAX (R, 2) = {216, 192, 168, 144, 120}.

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(ii) If −3 ≡ π 2 (mod P 3 ) then CYCL ∗ (R, 2) = {1, 2, 3, 4, 6, 8, 9, 12, 24}. Hence, by proposition 2.8(ii), CYCLMAX (R, 2) = {216, 192, 168, 144, 120, 81, 63, 45}.

4.2 Cycles for p = 3, e = 1, f = 1 Let R be such that p = 3, e = 1, f = 1. By proposition 2.8(vii) we have that {1, 2, 3, 4, 6} ⊂ CYCL ∗ (R, 2). By proposition 2.8(iii) we have 8 = 32 − 1 ∈ CYCL ∗ (R, 2). Lemma 4.6. Let R be such that p = 3, e = 1, f = 1. (i) There are no (*)-cycles of length 9 in R2 . Hence, in particular, in R2 there are no (*)-cycles of length 18, 36, 72. (ii) There are no (*)-cycles of length 12 in R2 . Hence, in particular, there are no (*)-cycles of length 24 in R2 . Proof. (i) By remark 2.2 it suﬃces to deal with 9. In this case R is embeddable into a ring S such that p = 3, e = 2, f = 1, π 2 ≡ 3 (mod P 3 ). As via this embedding a (*)-cycle in R2 is a (*)-cycle in S 2 by proposition 4.5(i) we get the assertion. (ii) By remark 2.2 it suﬃces to deal with 12. As 12 = 4 · 3 and 4 does not divide 1 3 − 1 we get by proposition 2.8(v) that in R2 there is a (*)-cycle of length 3 of the form ¯0 = x¯0 , x¯1 , x¯2 with w(¯ x1 ) ≥ 2 for a mapping Φ such that Φ (¯0) ≡ I (mod P ). Put x1 ≡ ¯0 (mod P w(¯x1 )+2 ) A = Φ (¯0). From A ≡ I (mod P ) we have (A − I)2 x¯1 , 3(A − I)¯ and, as by lemma 2.6(iv) ¯0 = x¯3 ≡ ((A − I)2 + 3(A − I) + 3I)¯ x1 (mod P w(¯x1 )+2 ), we obtain 3¯ x1 ≡ ¯0 (mod P w(¯x1 )+2 ) and, as w(3¯ x1 ) = 1 + w(¯ x1 ) < 2 + w(¯ x1 ), a contradiction follows. So by considerations in this subsection we obtained Proposition 4.7. Let R be such that p = 3, e = 1, f = 1. Then CYCL ∗ (R, 2) = {1, 2, 3, 4, 6, 8}. Hence, by proposition 2.8(ii), CYCLMAX (R, 2) = {72, 64, 56, 54, 48, 42, 40, 30}.

4.3 Evaluation of Γ(d, 3) Lemma 4.8. Let d be a square-free number. Then (i) Γ(d, 3) = {72, 64, 56, 54, 48, 42, 40, 30 and their divisors } for d ≡ 1 (mod 3); (ii) Γ(d, 3) = {192, 168, 144, 120, 108 and their divisors } for d ≡ 3 (mod 9); (iii) Γ(d, 3) = {192, 168, 144, 120, 108, 45 and their divisors } for d ≡ 6 (mod 9); (iv) Γ(d, 3) = D = {30n, 12n : 1 ≤ n ≤ 16; and their divisors } for d ≡ 2 (mod 3). Proof. (i) We use subsection 2.3 and obtain that 3ZK = p1 p2 for d = 1 and 3ZK = p1 for d = 1, where p1 , p2 are diﬀerent prime ideals of ZK such that (Z K )p1 , (ZK )p2 are isomorphic to R from subsection 4.2 and we use proposition 4.7 and the deﬁnition of Γ(d, p).

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√ (ii) In this case 3ZK = p2 and d may serve the role of a uniformizing parameter π 3 in (Z K )p. As π divides d − 3 in (ZK )p we see that (ZK )p is isomorphic to some R such that p = 3, e = 2, f = 1 and 3 ≡ π 2 (mod P 3 ), and we use proposition 4.5(i) and the deﬁnition of Γ(d, p). √ (iii) In this case 3ZK = p2 and d may serve the role of a uniformizing parameter π 3 in (Z K )p. As π divides d + 3 in (ZK )p we see that (ZK )p is isomorphic to some R such that p = 3, e = 2, f = 1 and −3 ≡ π 2 (mod P 3 ), and we use proposition 4.5(ii) and the deﬁnition of Γ(d, p). (iv) In this case 3ZK = p is prime and (Z K )p is isomorphic to some R such that p = 3, e = 1, f = 2, and we use lemma 3.10(iii).

5

Cycles for R with p = 5.

If R is such that p = 5, e = 1, f = 2 then by lemma 3.10(iii) CYCL(R, 2) ⊃ D. For ef ≤ 2 the remaining possibility is f = 1, e ≤ 2. If R is such that p = 5, f = 1, e ≤ 2 then, by proposition 2.8(iv), (*)-cycles in R2 may have lengths only of the form 5α , 2 · 5α , 3 · 5α , 4 · 5α , 6·5α , 8·5α , 12·5α , 24·5α . However unlike the case p = 2 and p = 3, e ≤ 2, f = 1 we will not ﬁnd a whole CYCL∗(R, 2) for R such that p = 5, e ≤ 2, f = 1 as in order to determine Γ(d, 5) ⊂ D an existence or non-existence of (*)-cycles of only certain particular lengths will have any signiﬁcance. The purpose of the next lemma is to ﬁnd those lengths. Lemma 5.1. Let R be such that p = 5, f = 1, e ≤ 2. Then (i) {480, 360, 300, 192, 168, 156, 144, 132, 108 and their divisors} ⊂ CYCL(R, 2); (ii) let k be one of the numbers 35, 55, 65, 70, 105, 110. Then k ∈ CYCL(R, 2) holds if and only if 5 ∈ CYCL ∗ (R, 2); (iii) let k be one of the numbers 130, 140. Then k ∈ CYCL(R, 2) holds if and only if 10 ∈ CYCL ∗ (R, 2); (iv) 210 ∈ CYCL(R, 2) holds if and only if there is a (*)-cycle of length 10 or 15 in R2 ; (v) let k be one of the numbers 135, 165, 195, 270, 330. Then k ∈ CYCL(R, 2) holds if and only if 15 ∈ CYCL ∗ (R, 2); (vi) 225 ∈ CYCL(R, 2) holds if and only if there is a (*)-cycle of length 15 or 25 in R2 ; (vii) 420 ∈ CYCL(R, 2) holds if and only if there is a (*)-cycle of length 20 or 30 in R2 ; (viii) 450 ∈ CYCL(R, 2) holds if and only if there is a (*)-cycle of length 25 or 30 in R2 ; (ix) 390 ∈ CYCL(R, 2) holds if and only if 30 ∈ CYCL ∗ (R, 2). Proof. (i) It follows from proposition 2.8(iii) that there exist in R2 (*)-cycles of length 24 = 52 − 1. By remark 2.2 the same holds for every divisor of 24. Since every integer listed in (i) can be written in the form a · b with a ≤ pf N = 51·2 = 25, b|24 the assertion follows from remark 1.1 and proposition 2.8(ii). (ii)-(ix) From proposition 2.8(iv) we obtain that (*)-cycles in R2 may have lengths only of the form 5α , 2 · 5α , 3 · 5α , 4 · 5α , 6 · 5α , 8 · 5α , 12 · 5α , 24 · 5α . Having in mind proposition 2.8(ii) we then represent every number from the set

