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BecmHUK CaMry — EcmecmeeHHOHaynHaa cepua. 2007. №7(57)
ALGEBRAIC TORI — THIRTY YEARS AFTER
©2007 B.Kunyavskii1
To my teacher Valentin Evgenyevich Voskresenskii, with gratitude and admiration
This paper is an extended version of my talk given at the International Conference "Algebra and Number Theory" dedicated to the 80th anniversary of Prof. V.E. Voskresenskii, which was held at the Samara State University in May 2007. The goal is to give an overview of results by V.E. Voskresenskii on arithmetic and birational properties of algebraic tori which culminated in his monograph [82] published 30 years ago. We put these results and ideas into somehow broader context and also to give a brief digest of the relevant activity related to the period after the English version of the monograph [83] has been published.
1. Rationality and nonrationality problems
A classical problem, going back to Pythagorean triples, of describing the set of solutions of a given system of polynomial equations by rational functions in a certain number of parameters (rationality problem) has been an attraction for many generations. Although a lot of various techniques have been used, one can notice that after all, to establish rationality, one usually has to exhibit some explicit parameterization such as that obtained by stereographic projection in the Pythagoras problem. The situation is drastically different if one wants to establish non-existence of such a parameterization (nonrationality problem): here one usually has to use some known (or even invent some new) birational invariant allowing one to detect nonrationality by comparing its value for the object under consideration with some "standard" one known to be zero; if the computation gives a nonzero value, we are done. Evidently, to be useful, such an invariant must be (relatively) easily computable. Note that the above mentioned subdivision to rationality and nonrationality problems is far from being absolute: given a class of objects, an ultimate goal could be to introduce some computable birational invariant giving necessary and sufficient conditions for rationality. Such a task may be not so hopeless, and some examples will be given below.
1 Kunyavskii Boris (kunyav@macs.biu.ac.il), Dept. of Mathematics, Bar-Ilan University, 52900 Ramat Gan, Israel.
In this section, we discuss several rationality and nonrationality problems related to algebraic tori. Since this is the main object of our consideration, for the reader's convenience we shall recall the definition.
Definition 1.1. Let k be a field. An algebraic k-torus T is an algebraic k-group such that over a (fixed) separable closure k of k it becomes isomorphic to a direct product of d copies of the multiplicative group:
T x kk = Gd k,
" m,k
wherein d is the dimension of T.
We shall repeatedly use the duality of categories of algebraic k-tori and finite-dimensional torsion-free Z-modules viewed together with the action of the Galois group g := Gal(k/k) which is given by associating to a torus T its g-module of characters T := Hom(T X k, Gm,k). Together with the fact that T splits over a finite extension of k (we shall denote by L the smallest of such extensions and call it the minimal splitting field of T), this allows us to reduce many problems to considering (conjugacy classes of) finite subgroups of GL(d, Z) corresponding to T.
1.1. Tori of small dimension
There are only two subgroups in GL(1,Z): {(1)} and {(1), (-1)}, both corresponding to k-rational tori: Gm,k and RL/kGm,L/Gm,k, respectively (here L is a separable quadratic extension of k and R-L/k stands for the Weil functor of restriction of scalars).
For d = 2 the situation was unclear until the breakthrough due to Voskre-senskii [77] who proved
Theorem 1.2. All two-dimensional tori are k-rational.
For d = 3 there exist nonrational tori, see [48] for birational classification.
For d = 4 there is no classification, some birational invariants were computed in [62].
Remark 1.3. The original proof of rationality of the two-dimensional tori in [77] is based on the classification of finite subgroups in GL(2, Z) and case-by-case analysis. In the monographs [82] and [83] a simplified proof is given, though using the classification of maximal finite subgroups in GL(2, Z). Merkurjev posed a question about the existence of a classification-free proof. (Note that one of his recent results on 3-dimensional tori [58] (see Section 3. below) was obtained without referring to the birational classification given in [48].) During discussions in Samara, Iskovskikh and Prokhorov showed me a proof only relying on an "easy" part of the classification theorem for rational surfaces. Remark 1.4. One should also note a recent application of Theorem 1.2 to the problem of classification of elements of prime order in the Cremona group Cr(2, Q) of birational transformations of plane (see [73, 6.9]). The problem was recently solved by Dolgachev and Iskovskikh [33]. See the above cited papers for more details.
