Multiplicative $f$-ic forms on algebraic varieties arising from Thaine's generalized Jacobi sums
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911652615553024 |
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| author | Hoshi, Akinari Kanai, Kazuki |
| author_facet | Hoshi, Akinari Kanai, Kazuki |
| contents | We study generalized Jacobi sums, cyclotomic numbers, and $d$-compositions in Thaine's framework, and prove new multiplicative identities extending Davenport and Hasse's lifting theorem from the classical prime-power setting to products of prime powers. As applications, we construct multiplicative forms of degree $f\ge2$, i.e. $f$-ic forms, on complete intersections of $f$-ics. This places Pfister's theory of multiplicative quadratic forms over fields within the broader setting of multiplicative $f$-ic forms on affine algebraic varieties, where new phenomena arise. Moreover, a dense open subset $W \subset V$ carries the structure of an algebraic torus, and the multiplicative form is compatible with the induced group law on $W$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_05039 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Multiplicative $f$-ic forms on algebraic varieties arising from Thaine's generalized Jacobi sums Hoshi, Akinari Kanai, Kazuki Number Theory Algebraic Geometry 11D57, 11R18, 11T22, 11T24, 14M10 We study generalized Jacobi sums, cyclotomic numbers, and $d$-compositions in Thaine's framework, and prove new multiplicative identities extending Davenport and Hasse's lifting theorem from the classical prime-power setting to products of prime powers. As applications, we construct multiplicative forms of degree $f\ge2$, i.e. $f$-ic forms, on complete intersections of $f$-ics. This places Pfister's theory of multiplicative quadratic forms over fields within the broader setting of multiplicative $f$-ic forms on affine algebraic varieties, where new phenomena arise. Moreover, a dense open subset $W \subset V$ carries the structure of an algebraic torus, and the multiplicative form is compatible with the induced group law on $W$. |
| title | Multiplicative $f$-ic forms on algebraic varieties arising from Thaine's generalized Jacobi sums |
| topic | Number Theory Algebraic Geometry 11D57, 11R18, 11T22, 11T24, 14M10 |
| url | https://arxiv.org/abs/2605.05039 |