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D \ {480, 360, 300, 192, 168, 156, 144, 132, 108 and their divisors} = {35, 55, 65, 70, 105, 110, 130, 135, 140, 165, 195, 210, 225, 270, 330, 390, 420, 450} in the form a · b with a ≤ pf N = 51·2 = 25 and b of one of the form 5α , 2 · 5α , 3 · 5α , 4 · 5α , 6 · 5α , 8 · 5α , 12 · 5α , 24 · 5α . We explain how to obtain statements (ii)-(ix) on one particular element of the set D \ {480, 360, 300, 192, 168, 156, 144, 132, 108 and their divisors } = { 35, 55, 65, 70, 105, 110, 130, 135, 140, 165, 195, 210, 225, 270, 330, 390, 420, 450}, namely let us consider the number 225 in more detail. We represent 225 as a multiplication a · b, with a, b as above and obtain that possible pairs (a, b) are as follows (a, b) = (3, 75), (9, 25), (15, 15). Thus in R2 there is a (*)cycle of length 75 or 25 or 15 if and only if 225 ∈ CYCL(R, 2). But by remark 2.2 if 75 ∈ CYCL ∗ (R, 2) then also 25 ∈ CYCL ∗ (R, 2). Thus 225 ∈ CYCL(R, 2) if and only if in R2 there is a (*)-cycle of length 15 or 25. Hence in this section we shall determine whether or not 5, 10, 15, 20, 25, 30 are lengths of suitable (*)-cycles in R2 for R such that p = 5, e ≤ 2, f = 1.

5.1 Cycles for p = 5, e = 2, f = 1 Lemma 5.2. Let R be such that p = 5, e = 2, f = 1, π 2 ≡ ±5 (mod P 3 ). Then in R2 there are (*)-cycles of length 20. Hence, in particular, in R2 there are (*)-cycles of length 10, 5. Proof. In this case 5 ≡ π 2 (mod P 3 ) or 5 ≡ (2π)2 (mod P 3 ). Hence, by Hensel’s √ lemma, the equation X 2 − 5 = 0 has a solution in R. So 5 ∈ R. Let i2 = −1 and ζ5 be the primitive root of unity of order 5. Note that i ∈ R. Then iζ5 is a primitive root of √ −1+ 5 2 unity of order √20. We easily see that iζ5 is a root of the equation X − (i 2 )X − 1 = 0 or X 2 − (i −1−2 5 )X − 1 = 0. So by proposition 2.8(vii) we get the statement. Lemma 5.3. Let R be such that p = 5, e = 2, f = 1. Then in R2 there are (*)-cycles of length 10. Hence, in R2 there are (*)-cycles of length 5. Proof. Let 5 = π 2 5 with 5 ≡ 0

(mod P ). Take a mappingΦ deﬁned by Φ(x, =   y)  π  −2π  (−x + y, π − y + mx3 + ax2 y). Then one easily sees that Φ2 (¯0) =  , Φ5 (¯0) ≡   0 π (mod P 2 ), hence we get Φ2 (¯0), Φ5 (¯0) = ¯0.   3  π K(m, a)  By a direct calculation we get Φ10 (¯0) =   with K(m, a) = 5 + 4m + π 5 L(m, a) πW1 (m, a) and L(m, a) = 12m2 + 6a5 + πW2 (m, a) for some polynomials W1 , W2 with coeﬃcients from R. So we have to ﬁnd m, a ∈ R such that K(m, a) = L(m, a) = 0. As K(5, 35) ≡

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







∂K ∂K ∂m ∂a

  4 0  (5, 35) ≡   (mod P ) and the last ∂L ∂L 245 65 ∂m ∂a matrix is invertible, the existence of suitable m, a follows from Hensel’s lemma. Having found such m, a in view of Φ2 (¯0), Φ5 (¯0) = ¯0 and simple properties of periodic points we obtain a (*)-cycle ¯0, Φ(¯0), ..., Φ9 (¯0) of length 10 for the mapping Φ. L(5, 35) ≡ 0

 (mod P ) and 

Lemma 5.4. Let R be such that p = 5, e = 2, f = 1. x1 ) = s ≥ 2 for a (i) In R2 there are no (*)-cycles of the form ¯0, x¯1 , ..., x¯4 with w(¯ ¯ ¯ mapping Φ satisfying Φ (0) ≡ I (mod P ) or Φ (0) (mod P ) having two diﬀerent eigenvalues. (ii) In R2 there are no (*)-cycles of length 25. (iii) In R2 there are no (*)-cycles of length 15, 30. Proof. (i) Suppose that we have such a (*)-cycle for Φ. First we consider the case πby +kx2 + lxy + my 2 + ..., y1 + Φ (¯0) ≡ I (mod P ). Write Φ(x, y) = (x1 + (1 + πa)x +  x1  πcx + (1 + πd)y + Kx2 + Lxy + M y 2 + ...) with s = w( ) ≥ 2. y1    5x1  By a straightforward calculation we obtain Φ5 (¯0) ≡   (mod P s+3 ). In view 5y1      x1   5x1  of w(5) = 2 < 3 we therefore obtain w(Φ5 (¯0)) = w( ) = 2 + w( ) = 2 + s < 5y1 y1 5 ¯ and then Φ (0) ¯ = 0. ¯ This is clearly a contradiction. 3 + s ≤ w(0) ¯ So Φ (0) (mod P ) has 2 diﬀerent eigenvalues. From lemma 2.7(v) it follows that ¯ Φ (0) (mod  P ) hasthe eigenvalue 1 + P . Thus by lemma 2.6(viii) we can assume that  1 + πa 0  Φ (¯0) =   for a suitable d ≡ 1 (mod P ). 0 d   s  uπ  Write then x¯1 =   and Φ(x, y) = (uπ s + (1 + πa)x + kx2 + gxy + hy 2 + ..., vπ s + vπ s   u dy + Kx2 + Gxy + Hy 2 + ...) with s ≥ 2 and w( ) = 0. By a straightforward v   ∗   calculation we then get ¯0 = Φ5 (¯0) ≡   (mod P s+1 ). As 2 3 4 s v(1 + d + d + d + d )π 1 + d + ... + d4 ≡ 0

(mod P ) we obtain π|v and from the very deﬁnition of w(¯ x1 ) we get

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π u.

