ON A CONJECTURE OF LE BRUYN

Anne Cortella and Boris Kunyavskii

Abstract. Given a generic field extension F/k of degree n > 3 (i.e. the Galois group of the normal closure of F is isomorphic to the symmetric group S_{n}), we prove that the norm torus, defined as the kernel of the norm map N: R_{F/k}(𝔾 _{textm}) → 𝔾_{m}, is not rational over k.

Given an arbitrary field k, we call a separable extension F/k of degree n generic if the Galois group G = Gal(L/k) of the normal closure L of F over k is isomorphic to the symmetric group S_{n}. We consider the norm map N: F^{∗} → k^{∗}. The kernel of N can be regarded as the set of k-points of an affine algebraic k-variety T called norm torus. Using the Weil symbol of restriction of scalars, we write T as the kernel of R_{F/k}(𝔾_{m}) → 𝔾_{m,k} where 𝔾_{m} stands for the multiplicative group. If the extension F/k is generic, the norm torus is also called generic and is denoted by T_{F/k}, or just T_{n} if it does not lead to any confusion.

In [LB], assuming n > 3, Le Bruyn proves that the generic norm torus T_{n} is non-rational over k whenever n is prime, and states a conjecture that T_{n} is never k-rational except, possibly, for n = 6. Our goal is to prove the above conjecture (including the case n = 6). Recall that T is called stably rational if there is a rational variety T′ such that T × T′ is rational.

Theorem. With the above notation, T_{n}(n > 3) is never stably rational over k.

Remark. The result might look a little bit surprising in view of good arithmetic properties of generic norm tori: in particular, if k is a number field, they are known to satisfy weak approximation property and their principal homogeneous spaces satisfy the Hasse principle. Moreover, for the case when n is prime, T_{n} is known to be a direct factor of a rational variety [CT/S2]. Note that the result cannot be ameliorated in the sense that for n = 2 or 3 the torus T_{n} is of dimensio n 1 or 2 and hence rational [V], 4.73, 4.74.

The proof follows from the lemmas below. Throughout we denote by M_{n} = Hom(T_{n}, 𝔾_{m}) the group of rational characters of T_{n} viewed as a G-module. By definition, there is an exact sequence of G-modules

0 → ℤ → P_{n} → M_{n} → 0  (1)

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 20G05.

Research of the second named author was partially supported by the Ministry of Absorption (Israel) and The Israel Science Foundation founded by the Israel Academy of Sciences and Humanities.

where P_{n} = ℤ[G/H] is a permutation module, G = S_{n}, H = Gal(L/F) is isomorphic to S_{n−1}. The following lemma is the key one.

Lemma 1. Let n = rs with arbitrary r, s > 1, and let F/k be a generic extension of degree n. If T_{F/k} = T_{n} is stably rational over k, there is a generic extension K/E of degree r such that T_{K/E} = T_{r} is stably rational over E.

Proof. Take a subgroup U = S_{r} ⊂ S_{n} embedded in such a way that P_{n} restricted to U is a direct sum \underbrace{P_{r} ⊕ ⋯ ⊕ P_{r}}_{s times}. (This simply means that we partition \{1, …, n\}

into s disjoint subset s of cardinality r and let U act in a standard way on each of these subsets.) We then regard (1) as a sequence of U-modules and notice that M_{n} restricted to U splits into a direct sum:

(M_{n})|_{U} = M_{r} ⊕ \underbrace{P_{r} ⊕ ⋯ ⊕ P_{r}}_{(s−1) times} .  (2)

In the language of tori, (2) reads as follows: let E = L^{U} be the fixed subfield of U, then the E-torus T_{E} = T ×_{k} E is isomorphic to a direct product of T_{r} = ker[R_{K/E}(𝔾_{m}) → 𝔾_{m,E}] and a quasi-split torus S = ∏_{i=1}^{s−1} R_{K/E}(𝔾_{m}) where K/E is a generic extension of degree r, K = L^{V}, V ⊂ U, V ≅ S_{r−1}. By assumption, T is stably rational over k, hence T_{E} is stably rational over E. Since any quasi-split torus is rational, we are done. □

Lemma 2 (Le Bruyn). If n > 3 is a prime number, T_{n} is not stably rational.

Proof. See [LB]. □

Before stating the next lemma, we recall that the group

� _{ω}^{2}(G, M) = ker[H^{2}(G, M) → ∏_{g∈G} H^{2}(⟨g⟩ , M)]

(where M stands for the character module of an algebraic torus T defined over k and split over L, G = Gal(L/k)), is a birational invariant of T. To be more precise, this group is zero whenever T is stably rational over k. Here is another useful description of the above invariant: consider a flasque resolution of M, i.e. an exact sequence of G-modules

0 → M → S → N → 0

where S is a permutation module and N is a flasque module (the latter means that H^{−1}(G′, N) = 0 for all subgroups G′ ⊆ G), then � _{ω}^{2}(G, M) ≅ H^{1}(G, N). See [V], 4.61, [CT/S1] for more details.

