The rationality problem for multinorm one tori
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866908780178964480 |
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| author | Hasegawa, Sumito Kanai, Kazuki Oki, Yasuhiro |
| author_facet | Hasegawa, Sumito Kanai, Kazuki Oki, Yasuhiro |
| contents | In this paper, we study the rationality problem for multinorm one tori, a natural generalization of norm one tori. For multinorm one tori that split over finite Galois extensions with nilpotent Galois group, we prove that stable rationality and retract rationality are equivalent, and give a criterion for the validity of the above two conditions. This generalizes the result of Endo (2011) on the rationality problem for norm one tori. To accomplish it, we introduce a generalization of character groups of multinorm one tori. Moreover, we establish systematic reduction methods originating in work of Endo (2001) for an investigation of the rationality problem for arbitrary multinorm one tori. In addition, we provide a new example for which the multinorm principle holds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_04078 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The rationality problem for multinorm one tori Hasegawa, Sumito Kanai, Kazuki Oki, Yasuhiro Algebraic Geometry Number Theory Rings and Algebras 11E72, 12F20, 14E08, 20C10, 20G15 In this paper, we study the rationality problem for multinorm one tori, a natural generalization of norm one tori. For multinorm one tori that split over finite Galois extensions with nilpotent Galois group, we prove that stable rationality and retract rationality are equivalent, and give a criterion for the validity of the above two conditions. This generalizes the result of Endo (2011) on the rationality problem for norm one tori. To accomplish it, we introduce a generalization of character groups of multinorm one tori. Moreover, we establish systematic reduction methods originating in work of Endo (2001) for an investigation of the rationality problem for arbitrary multinorm one tori. In addition, we provide a new example for which the multinorm principle holds. |
| title | The rationality problem for multinorm one tori |
| topic | Algebraic Geometry Number Theory Rings and Algebras 11E72, 12F20, 14E08, 20C10, 20G15 |
| url | https://arxiv.org/abs/2504.04078 |