325



 s

 5uπ  Now by a straightforward calculation we have ¯0 = Φ5 (¯0) ≡   ∗

(mod P s+3 ),

a contradiction as π u and w(5) = 2 < 3. (ii) Assume the contrary. By lemma 2.6(i), lemma 2.6(vii) and lemma 2.7(v) we can then assume that¯0 = x¯0 , x¯1 , ..., x¯24 is a (*)-cycle of length 25 in R2 for a mapping 1 Φ such that Φ (¯0) ≡  0

∗  ∗

(mod P ). Lemma 2.7(i) gives that w(¯ x5 ) ≥ 2 and ¯0 =

5 x¯0 , x¯5 , x¯10 , x¯15 , x¯20 would be   a (*)-cycle for a mapping Ψ = Φ satisfying, by lemma 2.5(i), 1 0 Ψ (¯0) ≡ (Φ (¯0))5 ≡   (mod P ). Clearly Ψ (¯0) ≡ I (mod P ) or Ψ (¯0) (mod P ) 0∗

has two diﬀerent eigenvalues, and we get a contradiction with (i). (iii) It suﬃces to deal with 15. As 3 51 − 1 we have by proposition 2.8(v) that in R2 there is a (*)-cycle of the form ¯0 = x¯0 , x¯1 , ..., x¯4 with w(¯ x1 ) ≥ 2 for a mapping Φ such ¯ that Φ (0) ≡ I (mod P ), which clearly contradicts (i). Lemma 5.5. Let R be such that p = 5, e = 2, f = 1, π 2 ≡ ±10 are no (*)-cycles of length 20 in R2 .

(mod P 3 ). Then there

Proof. Assume the contrary. Write 5 = π 2 5. So 5 ≡ 2, 3 (mod P ). By lemma 2.6(i) we can assume that in ¯ ¯ R2 there  isa (*)-cycle 0, x¯1 , ..., x¯19 of length 20 for a mapping Φ. Put Φ (0) = A and  xj  x¯j =  . yj >From lemma 2.6(iv) it follows that ¯0 = x¯20 ≡ (I + (Φ10 ) (¯0))¯ x10 ≡ (I + A10 )¯ x10 w(¯ x10 )+1 10 (mod P ). Hence by lemma 2.5(iv) the matrix (I + A ) (mod P ) = (I + A2 )5 (mod P ) = ((A − 2I)(A − 3I))5 (mod P ) is not invertible. Hence A (mod P ) has an eigenvalue equal to 2 + P or 3 + P . ¯ ¯ 1st case. Assume P)   that Φ (0) (mod P ) has two diﬀerent eigenvalues or Φ (0) (mod λ 0 λ 0 is of the form   . Then by lemma 2.6(viii) we can assume that Φ (¯0) ≡  0µ 0λ (mod P ) with λ, µ ∈ R and λ ≡ 2 (mod P ) or 3 (mod P ). 1st possibility, µ ≡ 2, 3, 4 (mod P ). Then I + A + A2 + A3 ≡ 0 (mod P ), and by lemma 2.6(iv) we obtain x¯4 ≡ (I + A + A2 + A3 )¯ x1 ≡ ¯0 (mod P w(¯x1 )+1 ) and w(¯ x4 ) ≥ 2 4 ¯ 4 follows. As by lemma 2.5(i) in this possibility (Φ ) (0) ≡ A ≡ I (mod P ) we obtain a (*)-cycle ¯0 = x¯0 , x¯4 , x¯8 , x¯12 , x¯16 for a mapping Ψ = Φ4 fulﬁlling Ψ (¯0) ≡ I (mod P ), which contradicts lemma 5.4(i).

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¯ 2nd possibility, µ ≡ 0 (modP ). Then  0 = x¯20 ≡ (I + A +   by  lemma  2.6(iv) 19  1 + λ + ... + λ 0  0 0 0 19 ... + A )¯ x1 ≡   x¯1 ≡   x¯1 ≡   (mod P w(¯x1 )+1 ). Thus 0 1 01 y1   0 x1 ) follows. So, w(y1 ) ≥ 2 and from   ≡ ¯0 (mod P w(¯x1 )+1 ) and w(y1 ) > w(¯ y1 x¯4 ≡ (I +A+A2 +A3 )¯ x1 (mod P w(¯x1 )+1 ) we easily get w(¯ x4 ) ≥ w(¯ x1 )+1 and w(¯ x4 ) ≥ 2 4 ¯ follows. But then 0 = x¯0 , x¯4 , ..., x¯16 is a (*)-cycle for a mapping Ψ = Φ . As w(¯ x4 ) ≥ 2  and, as Ψ (¯0) ≡ A4 ≡ 

1 0  00

(mod P ), the matrix Ψ (¯0) (mod P ) has two diﬀerent

eigenvalues and we get a contradiction with lemma 5.4(i). 3rd possibility, µ ≡ 1 (mod P ). By lemma 2.5(i) we have in this case (Φ4 ) (¯0) ≡ A4 ≡ I (mod P ) and ¯0 = x¯0 , x¯4 , x¯8 , x¯12 , x¯16 is a (*)-cycle for Φ4 such that (Φ4 ) (¯0) ≡ I 2.6(ii) also (mod P ). Hence by lemma 5.4(i) we must have w(¯ x4 ) = 1 andby lemma  0 w(¯ x1 ) = 1. By lemma 2.6(iv) we have x¯4 ≡ (I +A+A2 +A3 )¯ x1 ≡  0   0 In particular x¯4 ≡   (mod P 2 ). ∗

0  x¯1 4

(mod P 2 ).

Put Ψ = Φ4 . By lemma 2.4 we have that the coeﬃcients of y 2 in the ﬁrst component of Ψ and of xy in the second coordinate of Ψ are divisible by π. So we may write Ψ(x, y) = (uπ 2 + (1 + πa)x + πby + πhy 2 + ..., vπ + πcx + (1 + πd)y + πqxy + ry 2 + my 3 + ...) with π v ( as w(¯ x4 ) = 1).    By a straightforward calculation we get ¯0 = Ψ5 (¯0) ≡ 

∗ vπ(5 + (vr)2 π 2 )

 

(mod P 4 ), and therefore π 3 |5+v 2 r2 π 2 or equivalently π|5+(vr)2 , which is a contradiction since 5 ≡ 2, 3 (mod P ) and the equation X 2 ± 2 ≡ 0 (mod P ) has no solution in R. 2nd case. Assume that A = Φ (¯0) (modP ) has only one eigenvalue λ (mod P ) λ 0 and Φ (¯0) (mod P ) is not of the form  . >From the previous considerations we 0λ have λ ≡ 2, 3 (mod P ). By lemma and from (B −1 ◦ Φ ◦ B) (¯0) = B −1 Φ (¯0)B we  2.6(vii)  λ β  then can assume that Φ (¯0) ≡   (mod P ) with λ ≡ 2, 3 (mod P ) and β ≡ 0 0λ (mod P ).

