LINEAR REPRESENTATION OF QUASISPLIT TORI AND PROBLEM OF RATIONALITY

V. E. Voskresenskii ^{1}

Abstract

We study faithful representations of quasisplit algebraic tori and prove the oldstanding conjecture on rationality of a stably rational torus.

It is an important problem in the theory of linear algebraic groups defined over a nonclosed field to classify the varieties of these groups up to birational equivalence. This problem is still far from its solution. At present time we can only classify algebraic tori up to stable equivalence, and have a few partial results on their birational classification. In this paper we prove the old-standing conjecture on the rationality of stably rational tori. Consequently we get a very effective criterion allowing us to obtain definitive results in some of the classification problems, and to simplify some of the proofs.

Let us recall some facts and definitions. Let X be an irreducible algebraic variety defined over a field k. The variety X is called rational if the function field k(X) is purely transcendental over k, and it is called unirational if its function field is contained in a purely transcendental extension of k. The affine space A_{k}^{n} is rational over k, hence we will denote the purely transcendental extension of k of dimension n by k(A^{n}). The unirational varieties X naturally arise in the problems of the parametrisation of objects by rational functions of several variables. If dim X = 1, then X is rational over k (Lüroth’s theorem); if dim X = 2 and k is algebraically closed, then the variety X is also rational (Castelnuovo’s theorem). In any other case, an example of a non-rational unirational variety has been constructed. Since the variety of a connected linear group defined over a perfect field is known to be unirational over the field of definition [1], the classification problem of unirational varieties arises quite naturally in the theory of linear algebraic groups. Moreover, many of the important parameters of linear groups turn out to be of birational nature [2, 3]. The notions of stable rationality and stable equivalence appear to be rather useful in the classification problems of the varieties of connected linear groups. The k–varieties X_{1} and X_{2} are said to be stably equivalent if the products X_{1} ×_{k} A^{m} and X_{2} ×_{k} A^{p} are birationally isomorphic over k for some m and p. If there is a birational k–isomorphism X ×_{k} A^{m} ≅ A^{p}, then the variety X is said to be stably rational. Clearly, a stably rational variety is unirational. In the paper [4] in 1984, examples of stably rational varieties, which are not rational, have been constructed. If the field k is not algebraically closed, then such ”monsters” exist in any dimension bigger than 1; over the complex number field, there are such examples in any dimension bigger than 2. In the same paper [4], one can find the detailed discussion of this question and the exact bibliographical references.

Let us now turn to the variety of a connected linear algebraic k-group G. If k is algebraically closed, then G is rational, that is the field k(G) is purely transcendental. If k is perfect, then the variety G is unirational (Chevalley and Grothendieck, [1]). In the case of algebraic tori, we have the following more detailed classification. Let D_{k}(n) be a decomposable k–torus, that is a connected diagonal group defined over k; thus D_{k}(n) = Spec(k[M]), where M is a commutative multiplicative torsion free group of rank n, and k[M] stands for its group ring. The algebra k[M] is isomorphic to the Laurent ring k[M] = k[χ_{1}, …, χ_{n}, χ_{1}^{−1}, …, χ_{n}^{−1}], where χ_{i} are the generators of the group M. Let R be a commutative k–algebra; the set D_{k}(n)(R) of the R-points of the scheme D_{k}(n) can be easily described in terms of the group M, if one makes use of the functorial properties of the group ring

D_{k}(n)(R) = Hom_{k−ring}(k[M], R) = Hom_{gr}(M, R^{∗}) ≅ (R^{∗})^{n},

where R^{∗} stands for the multiplicative group of invertable elements of a ring R. Hence any R-point of the scheme D_{k}(n) can be represented by a diagonal matrix diag(a_{1}, …, a_{n}) with a_{i} ∈ R^{∗}. The product of matrices defines the group structure on the set D_{k}(n)(R), and the functor R → D_{k}(n)(R)

1Partially supported by a grant of RFBR (project No. 99-01-00014). defines the group structure on the scheme D_{k}(n). The Hopf operators of the Hopf algebra k[M] can be defined as follows

m^{∗}(χ_{i}) = χ_{i} ⊗ χ_{i} (coproduct),
i^{∗}(χ_{i}) = χ_{i}^{−1} (coinversion),
e^{∗}(χ_{i}) = 1 (coidentity).