Lemma 3. If n is a square, T_{n} is not stably rational.

Proof. Let n = m^{2}. Take a subgroup U = ℤ/mℤ × ℤ/mℤ ⊂ S_{n} embedded in such a way that the module P from sequence (1) viewed as a U-module is isomorphic to ℤ[U]. In other words, we choose U generated by

σ = (1 2 … m)(m + 1 m + 2 … 2m) … (n − m + 1 n − m + 2 … n),

τ = (1 m + 1 … n − m + 1)(2 m + 2 … n − m + 2) … (m 2m … m^{2}).

Then M_{n}, regarded as a U-module, is none other than Ĵ = ℤ[U]/ℤ, the character module of the norm torus J = ker[R_{L/E}(𝔾_{m}) → 𝔾_{m,E}] where E = L^{U}. It is well known that J is not rational over E because � _{ω}^{2}(U, Ĵ) = ℤ/pℤ. Since T_{n} ×_{k} E = J, we conclude that T_{n} cannot be stably rational over k. □

Corollary (Saltman, Snider). If n is divisible by a square, T_{n} is not stably rational.

Proof. Combine Lemma 1 and Lemma 3. □

Lemma 4. The torus T_{6} is not stably rational.

Proof. Take U = ℤ/2 × ℤ/2 ⊂ S_{6} generated by (12)(34) and (34)(56). We observe that U coincides with the Sylow 2-subgroup of the alternating group A_{4} embedded into S_{6} via its action on the edges of tetrahedron. Let M = M_{6} be the module of characters of T_{6} defined by sequence (1) with G = S_{6}, H = S_{5}. It is known that � _{ω}^{2}(A_{4}, M) = ℤ/2ℤ ([D/P], Lemma 13). This implies � _{ω}^{2}(U, M) ≠ 0. Indeed, assume the contrary. Then, since any Sylow 3-subgroup V of A_{4} is cyclic, one has � _{ω}^{2}(V, M) = 0, and vanishing of � _{ω}^{2}(U, M) would imply vanishing of � _{ω}^{2}(A_{4}, M) (one may apply the above interpretation of � _{ω}^{2}(G, M) as H^{1}(G, N) to the case G = A_{6} and use the fact that the restriction to a Sylow p-subgroup is injective on the p-component of H^{1}). □

Remark. Of course, one may give a more direct proof of Lemma 4 without referring to [D/P], either by a straightforward computation of � _{ω}^{2}(U, M) (which goes much simpler than for A_{4}), or by constructing an exact sequence of U-m odules

0 → M_{a} → M → ℤ ⊕ ℤ → 0

with M_{a} the character module of an anisotropic torus T_{a} which, in our case, turns out to be ℤ[U]/ℤ; by [V], 4.22, the latter exact sequence induces a birational equivalence of tori T_{n} ∼ T_{a} × 𝔾_{m}^{2}, whence the result.

Proof of the Theorem. We are now ready to prove the Theorem. Indeed, the above Corollary reduces the problem to the case when n is square-free, and Lemmas 1 and 2 englobe all n having a prime divisor greater than 3. We thus have to apply Lemma 4 for the only remaining case n = 6. □

Concluding remark. Our theorem can (and should) be viewed in a broader context. Namely, one can extend it to generic tori in (almost absolutely) simple groups. Indeed, the above result corresponds to the case of an inner form of a simply connected group of type A_{n−1}. Such a generalization to the other types of inner and outer forms of simply connected and adjoint groups is the subject of our forthcoming paper.

Acknowledgement. This work was done while the second named author was visiting Université de Franche-Comté at Besançon. B. Kunyavskii thanks this institution for its hospitality.

References

[CT/S1] J.-L. Colliot-Thélène and J.-J. Sansuc, La R-équivalence sur les tores, Ann. Sci. École Norm. Sup. 10 (1977), 175–229.

[CT/S2] , Principal homogeneous spaces under flasque tori: applications, J. Algebra 106

(1987), 148–205.

[D/P] Yu. I. Drakohrust and V. P. Platonov, The Hasse norm principle for algebraic number fields, Izv. Akad. Nauk SSSR Ser. Mat. 50 (1986), 946–968; English transl. in Math. USSR Izv. 29 (1987), 299–322.

[LB] L. Le Bruyn, Generic norm one tori, Nieuw Arch. Wiskd. IV Ser. 13 (1995), 401–407.

[V] V. E. Voskresenskiĭ, Algebraic Tori, Nauka, Moscow, 1977. (Russian)

A.C.: Université de Franche-Comté, Faculté des Sciences et des Techniques, Laboratoire de Mathématiques, 16 route de Gray, 25030 BESANÇON Cedex, FRANCE

   E-mail address: cortella@math.univ-fcomte.fr

B.K.: Bar-Ilan University, Department of Mathematics and Computer Science, 52900 RAMAT GAN, ISRAEL

   E-mail address: kunyav@macs.biu.ac.il