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



327





   x4   uπ  u Put w(¯ x4 ) = s and Φ4 = Ψ. Hence x¯4 =   =   with w( ) = 0. y4 vπ s v s

1st possibility, s ≥ 3. We see that Ψ = Φ4 can be written in the form Ψ(x, y) = (uπ s +(1+πa)x+by +kx2 +gxy +hy 2 +..., vπ s +πcx+(1+πd)y +Kx2 +Gxy +Hy 2 +...). By a straightforward calculation we get   (c2 b2 π 2 + 5)uπ s + (2cb2 aπ 2 + 2cb2 dπ 2 + 10b)vπ s  ¯0 = Φ20 (¯0) = Ψ5 (¯0) ≡    (mod P s+3 ). (c2 b2 π 2 + 5)vπ s Therefore (c2 b2 π 2 + 5)uπ s + (2cb2 aπ 2 + 2cb2 dπ 2 + 10b)vπ s ≡ 0

(mod P s+3 )

(10)

and (c2 b2 π 2 + 5)vπ s ≡ 0 (mod P s+3 ) follows. Because, like in the 1st case, we have π 3 5 + x2 for x ∈ R we obtain π|v and then by (10) (c2 b2 π 2 + 5)uπ s ≡ 0 (mod P s+3 ) and from π 3 5 + c2 b2 π 2 we obtain π|u, a u contradiction with 0 = w( ). v 2nd possibility, w(¯ x4 ) = s = 2. We write Ψ = Φ4 in the same form as in the 1st possibility. By an easy calculation we get that the coeﬃcient of x2 in the second component of Φ4 is divisible by π, i.e. π|K. Hence by a straightforward calculation we obtain   4 2 2 2 2 2 2 2 2 2  π (c b u + 2vcb a + 2vdcb + v kb + 3Gv b ) + π (5u + 10vb)  5 Ψ5 (¯0) ≡   (mod P ). π 4 (vc2 b2 ) + 5vπ 2 (11) 20 ¯ 5 ¯ 5 4 2 2 2 3 2 2 2 ¯ As 0 = Φ (0) = Ψ (0) we therefore obtain π |π (vc b ) + 5vπ and π |v(5 + c b π ) which, as we already know, is possible only when π|v. So π|v and therefore π u. Hence by (11) we obtain π 5 |uπ 2 (5 + c2 b2 π 2 ) and π 3 |5 + c2 b2 π 2 , which we already know to be impossible. x4 ) = 1. Then by x1 ) = 1. As by lemma 2.6(iv)  possibility, w(¯  3rd  lemma  2.6(ii) w(¯  x4  0 x1 ≡    = x¯4 ≡ (A3 + A2 + A + I)¯ y4 0

∗  x¯1 0

(mod P 2 ) we obtain that π 2 |y4 .

Write Φ(x, y) = (uπ + (λ + πa)x βy + kx2 + gxy + mx3 + ..., vπ + πcx + (λ + πd)y +  + u Kx2 + Gxy + M x3 + ...) with w( ) = 0 and β ≡ 0 (mod P ). v By lemma 2.4 we easily see that the coeﬃcient of x2 in the second component of Ψ = Φ4 is divisible by π. So we can write Ψ(x, y) = (U π + (1 + πa1 )x + β1 y + k1 x2 + g1 xy + m1 x3 + ..., V π 2 + πc1 x + (1 + πd1 )y + πK1 x2 + G1 xy + M1 x3 + ...) with, as π 2 |y4 and w(¯ x4 ) = 1, π U .

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20 ¯ 5 ¯ ¯  A direct calculation leads now to 0 = Φ (0) = Ψ(0) ≡ 3 2 2 3 2 3 2  5U π + π (c1 β1 U + M1 β1 U + k1 U + 3G1 c1 β1 U )   (mod P 4 ).  0

Thus π 4 |U π(5 + π 2 (c21 β12 + M1 β1 U 2 + k12 U 2 + 3G1 c1 β1 U )) and, as 5 = π 2 5, we obtain π|5+c21 β12 +M1 β1 U 2 +k12 U 2 +3G1 c1 β1 U or, equivalently, π|5+(c1 β1 +4G1 U )2 +U 2 (M1 β1 + k12 − G21 ). But a straightforward calculation gives M1 ≡ 4K 2 β (mod P ); β1 ≡ λβ (mod P ); k1 ≡ K(λ − 1)2λβ (mod P ) and G1 ≡ −2K(λ − 1)2λβ (mod P ). This implies π|M1 b1 + k12 − G21 , thus π|5 + (c1 b1 + 4G1 U )2 , a contradiction as in view of 5 ≡ 2, 3 (mod P ) a congruence X 2 + 5 ≡ 0 (mod P ) has no solution in R. So by lemmas 5.2,5.3,5.4 and 5.5 we obtained Proposition 5.6. Let R be such that p = 5, e = 2, f = 1. Then (i) if π 2 ≡ ±5 (mod P 3 ) then 5, 10, 20 are lengths of suitable (*)-cycles in R2 and 15, 25, 30 are not lengths of (*)-cycles in R2 ; (ii) if π 2 ≡ ±10 (mod P 3 ) then 5, 10 are lenghts of suitable (*)-cycles in R2 and 15, 20, 25, 30 are not lengths of (*)-cycles in R2 .

5.2

Cycles for p = 5, e = 1, f = 1

Proposition 5.7. Let R be such that p = 5, e = 1, f = 1. Then there are (*)-cycles of length 5 in R2 . There are no (*)-cycles of lengths 10, 15, 20, 25, 30 in R2 . Proof. As such R is embeddable into a ring S such that p = 5, e = 2, f = 1, π 2 ≡ ±10 (mod P 3 ) and via this embedding a (*)-cycle in R2 becomes a (*)-cycle in S 2 we get by proposition 5.6(ii) the statement concerning 15, 20, 25, 30. To ﬁnd a (*)-cycle of length 5 consider a mapping Φ deﬁned by Φ(x, y) = (x + y − 2 x + cx(y − 10)(y + 10), 5 + y − x2 + dx(y − 10)(y + 10)). Then clearly Φi (¯0) = ¯0 for i = 1, 2, 3, 4. As Φ5 (¯0) = (−225 − 1202 + c(−120)(−115)(−95), − 100 − 1202 + d(−120)(−115)(−95)) then for suitable c, d ∈ R we get Φ5 (¯0) = ¯0 and a tuple ¯0, Φ(¯0), ..., Φ4 (¯0) is a (*)-cycle of length 5 in R2 . Suppose that x¯0 , x¯1 , ..., x¯9 is a (*)-cycle of length 10 for a mapping Φ. By lemma 2.6(i) we can assume that ¯0 = x¯0 . Put A = Φ (¯0). Via embedding R into S a tuple ¯0, x¯2 , x¯4 , x¯6 , x¯8 would be a (*)-cycle for Φ2 with w1 (¯ x2 ) ≥ 2 ( here the function w1 is the counterpart of w referring to the ring S, in particular w1 (x) = 2w(x) for x ∈ R as the ramiﬁcation index of S over R is 2). Thus by lemma 5.4(i) we have that (Φ2 ) (¯0) (mod P ) has only one eigenvalue and (Φ2 ) (¯0) ≡ I (mod P ). As by lemma 2.6(iv) ¯0 = x¯10 ≡ (I + (Φ5 ) (¯0))¯ x5 ≡ (I + A5 )¯ x5 (mod P w(¯x5 )+1 ) we have that a matrix (I + A5 ) (mod P ) = (I + A)5 (mod P ) is not invertible. Thus A (mod P ) = Φ (¯0) (mod P ) has an eigenvalue −1 + P and by lemma 2.6(vii) we can assume that

