Rationality problem for norm one tori in small dimensions
HTML articles powered by AMS MathViewer
- by Sumito Hasegawa, Akinari Hoshi and Aiichi Yamasaki HTML | PDF
- Math. Comp. 89 (2020), 923-940 Request permission
Abstract:
We classify stably/retract rational norm one tori in dimension $n-1$ for $n=2^e$ $(e\geq 1)$ as a power of $2$ and $n=12, 14, 15$. Retract non-rationality of norm one tori for primitive $G\leq S_{2p}$ where $p$ is a prime number and for the five Mathieu groups $M_n\leq S_n$ $(n=11,12,22,23,24)$ is also given.References
- Greg Butler, The transitive groups of degree fourteen and fifteen, J. Symbolic Comput. 16 (1993), no. 5, 413–422. MR 1271082, DOI 10.1006/jsco.1993.1056
- Gregory Butler and John McKay, The transitive groups of degree up to eleven, Comm. Algebra 11 (1983), no. 8, 863–911. MR 695893, DOI 10.1080/00927878308822884
- Peter J. Cameron, Finite permutation groups and finite simple groups, Bull. London Math. Soc. 13 (1981), no. 1, 1–22. MR 599634, DOI 10.1112/blms/13.1.1
- John J. Cannon and Derek F. Holt, The transitive permutation groups of degree 32, Experiment. Math. 17 (2008), no. 3, 307–314. MR 2455702
- Henri Cartan and Samuel Eilenberg, Homological algebra, Princeton University Press, Princeton, N. J., 1956. MR 0077480
- Anne Cortella and Boris Kunyavskiĭ, Rationality problem for generic tori in simple groups, J. Algebra 225 (2000), no. 2, 771–793. MR 1741561, DOI 10.1006/jabr.1999.8150
- Jean-Louis Colliot-Thélène and Jean-Jacques Sansuc, La $R$-équivalence sur les tores, Ann. Sci. École Norm. Sup. (4) 10 (1977), no. 2, 175–229 (French). MR 450280
- Jean-Louis Colliot-Thélène and Jean-Jacques Sansuc, Principal homogeneous spaces under flasque tori: applications, J. Algebra 106 (1987), no. 1, 148–205. MR 878473, DOI 10.1016/0021-8693(87)90026-3
- John D. Dixon and Brian Mortimer, Permutation groups, Graduate Texts in Mathematics, vol. 163, Springer-Verlag, New York, 1996. MR 1409812, DOI 10.1007/978-1-4612-0731-3
- Amir Džambić and Gareth A. Jones, $p$-adic Hurwitz groups, J. Algebra 379 (2013), 179–207. MR 3019251, DOI 10.1016/j.jalgebra.2013.01.007
- Shizuo Endo, The rationality problem for norm one tori, Nagoya Math. J. 202 (2011), 83–106. MR 2804547, DOI 10.1215/00277630-1260459
- Shizuo Endo and Ming-Chang Kang, Function fields of algebraic tori revisited, Asian J. Math. 21 (2017), no. 2, 197–224. MR 3672258, DOI 10.4310/AJM.2017.v21.n2.a1
- Shizuo Endô and Takehiko Miyata, Invariants of finite abelian groups, J. Math. Soc. Japan 25 (1973), 7–26. MR 311754, DOI 10.2969/jmsj/02510007
- Shizuo Endô and Takehiko Miyata, On a classification of the function fields of algebraic tori, Nagoya Math. J. 56 (1975), 85–104. MR 364203
- Shizuo Endô and Takehiko Miyata, Integral representations with trivial first cohomology groups, Nagoya Math. J. 85 (1982), 231–240. MR 648425
- M. Florence, Non rationality of some norm one tori, preprint (2006).
- The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.8.10; 2018. (http://www.gap-system.org).
- S. Hasegawa, A. Hoshi, and A. Yamasaki, Rationality problem for norm one tori in small dimensions, arXiv:1811.02145 (the arXiv version of this paper).
- P. J. Hilton and U. Stammbach, A course in homological algebra, 2nd ed., Graduate Texts in Mathematics, vol. 4, Springer-Verlag, New York, 1997. MR 1438546, DOI 10.1007/978-1-4419-8566-8
- Akinari Hoshi, Ming-chang Kang, and Hidetaka Kitayama, Quasi-monomial actions and some 4-dimensional rationality problems, J. Algebra 403 (2014), 363–400. MR 3166080, DOI 10.1016/j.jalgebra.2014.01.019
- Akinari Hoshi and Aiichi Yamasaki, Rationality problem for algebraic tori, Mem. Amer. Math. Soc. 248 (2017), no. 1176, v+215. MR 3685951, DOI 10.1090/memo/1176
- A. Hoshi and A. Yamasaki, Rationality problem for norm one tori, arXiv:1811.01676.
- W. Hürlimann, On algebraic tori of norm type, Comment. Math. Helv. 59 (1984), no. 4, 539–549. MR 780075, DOI 10.1007/BF02566365
- Ming-chang Kang, Retract rational fields, J. Algebra 349 (2012), 22–37. MR 2853623, DOI 10.1016/j.jalgebra.2011.10.024
- B. E. Kunyavskii, Three-dimensional algebraic tori, Selecta Math. Soviet. 9 (1990) 1–21.
- B. E. Kunyavskii, Algebraic tori — thirty years after, Vestnik Samara State Univ. (2007) 198–214.