1.2. Invariants arising from resolutions
Starting from the pioneering works of Swan and Voskresenskii on Noether's problem (see Section 1.4.), it became clear that certain resolutions of the Galois module T play an important role in understanding birational properties of T. These ideas were further developed by Lenstra, Endo and Miyata, Col-liot-Thelene and Sansuc (see [82, 83] for an account of that period); they were pursued through several decades, and some far-reaching generalizations were obtained in more recent works (see the remarks at the end of this and the next sections).
For further explanations we need to recall some definitions. Further on "module" means a finitely generated Z-free g-module.
Definition 1.5. We say that M is a permutation module if it has a Z-basis permuted by g. We say that modules M\ and M2 are similar if there exist permutation modules S1 and S2 such that M\ ®S1 = M2 ®S2. We denote the similarity class of M by [M]. We say that M is a coflasque module if H 1(h, M) = 0 for all closed, subgroups h of g. We say that M is a flaisque module if its dual module M° := Hom(M,Z) is coflasque.
The following fact was first established in the case when k is a field of characteristic zero by Voskresenskii [78] by a geometric construction (see below) and then in [24, 34] in a purely algebraic way (the uniqueness of [F] was established independently by all above authors). Theorem 1.6. Any module M admits a resolution of the form
0 ^ M ^ S ^ F ^ 0, (1.1)
where S is a permutation module and F is a flasque module. The similarity class [F] is determined uniquely.
We denote [F] by p(M). We call (1.1) the flasque resolution of M. If T is a k-torus with character module M, the sequence of tori dual to (1.1) is called the flasque resolution of T.
Seeming strange at the first glance, the notion of flasque module (and the corresponding flasque torus) turned out to be very useful. Its meaning is clear from the following theorem due to Voskresenskii [82, 4.60]: Theorem 1.7. If tori Ti and T2 are birationally equivalent, then p(T\) = p(j2). Conversely, if p(T\) = p(T2), then Ti and T2 are stably equivalent, i.e. Ti X G^ is birationally equivalent to T2 X gJ k for some integers di, d2.
Theorem 1.7 provides a birational invariant of a torus which can be computed in a purely algebraic way. Moreover, it yields other invariants of co-homological nature which are even easier to compute. The most important among them is H 1(g ,F). One should note that it is well-defined in light of Shapiro's lemma (because H 1(g , S) = 0 for any permutation module S) and, in view of the inflation-restriction sequence, can be computed at a finite level: if L/k is the minimal splitting field of T and r = Gal(L/k), we have H 1(g, F) =
= H 1(r, F). The finite abelian group H1(r, F) is extremely important for birational geometry and arithmetic, see below.
Remark 1.8. Recently Colliot-Thelene [18] discovered a beautiful generalization of the flasque resolution in a much more general context, namely, for an arbitrary connected linear algebraic group G.
1.3. Geometric interpretation of the flasque resolution
As mentioned above, the flasque resolution (1.1) was originally constructed in a geometric way. Namely, assuming k is of characteristic zero, one can use Hironaka's resolution of singularities to embed a given k-torus T into a smooth complete k-variety V as an open subset and consider the exact sequence of g-modules
0 -» T -» PicF^ 0, (1.2)
where V = Vx^k, Pic V is the Picard module, and S is a permutation module (it is generated by the components of the divisors of V whose support is outside T). With another choice of an embedding T ^ V' we are led to an isomorphism Pic F©5"i = Pic V ©5"2, so the similarity class [Pic F] is well defined. Voskresenskii established the following property of the Picard module (see [82, 4.48]) which is of utmost importance for the whole theory: Theorem 1.9. The module Pic V is flasque.
As mentioned above, this result gave rise to a purely algebraic way of constructing the flasque resolutions, as well as other types of resolutions; flasque tori and torsors under such tori were objects of further thorough investigation [25, 26].
Remark 1.10. The geometric method of constructing flasque resolutions described above can be extended to arbitrary characteristic. This can be done in the most natural way after the gap in Brylinski's proof [12] of the existence of a smooth complete model of any torus have been filled in [20]. Remark 1.11. Recently Theorem 1.9 was extended to the Picard module of a smooth compactification of an arbitrary connected linear algebraic group [10] and, even more generally, of a homogeneous space of such a group with connected geometric stablizer [22]. In each case, there is a reduction to the case of tori (although that in [22] is involved enough).