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

329



 −1 ∗  A ≡   (mod P ) for a suitable µ ∈ R. As by lemma 2.5(i) (Φ2 ) (¯0) 0 µ (mod P ) = (Φ (¯0))2 (mod P ) and (Φ2 ) (¯0) (mod P )has only one eigenvalue and  −1 ∗  (Φ2 ) (¯0) ≡ I (mod P ) we obtain A = Φ (¯0) ≡   (mod P ). 0 −1 2 Put Φ2 = Ψ and write Ψ in the form Ψ(x, y) = (uπ s + (1 + πa)x + by + kx  + gxy +

u x2 ) = s and w( ) = 0. hy 2 +..., vπ s +πcx+(1+πd)y +Kx2 +Gxy +Hy 2 +...) with w(¯ v   s s  5uπ + 10bvπ  If s ≥ 2 then by a straightforward calculation we get ¯0 = Φ10 (¯0) = Ψ5 (¯0) ≡   5vπ s   u (mod P s+2 ). Hence, as w(5) = 1, π|u, v, a contradiction with w( ) = 0 follows. v   ∗ It remains to consider the case s = 1. By lemma 2.6(iv) we get x¯2 ≡   (mod P 2 ) 0 10 5 ¯ ¯ ¯ and π|v,  π u follows. Then by a straightforward calculation we get 0 = Φ (0) = Ψ (0) ≡  5πu    (mod P 3 ), a contradiction, because π u. 0

5.3 Evaluation of Γ(d, 5) Lemma 5.8. Let d be a square-free number. Then (i) Γ(d, 5) = {480, 360, 300, 192, 168, 156, 144, 132, 110, 108, 105, 70, 65 and their divisors } for d ≡ 1, 4 (mod 5); (ii) Γ(d, 5) = {480, 360, 300, 210, 192, 168, 156, 144, 140, 132, 130, 110, 108 and their divisors } for d ≡ 10, 15 (mod 25); (iii) Γ(d, 5) = {480, 420, 360, 300, 192, 168, 156, 144, 132, 130, 110, 108 and their divisors } for d ≡ 5, 20 (mod 25); (iv) Γ(d, 5) = D = {30n, 12n : 1 ≤ n ≤ 16; and their divisors } for d ≡ 2, 3 (mod 5). Proof. (i) We use subsection 2.3 and obtain that 5ZK = p1 p2 for d = 1 and 5ZK = p1 for d = 1, where p1 , p2 are diﬀerent prime ideals of ZK such that (Z K )p1 , (ZK )p2 are

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isomorphic to R from subsection 5.2 and we use proposition 5.7, lemma 5.1 and the deﬁnition of Γ(d, p). √ (ii) In this case 5ZK = p2 and d may serve the role of a uniformizing parameter π in 2 (Z (mod P 3 ) is equivalent to d ≡ ±10 (mod 25) we see that (Z K )p. As π ≡ ±10 K )p 2 3 is isomorphic to some R such that p = 2, e = 2, f = 1 and π ≡ ±10 (mod P ), and we use proposition 5.6(ii), lemma 5.1 and √ the deﬁnition of Γ(d, p). 2 (iii) In this case 5ZK = p and d may serve the role of a uniformizing parameter π in (ZK )p. As π 2 ≡ ±5 (mod P 3 ) is equivalent to d ≡ ±5 (mod 25) we see that (Z K )p 2 3 is isomorphic to some R such that p = 2, e = 2, f = 1 and π ≡ ±5 (mod P ), and we use proposition 5.6(i), lemma 5.1 and the deﬁnition of Γ(d, p). (iv) In this case 5ZK = p is prime and (Z K )p is isomorphic to some R such that p = 2, e = 1, f = 2, and we use lemma 3.10(iii).

6

Cycles for R with p = 7

If R is such that p = 7, e = 1, f = 2 then by lemma 3.10(iii) we have CYCL(R, 2) ⊃ D. Proposition 6.1. Let R be such that p = 7, f = 1, e ≤ 2. Then (i) {480, 420, 360, 300, 270, 192, 168, 156, 144, 132, 108 and their divisors} ⊂ CYCL(R, 2); (ii) 55, 65, 110, 130, 165, 195, 225, 330, 390, 450 ∈ / CYCL(R, 2). Proof. (i) In the case p = 7, e ≤ 2, f = 1 by proposition 2.8(iii) and remark 2.2 we have that p2f − 1 = 48 and all its divisors are lengths of suitable (*)-cycles in R2 . We easily represent the numbers from {480, 420, 360, 300, 270, 192, 168, 156, 144, 132, 108 and their divisors} in the form ab with a ≤ 49 and b|72 − 1 = 48. From proposition 2.8(ii) and pf N = 72 = 49 we thus obtain the statement. (ii) In the case p = 7, e ≤ 2, f = 1 by proposition 2.8(iv) (*)-cycles in R2 may have lengths only of the form a · 7α where a|72 − 1 = 48. However an easy check gives that no number from the set {55, 65, 110, 130, 165, 195, 225, 330, 390, 450} is representable in the form ab with a ≤ 49 and b dividing 48 · 7α for any α, and we use proposition 2.8(ii).

6.1 Evaluation of Γ(d, 7) Lemma 6.2. Let d be a square-free number. Then (i) Γ(d, 7) = {480, 420, 360, 300, 270, 192, 168, 156, 144, 132, 108 and their divisors } for d ≡ 0, 1, 2, 4 (mod 7); (ii) Γ(d, 7) = {480, 450, 420, 390, 360, 330, 300, 270, 192, 168, 156, 144, 140, 132, 108 and their divisors } = D for d ≡ 3, 5, 6 (mod 7). Proof. (i) We use subsection 2.3 and obtain that 7ZK = p1 p2 or 7ZK = p21 for d = 1 and

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7ZK = p1 for d = 1, where p1 , p2 are diﬀerent prime ideals of ZK such that (Z K )p1 , (ZK )p2 are isomorphic to some R such that p = 7, e ≤ 2, f = 1 and we use proposition 6.1 and the deﬁnition of Γ(d, p). (ii) In this case 7ZK = p is prime and (Z K )p is isomorphic to some R such that p = 2, e = 1, f = 2, and we use lemma 3.10(iii).

7

Proof of Theorem 1.3

Theorem 1.3 follows from lemma 3.10(ii) and lemmas 3.11, 4.8, 5.8, 6.2. We demonstrate the proof in more detail in cases (ii) and (xii). In the remaining cases we proceed in a similar manner. So let d ≡ 2 (mod 4). Then by lemma 3.10(ii) CYCL(ZK , 2) = Γ(d, 2) ∩ Γ(d, 3) ∩ Γ(d, 5) ∩ Γ(d, 7). >From lemma 3.11(ii) we have Γ(d, 2) = {32, 24, 18 and their divisors}. By lemmas 4.8, 5.8 and 6.2 one easily sees that for all d we have Γ(d, 3), Γ(d, 5), Γ(d, 7) ⊃ {32, 24, 18 and their divisors}. Now let d ≡ 53, 77, 197, 317, 413, 533, 557, 653 (mod 840). This is equivalent to d ≡ 5 (mod 8) and d ≡ 2 (mod 3) and d ≡ 2, 3 (mod 5) and d ≡ 0, 1, 2, 4 (mod 7). Thus by lemma 3.11(iv), lemma 4.8(iv), lemma 5.8(iv) and lemma 6.2(i) we obtain Γ(d, 2) = D, Γ(d, 3) = D, Γ(d, 5) = D and Γ(d, 7) = {480, 420, 360, 300, 270, 192, 168, 156, 144, 132, 108 and their divisors}. We then get the statement by lemma 3.10(ii).