- Nicole Lemire and Martin Lorenz, On certain lattices associated with generic division algebras, J. Group Theory 3 (2000), no. 4, 385–405. MR 1790337, DOI 10.1515/jgth.2000.031
- Nicole Lemire, Vladimir L. Popov, and Zinovy Reichstein, Cayley groups, J. Amer. Math. Soc. 19 (2006), no. 4, 921–967. MR 2219306, DOI 10.1090/S0894-0347-06-00522-4
- Lieven Le Bruyn, Generic norm one tori, Nieuw Arch. Wisk. (4) 13 (1995), no. 3, 401–407. MR 1378805
- H. W. Lenstra Jr., Rational functions invariant under a finite abelian group, Invent. Math. 25 (1974), 299–325. MR 347788, DOI 10.1007/BF01389732
- Martin W. Liebeck, Cheryl E. Praeger, and Jan Saxl, On the O’Nan-Scott theorem for finite primitive permutation groups, J. Austral. Math. Soc. Ser. A 44 (1988), no. 3, 389–396. MR 929529
- Martin W. Liebeck and Jan Saxl, Primitive permutation groups containing an element of large prime order, J. London Math. Soc. (2) 31 (1985), no. 2, 237–249. MR 809945, DOI 10.1112/jlms/s2-31.2.237
- Martin Lorenz, Multiplicative invariant theory, Encyclopaedia of Mathematical Sciences, vol. 135, Springer-Verlag, Berlin, 2005. Invariant Theory and Algebraic Transformation Groups, VI. MR 2131760
- Takashi Ono, Arithmetic of algebraic tori, Ann. of Math. (2) 74 (1961), 101–139. MR 124326, DOI 10.2307/1970307
- Gordon F. Royle, The transitive groups of degree twelve, J. Symbolic Comput. 4 (1987), no. 2, 255–268. MR 922391, DOI 10.1016/S0747-7171(87)80068-8
- David J. Saltman, Retract rational fields and cyclic Galois extensions, Israel J. Math. 47 (1984), no. 2-3, 165–215. MR 738167, DOI 10.1007/BF02760515
- John Shareshian, On the Möbius number of the subgroup lattice of the symmetric group, J. Combin. Theory Ser. A 78 (1997), no. 2, 236–267. MR 1445416, DOI 10.1006/jcta.1997.2762
- Richard G. Swan, Noether’s problem in Galois theory, Emmy Noether in Bryn Mawr (Bryn Mawr, Pa., 1982) Springer, New York-Berlin, 1983, pp. 21–40. MR 713790
- Richard G. Swan, The flabby class group of a finite cyclic group, Fourth International Congress of Chinese Mathematicians, AMS/IP Stud. Adv. Math., vol. 48, Amer. Math. Soc., Providence, RI, 2010, pp. 259–269. MR 2744226, DOI 10.1090/amsip/048/14
- V. E. Voskresenskiĭ, On two-dimensional algebraic tori. II, Izv. Akad. Nauk SSSR Ser. Mat. 31 (1967), 711–716 (Russian). MR 0214597
- V. E. Voskresenskiĭ, Birational properties of linear algebraic groups, Izv. Akad. Nauk SSSR Ser. Mat. 34 (1970), 3–19 (Russian). MR 0262251
- V. E. Voskresenskiĭ, Stable equivalence of algebraic tori, Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974), 3–10 (Russian). MR 0342515
- V. E. Voskresenskiĭ, Algebraic groups and their birational invariants, Translations of Mathematical Monographs, vol. 179, American Mathematical Society, Providence, RI, 1998. Translated from the Russian manuscript by Boris Kunyavski [Boris È. Kunyavskiĭ]. MR 1634406, DOI 10.1090/mmono/179
- Helmut Wielandt, Primitive Permutationsgruppen vom Grad $2p$, Math. Z. 63 (1956), 478–485 (German). MR 75200, DOI 10.1007/BF01187953
- Helmut Wielandt, Finite permutation groups, Academic Press, New York-London, 1964. Translated from the German by R. Bercov. MR 0183775
- Aiichi Yamasaki, Negative solutions to three-dimensional monomial Noether problem, J. Algebra 370 (2012), 46–78. MR 2966827, DOI 10.1016/j.jalgebra.2012.07.018
Additional Information
- Sumito Hasegawa
- Affiliation: Graduate School of Science and Technology, Niigata University, Niigata 950-2181, Japan
- Email: shasegawa@m.sc.niigata-u.ac.jp
- Akinari Hoshi
- Affiliation: Department of Mathematics, Niigata University, Niigata 950-2181, Japan
- MR Author ID: 714371
- Email: hoshi@math.sc.niigata-u.ac.jp
- Aiichi Yamasaki
- Affiliation: Department of Mathematics, Kyoto University, Kyoto 606-8502, Japan
- MR Author ID: 602892
- Email: aiichi.yamasaki@gmail.com
- Received by editor(s): January 1, 2019
- Received by editor(s) in revised form: April 14, 2019, May 14, 2019, and May 26, 2019
- Published electronically: September 12, 2019
- Additional Notes: This work was partially supported by JSPS KAKENHI Grant Numbers 25400027, 16K05059, 19K03418.
- © Copyright 2019 American Mathematical Society
- Journal: Math. Comp. 89 (2020), 923-940
- MSC (2010): Primary 11E72, 12F20, 13A50, 14E08, 20C10, 20G15
- DOI: https://doi.org/10.1090/mcom/3469
- MathSciNet review: 4044456