1.4. Noether's problem
One of the most striking applications of the birational invariant described above is a construction of counter-examples to a problem of E. Noether on rationality of the field of rational functions invariant under a finite group G of permutations. Such an example first appeared in a paper by Swan [76] where tori were not mentioned but a resolution of type (1.1) played a crucial role; at the same time Voskresenskii [78] considered the same resolution to prove that a certain torus is nonrational. In a later paper [79] he formulated in an
explicit way that the field of invariants under consideration is isomorphic to the function field of an algebraic torus. This discovery yielded a series of subsequent papers (Endo-Miyata, Lenstra, and Voskresenskii himself) which led to almost complete understanding of the case where the finite group G acting on the function field is abelian; see [82, Ch. VII] for a detailed account. Moreover, the idea of realizing some field of invariants as the function field of a certain torus proved useful in many other problems of the theory of invariants, see works of Beneish, Hajja, Kang, Lemire, Lorenz, Saltman, and others; an extensive bibliography can be found in the monograph [56], see [43] and references therein for some more recent development; note also an alternative approach to Noether's problem based on cohomological invariants (Serre, in [35]).
Note a significant difference in the proofs of nonrationality for the cases G = Z47 (the smallest counter-example for a cyclic group of prime order) and G = Zg (the smallest counter-example for an arbitrary cyclic group). If G is a cyclic group of order q = pn, and k is a field of characteristic different from p, the corresponding k-torus splits over the cyclotomic extension k(Zq). If p > 2, the extension k(Zq)/k is cyclic. According to [34], if a k-torus T splits over an extension L such that all Sylow subgroups of Gal(L/k) are cyclic, then the r-module F in the flasque resolution for T is a direct summand of a permutation module. Thus to prove that it is not a permutation module, one has to use subtle arguments. If p = 2 and n ^ 3, the Galois group r of k(Zq)/k may be noncyclic and contain a subgroup r' such that H1(r', F) + + 0 which guarantees that F is not a permutation module and hence T is not rational. This important observation was made in [80]. Another remark should be done here: for the tori appearing in Noether's problem over a field k with cyclic G such that char(k) is prime to the exponent of G, triviality of the similarity class [F] is necessary and sufficient condition for rationality of T (and hence of the corresponding field of invariants). This is an important instance of the following phenomenon: in a certain class of tori any stably rational torus is rational. The question whether this principle holds in general is known as Zariski's problem for tori and is left out of the scope of the present survey.
The group Hl(T, PicJQ, where X is a smooth compactification of a £-torus T7, admits another interpretation: it is isomorphic to BrX/Br k, where BrX stands for the Brauer-Grothendieck group of X. This birational invariant, later named the unramified Brauer group, played an important role in various problems, as well as its generalization for higher unramified cohomology (see [15, 69] for details). Let us only note that the main idea here is to avoid explicit construction of a smooth compactification of an affine variety V under consideration, trying to express BrX/Br k in terms of V itself. In the toric case this corresponds to the formula [26]
Br X/Br k = ker[H2(r, T) ^ ]"[ H2(C, T)], (1.3)
C
where the product is taken over all cyclic subgroups of r. Formulas of similar
flavour were obtained for the cases where V = G, an arbitrary connected linear algebraic group [9, 21], and V = G/H, a homogeneous space of a simply connected group G with connected stabilizer H [22]; the latter formula was used in [23] for proving nonrationality of the field extensions of the form k(V)/k(V)G, where k is an algebraically closed field of characteristic zero, G is a simple k-group of any type except for An, Cn, G2, and V is either the representation of G on itself by conjugation or the adjoint representation on its Lie algebra.
It is also interesting to note that the same invariant, the unramified Brauer group of the quotient space V/G, where G is a finite group and V its faithful complex representation, was used by Saltman [68] to produce the first counter-example to Noether's problem over C. In the same spirit as in formula (1.3) above, this invariant can be expressed solely in terms of G: it equals
Bo(G) := ker[H2(G, Q/Z) ^ ]"[H2(A, Q/Z)], (1.4)
A
where the product is taken over all abelian subgroups of G (and, in fact, may be taken over all bicyclic subgroups of G) [5]. This explicit formula yielded many new counter-examples (all arising for nilpotent groups G, particularly from ^-groups of nilpotency class 2). The reader interested in historical perspective and geometric context is referred to [7, 27, 74]. We only mention here some recent work [8, 49] showing that such counter-examples cannot occur if G is a simple group; this confirms a conjecture stated in [6].