Acknowledgements The computations needed in the proof of our results, in particular the calculation of iterations of polynomial mappings, have been performed with the use of GP/PARI CALCULATOR,Version 1.38(i386 version), Copyright 1989,1993 by C.Batut, D.Bernardi, H.Cohen and M.Olivier.

References [Ba] G.Baron: preprint, 1990. [Bo] J.Boduch: MA thesis, The University of Wroclaw, Wroclaw, 1990. [Pe1] T.Pezda: “On cycles and orbits of polynomial mappings Z 2 → Z 2 “, Acta Mathematica et Informatica Universitatis Ostraviensis, Vol. 10, (2002), pp. 95–102. [Pe2] T.Pezda: “Cycles of polynomial mappings in several variables over rings of integers in ﬁnite extensions of the rationals“, Acta Arith., Vol. 108, No. 2, (2003), pp. 127–146. [Pe3] T.Pezda: “Cycles of polynomial mappings in several variables over rings of integers in ﬁnite extensions of the rationals (II)“, submitted.

CEJM 2(2) 2004 332–338

Aﬃnely equivalent complete ﬂat manifolds Michal Sadowski

∗

Department of Mathematics, University of Gda´ nsk, 80-952 Gda´ nsk Wita Stwosza 57 Poland

Received 29 December 2003; revised 18 March 2004 Abstract: Let EAﬀ (Γ, G, m) be the set of aﬃne equivalence classes of m-dimensional complete ﬂat manifolds with a ﬁxed fundamental group Γ and a ﬁxed holonomy group G. Let n be the dimension of a closed ﬂat manifold whose fundamental group is isomorphic to Γ. We describe EAﬀ (Γ, G, m) in terms of equivalence classes of pairs (, ρ), consisting of epimorphisms of Γ onto G and representations of G in R m−n . As an application we give some estimates of card EAﬀ (Γ, G, m). c Central European Science Journals. All rights reserved. Keywords: Compete ﬂat manifold, aﬃne diﬀeomorphism, holonomy group MSC (2000): Primary: 53C99; Secondary: 53C25

1

Introduction

In this paper we deal with the set EAﬀ (Γ, G, m) of aﬃne equivalence classes of mdimensional complete ﬂat manifolds with a ﬁxed fundamental group Γ and a ﬁxed holonomy group G. We show that these classes correspond to appropriate equivalence classes of pairs (, ρ), consisting of epimorphisms of Γ onto G and representations of G in R m−n . Here n is the rank of the maximal abelian subgroup of Γ. Using this we prove that if G is ﬁnite, then card EAﬀ (Γ, G, m) < ∞ and if H1 (Γ, Q) = 0, then there are uncountably many aﬃne equivalence classes of complete ﬂat manifolds with fundamental groups isomorphic to Γ, whose holonomy groups are inﬁnite. A particular case of our results, when m = 3, can be derived from the classiﬁcation of complete ﬂat 3-manifolds (see e.g. [5, ch. 3, Theorem 3.5.1]). There are big diﬀerences between the theory of complete ﬂat manifolds and the theory ∗

E-mail: [email protected]

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of closed ﬂat manifolds. The Bieberbach theorems (cf. [1, ch. 2, § 5], [5, ch. 3, § 3.2]) show that there are ﬁnitely many aﬃne equivalence classes of closed ﬂat manifolds of the same dimension n, that closed ﬂat manifolds with isomorphic fundamental groups are aﬃnely diﬀeomorphic, and that holonomy groups are homotopical invariants. All these properties are not valid for complete ﬂat manifolds. To state our results we need some deﬁnitions. A complete m-dimensional ﬂat manifold M is the orbit space R m /Γ, where Γ is a discrete group of isometries of R m acting properly discontinuously and freely on R m . The holonomy homomorphism Φ : Γ → O(m) carries γ ∈ Γ onto its linear part Φ(γ) (cf. [5, ch. 3, Lemma 3.4.4]). There is a decomposition × U such that Γ(X) ⊆ X, Rm = X Φ(γ) = ΦX (γ) × ΦU (γ) and γ(x, u) = (γ|X (x), ΦU (γ)(u)) is for γ ∈ Γ, x ∈ V, u ∈ U (see the proof of Theorem 3.3.3 in [5, ch. 3]). The set X = X/Γ a closed ﬂat totally geodesic submanifold of M and the inclusion X → M is a homotopy equivalence. The manifold X and the holonomy group ΦX (Γ) of X are determined by Γ. If Γ is ﬁxed, then M is determined by the homomorphism ΦU : Γ → O(m − n). For a given topological space Y let Y be its universal covering space. The results of the paper will be derived from the following. Theorem 1.1. Let M and M be two complete m-dimensional ﬂat manifolds with isomorphic fundamental groups. Let X ⊆ M and X ⊆ M be totally geodesic subman= X × U and ifolds homotopy equivalent to M and M respectively. Assume that M = X × U . Let ΦU : π1 (X) → O(m − n), Φ : π1 (X ) → O(m − n) be the restricted M U holonomy homomorphisms described above. Then the following conditions are equivalent. (a) M is aﬃnely diﬀeomorphic to M , (b) there is an isomorphism f : π1 (X) → π1 (X ) and a linear isomorphism L : U → U such that for every γ ∈ π1 (X), ΦU (f (γ)) = L ◦ ΦU (γ) ◦ L−1 . For a ﬁxed discrete group G consider the set I(Γ, G, m) of all pairs (, ρ), where : Γ → G is an epimorphism and ρ : G → O(s) is a representation. Denote m − n by s. Two pairs (, ρ) and ( , ρ ) are said to be equivalent if there are f ∈ Aut(Γ) and a linear isomorphism L : R s → R s such that L(ρ) ◦ = ρ ◦ ◦ f, where L(ρ)(g) = L ◦ ρ(g) ◦ L−1 . Let Inv(Γ, G, m) be the set of equivalence classes of the elements of I(Γ, G, m), let Epi(Γ, G) be the set of epimorphisms from Γ to G, and let Rep(G, s) be the set of conjugacy classes of representations of G in R s . Applying Theorem 1.1 we have.