An element χ of the group M can be identified with a homomorphism of the group D_{k}(n) to the group D_{k}(1) = G_{m,k} :

M = D̂_{k}(n) = Hom_{gr}(D_{k}(n), G_{m,k});

such an element χ is called a character of the group D_{k}(n). On the R-points, the basic characters χ_{i} act naturally as projections: χ_{i}(a_{1}, …, a_{n}) = a_{i}. There are some reasons to write the group of characters D̂_{k}(n) additively, the multiplicative notation is used only to avoid possible misunderstanding; for example, the latter notation is used in description of the group ring. In the additive notation, the group D̂_{k}(n) coincides with Z^{n}. A decomposable torus D_{k}(n) is a special case of a general diagonal group D_{k}(M) = Spec k[M], where M is an arbitrary commutative group. If M is a finitely generated, then the k-scheme D_{k}(M) is of finite type. The category of diagonal k-groups of finite type is known to be an abelian category, dual to the category of finitely generated commutative groups [5]. The duality is established by means of the correspondence M → D_{k}(M), or the inverse correspondence D_{k}(M) → D̂_{k}(M). An exact sequence of k–groups

1 → D_{k}(M′′) → D_{k}(M) → D_{k}(M′) → 1

uniquely determines the exact sequence of the abelian groups

0 → M′ → M → M′′ → 0,

and conversely. Hence a curious fact: a homomorphic image of a diagonal k–torus D_{k}(n) is a diagonal k-torus again. In particular, the image of a diagonal torus under an isogeny is isomorphic to that torus.

Let us now recall a few results from the general theory of k-tori. Denote by k̄ the algebraic closure of the field k.

Definition. An algebraic k–group T is called a k–torus if the k-group T ⊗_{k} k̄ is isomorphic to a connected diagonal group D_{k}(n) ⊗_{k} k̄ = Dk̄(n), n := dim T.

A field L containing k is called a splitting field of the torus T if T ⊗_{k} L ≅ D_{L}(n). Any k–torus splits over a finite separable extension of the field k [1]. Let T̂ = Hom_{gr}(T_{k_{s}}, G_{m,k_{s}}) be the group of characters of the torus T, where k_{s} denotes the separable closure of the field k. The Galois group G of the extension k_{s}/k acts on the lattice T̂ in a natural way. The group T̂ is of finite rank, equal to the dimention of T. Hence there is an integral representation φ : G → GL(n, Z), n = dim_{k} T. The kernel Ker(φ) of this map is a normal subgroup of a finite index in G, and the field of invariants L = k_{s}^{Ker(φ)} is the Galois extension of k of the lowest degree such that the torus T splits over L. The image Im(φ) is a finite subgroup of the group GL(n, Z), which determines the symmetries of the lattice T̂ and provides important information about the structure of the torus T. Let Π = Gal (L/k), then T = Spec(L(T̂)^{Π}). The permutation Π-modules Ŝ, that is those modules with a finite Z–basis whose elements are permuted by the group Π, play a special role in the birational geometry of algebraic tori. A torus S defined over k by a permutation Π-module Ŝ is called a quasisplit torus. Only the k–decomposable tori are simpler than the quasisplit ones. On decomposing a basis of the Π-module Ŝ, whose elements are permuted by the group Π, in Π-orbits, one defines a decomposition of the module Ŝ in a direct sum ⊕_{i=1}^{s}Z[Π/π_{i}], where π_{i} is a subgroup of the group Π, 1 ≤ i ≤ s. The torus S with Ŝ = Z[Π/π] can be easily described. Let F = L^{π} be the field of invariants of such a torus, and let ω_{1}, …, ω_{m} be a basis of the extension L/k. Consider the regular representation of the field F in the algebra M(m, k); this representation defines the group F^{∗} as a subgroup in the group GL(m, k). The torus S may be regarded as the maximal k-torus of the group GL_{k}(m) satisfying the condition S(k) = F^{∗}; it is usually denoted by S = R_{F/k}(G_{m}). Thus any quasisplit torus L/ktorus is isomorphic to the product of the tori R_{F_{i}/k}(G_{m}), where k ⊂ F_{i} ⊂ L. If x_{1}, x_{2}, …, x_{m} are the coordinates of a general element of the field F with respect to the basis ω_{1}, …, ω_{n}, then the coordinate ring of the torus R_{F/k}(G_{m}) can be given explicitly: k[R_{F/k}(G_{m})] = k[x_{1}, …, x_{m}, y^{−1}], where y = N_{F/k}(x_{1}ω_{1} + … + x_{m}ω_{m}). The embedding k[x_{1}, …, x_{m}] ⊂ k[x_{1}, …, x_{m}, y^{−1}] determines a faithful action of the group R_{F/k}(G_{m}) on the space R_{F/k}(A_{F}^{1} ) ≅ A_{k}^{m}, and embeds the variety R_{F/k}(G_{m}) in A^{m} as an open orbit. Making use of the described direct product decomposition of a quasisplit torus, one concludes that any quasisplit k-torus of dimension n can be realised as a maximal k-torus of the group GL_{k}(A^{n}), and that the variety S may be regarded as an open orbit in the affine space A_{k}^{n}. A quasisplit torus S is k–rational. The following notion is important when discussing the problems of rationality.