1.5. Generic tori
Having at our disposal examples of tori with "good" and "bad" birational properties, it is natural to ask what type of behaviour is typical. Questions of such "nonbinary" type, which do not admit an answer of the form "yes-no", have been considered by many mathematicians, from Poincare to Arnold, as the most interesting ones. In the toric context, the starting point was the famous Chevalley-Grothendieck theorem stating that the variety of maximal tori in a connected linear algebraic group G is rational. If G is defined over an algebraically closed field, its underlying variety is rational. However, if k is not algebraically closed, k-rationality (or nonrationality) of G is a hard problem. The Chevalley-Grothendieck theorem gives a motivation for studying generic tori in G: if this torus is rational, it gives the k-rationality of G. The notion of generic torus can be expressed in Galois-theoretic terms: these are tori whose minimal splitting field has "maximal possible" Galois group r (i.e. r lies between W(R) and Aut(R) where R stands for the root system of G). This result, going back to E. Cartan, was proved in [81]. In this way, Voskresenskii and Klyachko [84] proved the rationality of all adjoint groups of type A2n. The rationality was earlier known for the adjoint groups of type Bn, the simply connected groups of type Cn, the inner forms of the adjoint groups of type An, and (after Theorem 1.2) for all groups of rank at most two. It turned out that for the adjoint and simply connected groups of all remaining types the
generic torus is not even stably rational [29]; in most cases this was proved by computing the birational invariant Hl(T',F) for certain subgroups Г' с Г. In the case of inner forms of simply connected groups of type An, corresponding to generic norm one tori, this confirms a conjecture by Le Bruyn [53] (independently proved later in [54]). The above theorem was extended to groups which are neither simply connected nor adjoint and heavily used in the classification of linear algebraic groups admitting a rational parameterization of Cayley type
[55].
The above mentioned results may give an impression that except for certain types of groups the behaviour of generic tori is "bad" from birational point of view. However, there is also a positive result: if T is a generic torus in G, then H 1(Г, F) = 0, This was first proved in [85] for generic tori in the classical simply connected (type An) and adjoint groups, and in [46] in the general case. (An independent proof for the simply connected case was communicated to the author by M. Borovoi.) This result has several number-theoretic applications, see Section 2.
Remark 1.12. Yet another approach to the notion of generic torus was developed in [41] where the author, with an eye towards arithmetic applications, considered maximal tori in semisimple simply connected groups arising as the centralizers of regular semisimple stable conjugacy classes.
2. Relationship to arithmetic
2.1. Global fields: Hasse principle and weak approximation
According to a general principle formulated in [57], the influence of (birational) geometry of a variety on its arithmetic (diophantine) properties may often be revealed via some algebraic (Galois-cohomological) invariants. In the toric case such a relationship was obtained by Voskresenskil and formulated as the exact sequence (see [82, 6.38])
0 -» A(T) -» Hl(k,?icV)~^ Sha(r) -» 0, (2.1)
where к is a number field, T is a k-torus, V is a smooth compactification of T, A(T) is the defect of weak approximation, Sha(T) is the Shafarevich - Tate group, and ~ stands for the Pontrjagin duality of abelian groups. The cohomo-logical invariant in the middle, being a purely algebraic object, governs arithmetic properties of T.
On specializing T to be the norm one torus corresponding to a finite field extension K/к, we get a convenient algebraic condition sufficient for the Hasse norm principle to hold for K/к. In particular, together with results described in Section 1.5., this shows that the Hasse norm principle holds "generically", i.e. for any field extension K/к of degree n such that the Galois group of the normal closure is the symmetric group Sn [85]. (Another proof was independently found for n > 7 by Yu. A. Drakokhrust.) Another application was found
in [51]: combining the above mentioned theorem of Klyachko with the Cheval-ley-Grothendieck theorem and Hilbert's irreducibility theorem, one can produce a uniform proof of weak approximation property for all simply connected, adjoint and absolutely almost simple groups. (Another uniform proof was found by Harari [42] as a consequence of a stronger result on the uniqueness of the Brauer-Manin obstruction, see the next paragraph.) Yet another interesting application of sequence (2.1) refers to counting points of bounded height on smooth compactifications of tori [2, 3]: the constant appearing in the asymptotic formula of Peyre [60] must be corrected by a factor equal to the order of Hl(k, Pic F) and arising on the proof as the product of the orders of A(T) and Sha(T) (I thank J.-L. Colliot-Thelene for this remark).
