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| Format: | Preprint |
| Published: |
2017
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/1703.06089 |
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| _version_ | 1866910449930338304 |
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| author | Barańczuk, Stefan |
| author_facet | Barańczuk, Stefan |
| contents | Consider groups such as Mordell-Weil groups of abelian varieties over number fields, odd algebraic $K$-theory groups of number fields, or finitely generated subgroups of the multiplicative groups of number fields. They are all equipped with systems of reduction maps; thus, one can investigate the Hasse-Minkowski theorem for quadratic forms with coefficients in such groups. In this paper, we prove that the theorem holds for the forms whose rank equals $2$ or $3$, and we demonstrate that it does not hold for higher ranks by providing a counterexample. We also show that our results constitute a generalization of the classic Hasse-Minkowski theorem for binary and ternary integral forms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1703_06089 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | Hasse-Minkowski theorem for quadratic forms on groups Barańczuk, Stefan Number Theory 14G12, 14K15, 11R70, 11R04 Consider groups such as Mordell-Weil groups of abelian varieties over number fields, odd algebraic $K$-theory groups of number fields, or finitely generated subgroups of the multiplicative groups of number fields. They are all equipped with systems of reduction maps; thus, one can investigate the Hasse-Minkowski theorem for quadratic forms with coefficients in such groups. In this paper, we prove that the theorem holds for the forms whose rank equals $2$ or $3$, and we demonstrate that it does not hold for higher ranks by providing a counterexample. We also show that our results constitute a generalization of the classic Hasse-Minkowski theorem for binary and ternary integral forms. |
| title | Hasse-Minkowski theorem for quadratic forms on groups |
| topic | Number Theory 14G12, 14K15, 11R70, 11R04 |
| url | https://arxiv.org/abs/1703.06089 |