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The rationality problem for norm one tori. (English) Zbl 1223.14017
This paper is on the rationality problem of algebraic tori of norm type. It contains several very significant results with elegant proofs.
Let \(K/k\) be a finite separable field extension, \(R_{K/k}(G_m)\) be the Weil restriction of \(G_{m,K}\) to \(k\). Define the norm one torus \(R^1_{K/k}(G_m):= \ker \{ \phi : R_{K/k}(G_m) \rightarrow G_{m,k} \}\) where \(\phi\) is the norm map \(\phi(\alpha)=N_{K/k}(\alpha)\) for any \(\alpha \in K^{\times}\). The main question is to study under what situation the torus \(R^1_{K/k}(G_m)\) is rational over \(k\).
The function field of \(R^1_{K/k}(G_m)\) may be represented as follow. Let \(L/k\) be the Galois closure of \(K/k\), \(G=\mathrm{Gal}(L/k)\), and \(H\) be the subgroup such that \(L^H=K\). Consider the augmentation map of \(G\)-lattices \(\epsilon : Z[G/H] \rightarrow Z\). Let \(I_{G/H}= \ker(\epsilon)\), and \(J_{G/H}=\operatorname{Hom} (I_{G/H}, Z)\). Then the character group and the function field of \(R^1_{K/k}(G_m)\) are \(J_{G/H}\) and \(L(J_{G/H})^G\) respectively.
From the theory of flabby classes the rationality problem can be reduced to
(1) \(L(J_{G/H})^G\) is stably rational over \(k\) if and only if \(J_{G/H}\) is quasi-permutation, i.e. there is a short exact sequence of \(Z[G]\)-modules \(0 \rightarrow J_{G/H} \rightarrow P_1 \rightarrow P_2 \rightarrow 0\) where \(P_1,P_2\) are permutation \(G\)-lattices, and
(2) \(L(J_{G/H})^G\) is retract rational over \(k\) if and only if \(J_{G/H}\) is quasi-invertible, i.e. there is a short exact sequence of \(Z[G]\)-modules \(0 \rightarrow J_{G/H} \rightarrow P \rightarrow E \rightarrow 0\) where \(P\) is a permutation \(G\)-lattice and \(E\) is an invertible \(G\)-lattice (see Theorem 1.1).
The case when \(K/k\) is Galois has been solved by S. Endo and T. Miyata [Nagoya Math. J. 56, 85–104 (1975; Zbl 0301.14008)]; also see Theorem 1.2 of this paper. Thus only the non-Galois case remains, i.e. \(H\) is a non-normal non-trivial subgroup of \(G\) (in particular, \(G\) is non-abelian).
The first two main results are the following.
Theorem 1. Let \(G\) be a non-abelian nilpotent group and \(H\) be a non-normal non-trivial subgroup of \(G\). Then \(J_{G/H}\) is not quasi-invertible (see Theorem 2.1).
Theorem 2. Let \(G\) be a non-abelian group and \(H\) be a non-normal non-trivial subgroup of \(G\). Then \(J_{G/H}\) is quasi-invertible if and only if all the Sylow subgroups of \(G\) are cyclic. In this situation, \(J_{G/H}\) is quasi-permutation if and only if \(G \simeq C_m \times D_n\) (\(m,n\) are odd, gcd\( \{m,n \}=1\), \(n \geq 3\)) and \(H \subset D_n\) is of order 2 (see Theorem 3.1).
The author then consider the generic tori, i.e. \(G=S_n\) and \(H=S_{n-1}\) where \(S_n\) denotes the symmetric group of degree \(n\).
Theorem 3. For \(n \geq 2\), \(J_{S_n/S_{n-1}}\) is quasi-invertible if and \(n\) is a prime number. Moreover, it is quasi-permutation if and only if \(n=2,3\) (see Theorem 4.3). For previous partial results of the above theorem, see Remark 2 of this paper and Section 8 of the paper by A. Cortella and B. Kunyavskii [J. Algebra 225, No. 2, 771–793 (2000; Zbl 1054.14507)]. Finally consider the alternating groups.
Theorem 4. For \(n \geq 3\), \(J_{A_n/A_{n-1}}\) is quasi-invertible if and \(n\) is a prime number. Moreover, the direct sum \([J_{A_n/A_{n-1}}]^{(t)}\) is quasi-permutation for some \(t \geq 1\) if and only if \(n=3, 5\) (see Theorem 4.5).
{Remarks}: The reader is warned that the definition of a meta-cyclic group is different from the usual one. The usual definition of a meta-cyclic group is an extension of a cyclic group by another cyclic group (see, for example, [I. M. Isaacs, Finite Group Theory. Graduate Studies in Mathematics 92. Providence, RI: American Mathematical Society (AMS). (page 160) (2008; Zbl 1169.20001)]). In this paper, a group \(G\) is called meta-cyclic if all the Sylow subgroups of \(G\) are cyclic, equivalently, \(G \simeq C_m \cdot C_n\) (\(m\) is odd and gcd\(\{m,n \}=1\)), where \(C_m \cdot C_n\) denotes the semi-direct product with \(C_m\) being a normal subgroup there.

MSC:
14E08 Rationality questions in algebraic geometry
20C10 Integral representations of finite groups
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