Definition. The Π-modules A and B are said to be similar, if there are permutation Π-modules Ŝ_{1} and Ŝ_{2} such that the direct sum A ⊕ Ŝ_{1} is isomorphic to the direct sum B ⊕ Ŝ_{2}.

Let us denote by [A] the similarity class of a module A. The Π-module T̂ of characters of a torus T has a flasque resolvent, that is there is an exact sequence of torsion-free Π-modules of finite rank

0 → T̂ → Ŝ → N̂ → 0,

such that Ŝ is a permutation Π-module, and N̂ is a flasque module (a module N is said to be flasque if the condition H^{−1}(π, N̂) = 0 holds for any subgroup π of Π). Although a flasque resolvent of T̂ is not uniquely defined, the similarity class [N̂] of N̂ is determined uniquely by the module T̂; we denote this class by p(T̂) or p(T). The class p(T) is a birational invariant of the k−torus T [2,3]. Let X be a smooth projective completion of the torus T over k, the class p(T) coincides with the class [P icX_{L}] of the Picard module P icX_{L}. Let C(L/k) be the category of algebraic k-tori decomposable over L, and let P (Π) be the semigroup of the similarity classes of Π-modules generated by the flasque modules. The sum of the classes [N̂_{1}]+[N̂_{2}] is defined to be equal to the class [N̂_{1} ⊕N̂_{2}]. Let us define a semigroup Z(L/k) generated by the classes of stably equivalent tori lying in C(L/k), the product of two classes being given as follows:

{T_{1}} × {T_{2}} = {T_{1} ×_{k} T_{2}}.

We recall that the k-varieties X_{1} and X_{2} are said to be stably equivalent, if the products X_{1} ×_{k} A^{m} and X_{2} ×_{k} A^{p} are birationally isomorphic over k for some m and p. The following result can be found in [3].

Theorem 1. The correspondence T → p(T) defines an isomorphism of the semigroups Z(L/k) and P (Π).

This isomorphism may be regarded as a first approximation to the problem of birational classification of tori lying in C(L/k). As one may expect, tori T with p(T) = [0] have some special features. Such a torus T can be described as the following factor-group:

1 → S_{1} →^{α} S_{2} →^{β} T → 1,  (1)

where S_{1} and S_{2} are quasisplit k-tori. Since the map β is surjective, it follows that the field k(T) is contained in k(S_{2}); therefore any k-torus is an unirational variety. A quasisplit torus S satisfies the following important condition: H^{1}(M/M_{1}, S(M)) = 0 for any Galois extension M/M_{1} with M_{1} ⊃ k. Consequently the map β of the sequence (1) has a k-rational section γ : T → S_{2} with βγ = 1. Thus there is a k-rational map

φ = (α, γ) : S_{1} ×_{k} T → S_{2},  (2)

given by

φ(s, t) = α(s)γ(t), s ∈ S_{1}(k_{s}), t ∈ T (k_{s}).