The sequence (2.1) was extended by Sansuc [70] to the case of arbitrary linear algebraic groups. On identifying the invariant in the middle with Br V/Br k, as in Section 1.4., one can put this result into more general context of the so-called Brauer-Manin obstruction to the Hasse principle and weak approximation (which is thus the only one for principal homogeneous spaces of linear algebraic groups). This research, started in [57], gave many impressive results. It is beyond the scope of the present survey.
Remark 2.1. Other types of approximation properties for tori have been considered in [28] (weaker than weak approximation), [63, 64] (strong approximation with respect to certain infinite sets of primes with infinite complements — so-called generalized arithmetic progressions). In the latter paper generic tori described above also played an important role. They were also used in [14] in a quite different arithmetic context.
2.2. Arithmetic of tori over more general fields
Approximation properties and local-global principles were studied for some function fields. In [1] the exact sequence (2.1) has been extended to the case where the ground field k is pseudoglobal, i.e. k is a function field in one variable whose field of constants K is pseudofinite (this means that K has exactly one extension of degree n for every n and every absolutely irreducible affine K-variety has a K-rational point). In [16] weak approximation and the Hasse principle were established for any torus defined over R(X where X is an irreducible real curve. This allows one to establish these properties for arbitrary groups over such fields, and, more generally, over the fields of virtual cohomo-logical dimension 1 [71]. The same properties for tori defined over some geometric fields of dimension 2 (such as a function field in two variables over an algebraically closed field of characteristic zero, or the fraction field of a two-dimensional excellent, Henselian, local domain with algebraically closed residue field, or the field of Laurent series in one variable over a field of characteristic zero and cohomological dimension one) were considered in [19]. Here one can note an interesting phenomenon: there are counter-examples to weak approximation but no counter-example to the Hasse principle is known. One can
ask whether there exists some Galois-cohomological invariant of tori defined over more general fields whose vanishing would guarantee weak approximation property for the torus under consideration. Apart from the geometric fields considered in [19], another interesting case could be k = Qp(X), where X is an irreducible Qp-curve; here some useful cohomological machinery has been developed in [72].
2.3. Integral models and class numbers of tori
The theory of integral models of tori, started by Raynaud who constructed an analogue of the Neron smooth model [11], has been extensively studied during the past years, and some interesting applications were found using both Neron-Raynaud models and Voskresenskii's "standard" models. The interested reader is referred to the bibliography in [86]. Some more recent works include standard integral models of toric varieties [50] and formal models for some classes of tori [30].
Main results on class numbers of algebraic tori are summarized in [83]. One can only add that the toric analogue of Dirichlet's class number formula established in [75] suggests that a toric analogue of the Brauer-Siegel theorem may also exist. A conjectural formula of the Brauer-Siegel type for constant tori defined over a global function field can be found in a recent paper [52].
3. R-equivalence and zero-cycles
R-equivalence on the set of rational points of an algebraic variety introduced in [57] turned out to be an extremely powerful birational invariant. Its study in the context of algebraic groups, initiated in [24], yielded many striking achievements. We shall only recall here that the first example of a simply connected group whose underlying variety is not k-rational is a consequence of an isomorphism, established by Voskresenskii, between G(k)/R, where G = = SL(1, D), the group of norm 1 elements in a division algebra over k, with the reduced Whitehead group S^(D); as the latter group may be nonzero because of a theorem by Platonov [61], this gives the needed nonrationality of G. This breakthrough gave rise to dozens of papers on the topic certainly deserving a separate survey. For the lack of such, the interested reader is referred to [36. Sections 24-33; 37; 83. Ch. 6].