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Theorem 1.2. If Γ be a Bieberbach group, then there is a bijection ν : Inv(Γ, G, m) → EAﬀ (Γ, G, m). Corollary 1.3. If |G| < ∞, then card EAﬀ (Γ, G, m) ≤ card Epi(Γ, G) card Rep(G, s). Recall that a ﬁnitely generated discrete group Γ is a Bieberbach group if Γ is torsion free and a ﬁnite index subgroup of Γ is free abelian. Bieberbach groups are exactly the fundamental groups of complete ﬂat manifolds (cf. [5, ch. 3, Theorems 3.1.3 and 3.2.9]). Let EAﬀ (Γ, m) be the set of aﬃne diﬀeomorphism classes of m-dimensional complete ﬂat manifolds with the same fundamental group Γ and let n be as above. Theorem 1.4. If m ≥ n + 2 and H1 (Γ, Z) is inﬁnite, then EAﬀ (Γ, m) is uncountable. If H1 (Γ, Z) is ﬁnite, then the problem as to when EAﬀ (Γ, m) is uncountable is more has an invariant diﬃcult. We consider only the particular case when the action of Γ on X one-dimensional subspace. This condition is satisﬁed if X is a generalized HantzscheWendt manifold i.e. if X = R n /γ1 , . . . , γn−1 , where n is odd, and γ1 : (x1 , x2 , x3 , ..., xn ) → (x1 + 1/2, −x2 + 1/2, −x3 , ..., −xn ) , ..., γn−1 : (x1 , x2 , .., xn−1 , xn ) → (−x1 , ..., xn−1 + 1/2, −xn + 1/2) (see e.g. [4]). We shall prove the following. Proposition 1.5. a) Let X be a closed n-dimensional ﬂat manifold such that the action has a one-dimensional invariant subspace and let m ≥ n + 2. of the deck group Γ on X Then EAﬀ (Γ, m) is uncountable. b) If Γ is the deck group of the n-dimensional generalized Hantzsche-Wendt manifold and m ≥ n + 2, then H1 (Γ, Q) = {0} and EAﬀ (Γ, m) is uncountable. Throughout this paper the following notation will be used. X, U, ΦX , ΦU , EAﬀ (Γ, G, m), EAﬀ (Γ, m), I(Γ, G, m), Inv(Γ, G, m), Y , m, and n are as above. The letter ta stands for the translation by a and I(V ) for the isometry group of a Riemannian manifold V. If γ ∈ Γ, then γ will be written as tv(γ) ◦ Φ(γ) and γX = γ|X .

2

Aﬃnely equivalent complete ﬂat manifolds

Proof of Theorem 1.1. (a) ⇒ (b). Let F : M → M be an aﬃne diﬀeomorphism and and M . We can →M be the lift of F to the universal covering spaces M let F : M assume that F(0) = 0 so that F is a linear map. Let Φ , ΦX , and ΦU , be the holonomy homomorphisms for M and let Γ, Γ be the deck groups of M and M respectively. Take γ = tv(γ) ◦ Φ(γ) ∈ Γ. If f : Γ → Γ is the isomorphism induced by F , then F ◦ γ = f (γ) ◦ F.

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We have Φ (f (γ) = Φ (F ◦ γ ◦ F−1 ) = Φ (F ◦ tv(γ) ◦ Φ(γ) ◦ F−1 ) = Φ (tF (v(γ)) ◦ F ◦ Φ(γ) ◦ F−1 ) = F ◦ Φ(γ) ◦ F−1 . To see that F (X) = X take the maximal abelian subgroups Γ0 and Γ0 of Γ and joining 0 with δ(0). The projection c of Γ , δ ∈ Γ0 and the segment c:I →X c onto M belongs to the homotopy class corresponding to δ. The segment F ◦ c joins 0 with . The set Γ0 f (δ)(0). Since X → M is a homotopy equivalence F ◦ c is contained in X and so F(X) =X and F (X) = X , as claimed. Let FX : X → X , contains a basis of X → X be the restrictions of F and F to X and X respectively. The equality FX : X −1 Φ (f (γ)) = F ◦ Φ(γ) ◦ F applied to X implies that ΦX (f (γ)) = FX ◦ ΦX (γ) ◦ FX−1 . × U → U is a linear map. Since We have F(x, u) = (FX (x), L(x, u)), where L : X F(x, 0) = FX (x), the map L depends only on u. Hence (f (γ)FX (x), L(ΦU (γ)u)) = F(γx, ΦU (γ)u) = F(γ(x, u)) = f (γ)(FX (x), L(u)) = (f (γ)FX (x), ΦU (f (γ))L(u)). In particular, ΦU (f (γ)) = L ◦ ΦU (γ) ◦ L−1 . (b) ⇒ (a). By Bieberbach theorem (see e.g. [5, ch. 3, Theorem 3.2.2]), there is an aﬃne diﬀeomorphism FX : X → X such that the induced homomorphism (FX )∗ : π1 (X) → →X of FX is a linear map. π1 (X ) coincides with f. One can assume that the lift FX : X →M by the formula Deﬁne F : M F(x, u) = (FX (x), L(u)). Let γ ∈ Γ. By the assumption, L ◦ ΦU (γ) = ΦU (f (γ)) ◦ L and thus F(γ(x, u)) = (FX (γ(x)), (L ◦ ΦU (γ))u) = (f (γ)FX (x), ΦU (f (γ))L(u)) = f (γ)(FX (x), L(u)) = f (γ)F(x, u). The last equality implies that F determines an aﬃne diﬀeomorphism F : M → M . This ﬁnishes the proof of Theorem 1.1.

3

Algebraic invariants associated with aﬃne equivalences

Proof of Theorem 1.2. Let (, ρ) ∈ I(Γ, m, G), let Π : G → G/ker ρ be the projection, and let φU = ρ ◦ . Fix an n-dimensional closed ﬂat manifold X with the deck group

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given by the action of γ ∈ Γ on X. isomorphic to Γ. Denote by γX the diﬀeomorphism of X Let M (, ρ) = (R n ×R s )/Γ, where γ(x, u) = (γX (x), φU (γ)u). The action of Γ on M (, ρ) is properly discontinuous and free, because the action of Γ on X is properly discontinuous and free. A ﬂat Riemannian metric on X and a ρ(G)-invariant scalar product in R s determine a ﬂat Riemannian metric on M so that M (, ρ) has the structure of a complete ﬂat manifold. Applying Theorem 1.1, it is easy to see that the map (, ρ) → M (, ρ) determines a well deﬁned injection ν : Inv(Γ, G, m) → EAﬀ (Γ, G, m). To see that ν is a surjection take an m-dimensional complete ﬂat manifold M with the deck group ΓM isomorphic to Γ and the holonomy group Φ(Γ) isomorphic to G. Consider the commutative diagram Φ

ΓM −→  f Π

im Φ  f 1

id

ϕ

U −→ im Φ −→ O(s)   f  1 id

I

Γ −→ Γ/f (ker Φ) −→ G

ϕU ◦f1−1

−→ O(s)

Here Π is the projection, f and I are ﬁxed isomorphisms, f1 carries Φ(γ) onto Π(f (γ)), ϕU carries Φ(γ) onto ΦU (γ), and f1 = I ◦ f1 . Let = I ◦ Π and ρ = ϕu ◦ f1−1 . Clearly ρ ◦ ◦ f = ΦU . By Theorem 1.1, M (, ρ) is aﬃnely diﬀeomorphic to M. This completes the proof of Theorem 1.2. Proof of Corollary 1.3. Take (, ρ) ∈ I(Γ, m, G) and (, ρ ) ∈ I(Γ, m, G) such that ρ is conjugate to ρ . The arguments given in the proof of Theorem 1.2 show that M (, ρ) is aﬃnely diﬀeomorphic to M (, ρ ) so that card EAﬀ (Γ, G, m) ≤ card Epi(Γ, G) card Rep(G, s).