The map φ is birational. For let α(s)γ(t) = α(s_{1})γ(t_{1}), then β[α(s)γ(t)] = β[α(s_{1})γ(t_{1})]; hence t = t_{1}, and therefore s = s_{1}. The torus T with the property p(T) = [0] is stably rational over k. The tori S_{1} and S_{2} can be realised as maximal k-tori in the groups GL(V_{1}) and GL(V_{2}), dim V_{i} = dim S_{i}. Thus we see that the function field k(T) of the torus T is isomorphic to the field of invariants k(V_{2})^{S_{1}} of the group S_{1}, which acts linearly and faithfully on V_{2} by means of the representation α. We can rewrite the birational map φ defined by (2) as follows:

φ : V_{1} ×_{k} T → V_{2}.  (3)

Let now S be any maximal k-torus in GL(V_{1}). We can define an action of S on V_{1} ×_{k} T by letting S act linearly and faithfully on V_{1}, and trivially on T (it is a reflection with respect to the the space T, in a generalised sense). The group S acts then birationally on V_{2} in the obvious way: g → φgφ^{−1}. Curiously enough, the field k(T) can be regarded as the field of invariants k(V_{2})^{S} for any maximal k-torus S of the group GL(V_{1}), assuming S acts on V_{2} in the way we have just described. One can now take S to be equal to a maximal diagonalizable subgroup D ⊂ GL(V_{1}). It suffices to show that the group D acts linearly on the field k(V_{2}), that is there is a basis {f_{1}, ..., f_{n}} of the field k(V_{2}) such that D acts linearly on the k-vector space U =< f_{1}, ..., f_{n} >, generated by this basis. For then D is contained in the maximal k-decomposable torus D_{2} of the group GL(U), and therefore the field k(T) = k(V_{2})^{D} = k(D_{2}/D) is a purely transcendental extension of k. Let us study the representation α in more detail. The representation α of the group S_{1} in the group S_{2} ⊂ GL(V_{2}) defines on V_{2} a structure of an S_{1}-module. Let us decompose the module V_{2} in a direct sum of indecomposible S_{1}-modules. Without loss of generality, it can be assumed that this decomposition contains no trivial S_{1}-component, for if V_{2} = U ⊕ W and the group S_{1} acts trivially on the k-vector space W, then U is a faithful S_{1}-module and therefore the field of invariants k(V_{2})^{S_{1}} coincides with the purely transcendental extension k(U)^{S_{1}}(W) of the field k(U)^{S_{1}}, so that the field k(V_{2})^{S_{1}} is rational as soon as k(U)^{S_{1}} is. Similarly, one can assume that the chosen decomposition of the S_{1}-module V_{2} contains no multiple components. Thus we have reduced the problem of the k-rationality of a stably rational k-torus to the problem of rationality of the field of invariants k(V_{2})^{S_{1}} of a quasisplit k-torus S_{1}, acting linearly and faithfully on a k-vector space V_{2}; moreover, it may be assumed that the space V_{2} contains no S_{1}-subspace U with a faithful action of S_{1}. Let us call such a representation minimal and describe its structure. Consider at first the simplest case when S_{1} = R_{F/k}(G_{m}). Let L be the normal closure of F in k_{s}, and let

Π = Gal (L/k), Π_{1} = Gal (L/F), Ŝ = Z[Π/Π_{1}] = Z[Π] ⊗_{Π_{1}} Z.

Let us choose a basis χ_{1}, .., χ_{d} of the group Ŝ_{1} transitively permuted by the group Π, and let Π_{1} denote the stabilizer of the character χ_{1}. There is an one-dimensional faithful linear representation of the group G_{m,F} on the vector space A_{F}^{1} . Then V_{1} = R_{F/k}(A_{F}^{1} ) is a space of the regular representation of the group S_{1} = R_{F/k}(G_{m}). The trace of this representation is equal to the sum χ_{1} + ... + χ_{d}. Let e_{1}, ..., e_{d} be a basis of the space V_{1} over k, let Θ_{1} = α_{1}e_{1} + ... + α_{d}e_{d} be the eigenvector corresponding to the character χ_{1}, α_{i} ∈ F, Θ_{1} ∈ F ⊗_{k} V_{1}. The other characters χ_{i} and their eigenvectors Θ_{i} can be obtained acting on χ_{1}, and Θ_{1} by the Galois group. The module V_{2} can be decomposed in a direct sum of indecomposable S_{1}-modules. Let V be one of the indecomposable components of the S_{1}-module V_{2}. Let us describe its structure. The group S_{1} is diagonalizable over L, therefore the L-vector space V ⊗_{k} L is isomorphic to a direct sum of one-dimensional components V_{ξ} transitively permuted by the Galois group Π : σV_{ξ} = V_{σ(ξ)}. Each of the components V_{ξ} affords an one-dimensional representation of the group S_{1,L} belonging to the character ξ. It follows therefore that the trace of restriction on V of the representation α is equal to

tr (α | V) = ξ_{1} + ... + ξ_{r}, r = dim V.

