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arXiv · 2605.05039

Multiplicative $f$-ic forms on algebraic varieties arising from Thaine's generalized Jacobi sums

Akinari Hoshi·Kazuki Kanai
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Abstract

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 $W$.

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Akinari Hoshi, Kazuki Kanai. 2026-05-06. Multiplicative $f$-ic forms on algebraic varieties arising from Thaine's generalized Jacobi sums. https://arxiv.org/abs/2605.05039

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