The rationality problem for norm one tori.

*(English)*Zbl 1223.14017This 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.

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.

Reviewer: Ming-Chang Kang (Taipei)

##### MSC:

14E08 | Rationality questions in algebraic geometry |

20C10 | Integral representations of finite groups |

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##### References:

[1] | Y. Berkovich, Groups of Prime Power Order, I , de Gruyter, Berlin, 2008. |

[2] | K. S. Brown, Cohomology of Groups , Grad. Texts in Math. 87 , Springer, New York, 1982. |

[3] | J.-L. Colliot-Thélène and J.-J. Sansuc, La R-équivalence sur les tores , Ann. Sci. Éc. Norm. Supér. (4) 10 (1977), 175-230. · Zbl 0356.14007 |

[4] | J.-L. Colliot-Thélène and J.-J. Sansuc, Principal homogeneous spaces under flasque tori: Applications , J. Algebra 106 (1987), 148-205. · Zbl 0597.14014 |

[5] | A. Cortella and B. Kunyavskiĭ, Rationality problem for generic tori in simple groups , J. Algebra 225 (2000), 771-793. · Zbl 1054.14507 |

[6] | C. W. Curtis and I. Reiner, Methods of Representation Theory, I , John Wiley, New York, 1981. |

[7] | A. Dress, The permutation class group of a finite group , J. Pure Appl. Algebra 6 (1975), 1-12. · Zbl 0302.20010 |

[8] | S. Endo, On the rationality of algebraic tori of norm type , J. Algebra 235 (2001), 27-35. · Zbl 1062.20505 |

[9] | S. Endo and T. Miyata, Invariants of finite abelian groups , J. Math. Soc. Japan 25 (1973), 7-26. · Zbl 0245.20007 |

[10] | S. Endo and T. Miyata, On a classification of the function fields of algebraic tori , Nagoya Math. J. 56 (1975), 85-104; Correction , Nagoya Math. J. 79 (1980), 187-190. · Zbl 0301.14008 |

[11] | M. Florence, Non rationality of some norm one tori , preprint, 2006. |

[12] | L. Le Bruyn, Generic norm one tori , Nieuw Arch. Wiskd. (5) 13 (1995), 401-407. · Zbl 0872.20043 |

[13] | N. Lemire and M. Lorenz, On certain lattices associated with generic division algebras , J. Group Theory 3 (2000), 385-405. · Zbl 0998.16012 |

[14] | M. Rosen, Representations of twisted group rings , Ph.D. dissertation, Princeton University, Princeton, N. J., 1963. · Zbl 0115.40602 |

[15] | D. J. Saltman, Retract rational fields and cyclic Galois extensions , Israel J. Math. 47 (1984), 165-215. · Zbl 0546.14013 |

[16] | V. E. Voskresenskiĭ, Algebraic Groups and Their Birational Invariants , Transl. Math. Monogr. 179 , Amer. Math. Soc., Providence, 1998. · Zbl 0974.14034 |

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