We call the characters ξ_{i} the weights of the representation of the group S_{1,L}. Let ξ_{1} = χ_{1}^{a_{1}} ⋯ χ_{d}^{a_{d}} be one of the characters. Then any of the weights ξ_{i} can be obtained from ξ_{1} by the permuting the exponents; these permutations are given by the action of the Galois group on the element ξ_{1}. The one-dimensional component V_{ξ_{1}} is generated by the eigenvector Θ_{1}^{a_{1}} ⋯ Θ_{d}^{a_{d}} belonging to the character ξ_{1}; here Θ_{1}^{a_{1}} ⋯ Θ_{d}^{a_{d}} is regarded as an element of the field of fractions L(V_{1}) of the symmetric algebra L[V_{1}] on the space V_{1} ⊗_{k} L. Let π be the stabiliser of the character ξ_{1} in the Galois group Π, and let F_{1} = L^{π} be its field of invariants. The space V of the linear representation of S_{1} is generated over k by the k-rational functions P_{i}(e_{1}, ..., e_{d}), 1 ≤ i ≤ r, defined by the decomposition

Θ_{1}^{a_{1}} ⋯ Θ_{d}^{a_{d}} = P_{1}ω_{1} + ⋯ + P_{r}ω_{r},  (4),

where ω_{1}, ..., ω_{r} is a basis of the extension F_{1}/k, and

σ(Θ_{1}^{a_{1}} ⋯ Θ_{d}^{a_{d}}) = P_{1}σ(ω_{1}) + ⋯ + P_{r}σ(ω_{r}), σ ∈ Π.

Thus the S_{1}-module V_{2} can be realized on functions from the field of fractions k(V_{1}). Let P_{1}, ..., P_{n} be the sequence of functions obtained from (4) for all the components of the k-vector space V_{2}; it follows from the minimality of the S_{1}-module V_{2} that these functions are linearly independent over k. Thus the sequence of functions P_{1}, ..., P_{n} is a k-basis of the vector space V_{2}. Since α : S_{1} → S_{2} is a monomorphism, the characters of the torus S_{1}, entering into the sum for tr(α | V_{2}) generate the group of characters of the torus S_{1}. But then the system of functions {P_{1}, ..., P_{n}} generates the field k(V_{1}), and in particular, each of the elements e_{1}, ..., e_{d} can be written as a rational function of P_{1}, ..., P_{n}. We associate to each Laurent monomial Θ_{1}^{b_{1}} ⋯ Θ_{d}^{b_{d}}, coinciding with the eigenvector of a character ξ entering into the sum for tr(α), the monomial e_{1}^{b_{1}} ⋯ e_{d}^{b_{d}} (forgetting about the action of the Galois group). There are exactly n such monomials, and they are pairwise distinct; let us list these monomials in some order: f_{1}, ..., f_{n}. The above argument shows also that the system of functions f_{1}, ..., f_{n} generates the field k(V_{1}). Therefore each of the functions f_{1}, ..., f_{n} can be rationally expressed in terms of the functions P_{1}, ..., P_{n} and vice versa: f_{i} = R_{i}(P_{1}, ..., P_{n}), P_{i} = Q_{i}(f_{1}, ..., f_{n}); here R_{i} and Q_{i} are rational functions of n independent variables with coefficients in k. Let D_{1} be a maximal diagonal k-torus in the group GL(V_{1}) acting on V_{1} as follows: g(e_{i}) = λ_{i}e_{i}, g ∈ D_{1}(k_{s}), λ_{i} ∈ k_{s}^{∗}. Then this action can be naturally extended to a linear diagonal action on the space < f_{1}, ..., f_{n} > because the basic monomials are mapped to the proportional ones. The vector space V_{2} is generated by the basis {P_{1}, ..., P_{n}}; let k[V_{2}] be the symmetric algebra on the vector space V_{2}, and let k(V_{2}) be its field of fractions. One remarks that the field k(V_{2}) can be also described as the field of fractions of the symmetric algebra k[f_{1}, …, f_{n}], and that the group D_{1} acts diagonally on the elements of the basis {f_{1}, …, f_{n}}. This has as a consequence that there is an n−dimensional subtorus D_{2} of the group of all the birational k-transformations of the vector space V_{2}, which contains the torus D_{1} and splits over k. As we have already remarked, this implies that the variety T is birationally equivalent over k to the variety D_{2}/D_{1}, which is k−rational. Thus if S_{1} = R_{F/k}(G_{m}) and V_{2} is a faithful S_{1}−module, then the field k(V_{2})^{S_{1}} is a purely transcendental extension of k. A general quasisplit k−torus S_{1} is isomorphic to the product of the tori of the form R_{F_{i}/k}(G_{m}), and V_{1} is equal to a direct sum of the invariant subspaces of the form R_{F_{i}/k}(A^{1}) = V_{1,i}. In each of the subspaces V_{1,i}, one chooses a k−basis as above, and a permutation basis L−basis Θ_{1}^{(i)}, ..., Θ_{d}^{(i)} of the L[Π]−module L ⊗_{k} V_{1,i}. Then again the S_{1}−module V_{2} can be generated by a system of functions from the field k(S_{1}) = k(V_{1}) :