As to R-equivalence on tori, the most intriguing question concerns relationship between T(k)/R and the group Ao(X) of classes of 0-cycles of degree 0 on a smooth compactification X of T. In a recent paper [58] Merkurjev proved that these two abelian groups are isomorphic if T is of dimension 3 (for tori of dimension at most 2 both groups are zero because of their birational invariance and Theorem 1.2) For such tori he also obtained a beautiful formula expressing T(k)/R in "intrinsic" terms, which does not require constructing X as above nor a flasque resolution of T as in [24]: T(k)/R = H 1(k, T°)/R, where
T° denotes the dual torus (i.e. T° = Hom(T, Z)). (In the general case, it is not even known whether the map X(k)/R ^ A0(X) is surjective, see [17, §4] for more details.) As a consequence, Merkurjev obtained an explicit formula for the Chow group CH0(T) of classes of 0-cycles on a torus T of dimension at most 3: CH0(T) = T(k)/R ® ZiT where if denotes the greatest common divisor of the degrees of all field extensions L/k such that the torus TL is isotropic. The proofs, among other things, use earlier results [45], [59] on the ^-theory of toric varieties. As mentioned above, they do not rely on the classification of 3-dimensional tori.
To conclude this section, one can also add the same references [1, 19] as in Section 2.2. for R-equivalence on tori over more general fields.
4. Applications in information theory
4.1. Primality testing
One should mention here several recent papers [40, 44] trying to interpret in toric terms some known methods for checking whether a given integer n is a prime. In fact, this approach goes back to a much older paper [13] where the authors noticed symmetries in the sequences of Lucas type used in such tests (though algebraic tori do not explicitly show up in [13]).
4.2. Public-key cryptography
A new cryptosystem based on the discrete logarithm problem in the group of rational points of an algebraic torus T defined over a finite field was recently invented by Rubin and Silverberg [65, 66, 67]. Since this cryptosystem possesses a compression property, i.e. allows one to use less memory at the same security level, it drew serious attention of applied cryptographers and yielded a series of papers devoted to implementation issues [31, 32, 39, 47] (in the latter paper another interesting approach is suggested based on representing a given torus as a quotient of the generalized jacobian of a singular hyperelliptic curve). In [38] the authors propose to use a similar idea of compression for using tori in an even more recent cryptographic protocol (so-called pairing-based cryptography). It is interesting to note that the efficiency (compression factor) of the above mentioned cryptosystems heavily depends on rationality of tori under consideration (more precisely, on an explicit rational parameterization of the underlying variety). As the tori used by Rubin and Silverberg are known to be stably rational, the seemingly abstract question on rationality of a given stably rational torus is moving to the area of applied mathematics. The first challenging problem here is to obtain an explicit rational parameterization of the 8-dimensional torus T30, defined over a finite field k and splitting over its cyclic extension L of degree 30, whose character module T30 is isomorphic to Z[^30], where Z[^30] stands for a primitive 30th root of unity. (Here is an al-
ternative description of T30: it is a maximal torus in E% such that Gal(L/k) acts on T30 as the Coxeter element of W(Eg); this can be checked by a direct computation or using [4].)
This is a particular case of a problem posed by Voskresenskii [82, Problem 5.12] 30 years ago. Let us hope that we will not have to wait another 30 years for answering this question on a degree 30 extension.
Acknowledgements. The author was supported in part by the Russia-Israel Scientific Research Cooperation Project 3-3578 and by the Minerva Foundation through the Emmy Noether Research Institute of Mathematics. This paper was written during the visit to the MPIM (Bonn) in August-September 2007. The support of these institutions is highly appreciated. I thank J.-L. Colliot-Thelene for many helpful remarks.
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Paper received 17/1X/2007. Paper accepted 17/1X/2007.
АЛГЕБРАИЧЕСКИЕ ТОРЫ — ТРИДЦАТЬ ЛЕТ СПУСТЯ
© 2007 Б.Э.Кунявский2
Цель статьи — дать обзор результатов В.Е. Воскресенского по арифметическим и бирациональным свойствам алгебраических торов, получивших наибольшее развитие в его монографии [82], опубликованной 30 лет назад. Я постараюсь представить эти результаты и идеи в некотором общем контексте, а также дать краткий обзор продвижений в этой области после выхода английской версии монографии
[83].
Поступила в редакцию 17/1Х/2007; в окончательном варианте — 17/1Х/2007.
2Кунявский Борис Эммануилович (kunyav@macs.biu.ac.il), кафедра математики университета Бар-Илан, 52900, Рамат Ган, Израиль.