4

Inﬁnite families of elements of EAﬀ (Γ, m)

Proof of Theorem 1.4. Since H1 (Γ, Z) = Γ/[Γ, Γ] is inﬁnite it can be written as a ⊕ B, where a is an inﬁnite cyclic group generated by some a ∈ H1 (Γ, Z) so that every element of a can be written as la + b for some l ∈ Z, b ∈ B. Let Πab : Γ → H1 (Γ, Z) be the projection. By the assumption that m ≥ n + 2, the group SO(s) contains a subgroup isomorphic to S 1 . For every t ∈ R take ht : H1 (Γ, Z) → S 1 deﬁned by the formula ht (la + b) = e2πilt . Let ht = ht ◦ Πab . Fix a closed ﬂat manifold X = R n /ΓX with the deck group ΓX = {γX : γ ∈ Γ} isomorphic to Γ. Consider the action of Γ on R n × R s , given by the formula γt (x, u) = (γX (x), ht (γ)u), and the arising orbit space Mt . Clearly Mt = M (t , ρt ), where t : γ → ΦX (γ) × ht (γ) and ρt : ΦX (γ) × ht (γ) → ht (γ).

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If Mt is aﬃnely diﬀeomorphic to Mt , then there are f ∈ Aut(Γ) and a linear isomorphism L : R s → R s such that ht (f (γ)) = L ◦ ht (γ) ◦ L−1 for γ ∈ Γ. Since ht (f (γ)) and ht (γ) have the same eigenvalue, ht (f (γ)) = ht (γ) so that the aﬃne equivalence classes of Mt correspond to the orbits [ht ] of the action of Aut(Γ) on {ht : t ∈ R} given by (f, ht ) → ht ◦ f −1 . Since every orbit is ﬁnite or countable, the number of the orbits is uncountable. This completes the proof of Theorem 1.4. Remark 4.1. a) card EAﬀ (Γ, n + 1) ≤ card Hom (Γ, O(1)) < ∞. b) It is easy to see that Mt and Mt are not aﬃnely diﬀeomorphic if |t − t | is suﬃciently small. Remark 4.2. a) The manifolds Mt , considered in the proof of Theorem 1.4, are diﬀeo × R = R n × R s × R, the action morphic. To see this consider M × R (x, u, t) → (γX (x), ht (γ)u, t) ∈ M × R, γR : M × R)/ΓR . The projection f : M ×R → R ΓR = {γR : γ ∈ Γ}, and the orbit space (M × R)/ΓR . As this function has not critical points, determines a Morse function f on (M all Mt = f −1 (t) are diﬀeomorphic. b) The simplest example of an uncountable family of the elements of EAﬀ (Γ, m) arises in the classiﬁcation of 3-dimensional complete ﬂat manifolds (cf. [5, ch. 3, Theorem 3.5.1]). It can be described as follows. Let α ∈ [0, π), let Oα : R 2 → R 2 be the rotation by α, and let γα : R 3 (x, y, z) → (x + 1, Oα (y, z)) ∈ R 3 . The group Γα = γα acts freely and properly discontinuously on R 3 so that Mα = R 3 /Γα is a complete ﬂat manifold. It is not diﬃcult to see that Mα is not aﬃnely diﬀeomorphic to Mβ for α = β (cf. Theorem 1.1 see also [5, ch. 3, Theorem 3.5.1]). c) There are more nontrivial examples of complete ﬂat manifolds, that are diﬀeomorphic, but not aﬃnely diﬀeomorphic. For example there are two diﬀeomorphism classes of complete noncompact m-dimensional ﬂat manifolds, whose fundamental groups are isomorphic to the same Bieberbach group Γ and whose holonomy groups are cyclic (see [3]). Proof of Proposition 1.5. a) It suﬃces to prove that EAﬀ (Γ, n + 2) is uncountable. Identify the one-dimensional invariant subspace with R. Let F : Γ → I(R) be the homomorphism carrying γX ∈ Γ onto γR = γX |R and let ∆ = F (Γ). For every h ∈ ∆, h2 is a translation so that ∆ is an inﬁnite cyclic group or ∆ is generated by a translation τ and L = −idR . The conclusion of Proposition 1.5 in the ﬁrst case follows from Theorem 1.4. In the second case note that ∆ is the group with two generators L and τ satisfying the relations Lτ = τ −1 L and L2 = 1. The same description has the group fβ , g generated by g : C z → z ∈ C and fβ : C z → eiβ z ∈ C, where β is any irrational multiple of 2π. Clearly fβ , g ⊂ O(2) so that the composition of F with the identiﬁcation of ∆ with fβ , g deﬁnes a representation ρβ : Γ → O(2).

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For every β ∈ R − Q and γX ∈ Γ consider γβ : R n+2 (x, u) → (γX (x), ρβ (γ)u) ∈ R n+2 . Let Γβ = {γβ : γX ∈ Γ} and let Mβ = R n /Γβ . Then Mβ is a complete ﬂat manifold homotopy equivalent to X. Let ΓR = {γR : γX ∈ Γ} and let fR : γR → f (γ)R . If Mβ is aﬃnely diﬀeomorphic to Mβ , then there are f ∈ Aut(Γ) and a linear isomorphism L : R 2 → R 2 such that ρβ (f (γX )) = L ◦ ρβ (γX ) ◦ L−1 for γX ∈ Γ. In particular, ρβ (fR (τ )) = L ◦ ρβ (τ ) ◦ L−1 . After the complexiﬁcation, the maps ρβ (fR (τ )), ρβ (τ ), and L ◦ ρβ (τ ) ◦ L−1 have the same eigenvectors so that L ◦ ρβ (τ ) ◦ L−1 = ρβ (τ ). A similar argument to that given in the proof of Theorem 1.4 shows that the set EAﬀ (Γ, n + 2) is uncountable. b) The part b) follows from a) and from the results of [4]. It is easy to show that for any Bieberbach group Γ there is an inﬁnite sequence of complete ﬂat manifolds whose holonomy groups form an increasing sequence of ﬁnite groups. This is a consequence of the following lemma. Lemma 4.3. If Γ is a Bieberbach group, then there is an inﬁnite sequence of representations ρk : Γ → O(sk ) such that |ρk (Γ)| = k n |ρ1 (Γ)|. Proof. Let A ∼ = Zn be the maximal abelian subgroup of Γ. It is known that A is the set of all translations in Γ (see e.g. [5, Theorem 3.3.2]). Take Ak = {ak : a ∈ A}, a ∈ A, and γ ∈ Γ. Since A is a normal subgroup of Γ, γaγ −1 = b for some b ∈ A. Using the fact that γ is the composition of a linear map with a translation we see that γak γ −1 = bk and consequently Ak is a normal subgroup of Γ. Take Gk = Γ/Ak , a monomorphism ρk : Gk → O(sk ) ([2, ch. 3, Theorem 4.1]), and the composition ρk of the projection of Γ onto Gk with ρk . Then ρk (Γ) ∼ = Gk and |Gk | = k n |G/A|.

Acknowledgements I am grateful to Karel Dekimpe, who pointed out an error in the manuscript and helped to prove Proposition 1.5. I would like to thank Andrzej Szczepa´ nski for his useful remarks.

References [1] L. Charlap: Bieberbach groups and ﬂat manifolds, Springer-Verlag, New York, 1986. [2] T. tom Dieck: Transformation groups, de Gruyter, Berlin 1987. [3] M. Sadowski: Topological structure of complete ﬂat manifolds with cyclic holonomy groups, to appear. [4] A. Szczepa´ nski: Aspherical manifolds with the Q-homology of a sphere, Mathematika, Vol. 30, (1983), pp. 291–294. [5] J. Wolf: Spaces of constant curvature, McGraw-Hill, 1967.