V_{2} =< P_{1}, ..., P_{n} >, n = dim_{k}V_{2},

with P_{1}, ..., P_{n} linearly independent over k. The rest of argument remains the same, as in the special case considered above. We have just obtained the main result of this article.

Theorem 2 (Main Theorem). Any stably rational k-torus is rational over k.

Thus the condition p(T) = [0] characterises the algebraic k-tori rational over k.

Corollary 1. Let α : S → GL(V) be a faithful linear representation of a quasisplit torus S. There is a birational transformation of the vector space V into a k−vector space W, such that the action of the group S on W is quasiregular, that is the S-module W is isomorphic to the direct sum W = W_{1} ⊕ W_{2} such that W_{1} supports the regular representation of the torus S, while S acts trivially on W_{2}.

Suppose now that L/k is a cyclic extension of degree (L : k) = n, and let Π = Gal (L/k). The ring of integers Z[ζ_{n}] of the cyclotomic field Q(ζ_{n}) can be regarded as a Π-module on letting σ(α) = ζ_{n}α for α ∈ Z[ζ_{n}], where ζ_{n} is a primitive n−th root of 1 and σ is a generator of Π. Let T_{n} be the k-torus, having the Π-module T̂_{n} = Z[ζ_{n}] as its module of rational characters. Since such a torus T_{n} is stably rational over k [3], we have the following result.

Corollary 2. The torus T_{n} is rational over its field of definition, for any n.

A direct proof of the rationality of the tori T_{n} has been given in the author’s lecture at the Journées Arithméques in Limoges [6].

The problem of the reduction of a pair matrices to a canonical form gives us another series of stably rational varieties. Let M(n) = M(n, C) be the ring of square matrices of order n with complex coefficients, and let V_{m,n} = M(n) ⊕ ... ⊕ M(n) be the direct sum of m copies of the vector space M(n). The group GL(n, C) acts on V_{m,n} by the conjugations. Let K_{m,n} = C(V_{m,n})^{GL_{n}} be its field of invariants. It is comparatively easy to show that the field K_{m,n} is a purely transcendental extension of the field K_{2,n}, m ≥ 2. The problem of rationality of the field K_{2,n} is an old-standing problem. Sylvester had already proved rationality of the field K_{2,2}. Formanek [7,8] has proved that the fields K_{2,3} and K_{2,4} are rational. He has also shown that the field K_{2,n} can be realized as a function field of an explicitly described algebraic torus. This line of thought is followed in [9], where it has been proved that the tori corresponding to the fields K_{2,5} and K_{2,7} are stably rational over C. Since stable rationality of the fields K_{2,a} and K_{2,b} implies stable rationality of the field K_{2,ab} if the integers a and b are relatively prime, it follows that the field K_{2,d} is stably rational for any d dividing 420. We have a stronger result.

Corollary 3. If d | 420, then the field K_{2,d} is rational over C.

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Samara State University

Faculty of Mathematics and Mechanics

ul. akad. Pavlova 1

Samara 443011, Russia

e-mail: voskres@info.ssu.samara.ru

Translated from the Russian by S.Yu.Popov and B.Z.Moroz