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

TL;DR

This paper extends classical number theory identities to products of prime powers and constructs multiplicative forms on algebraic varieties, connecting them with Pfister's quadratic forms and algebraic tori.

Contribution

It introduces new multiplicative identities for generalized Jacobi sums and constructs multiplicative $f$-ic forms on algebraic varieties, broadening the scope of Pfister's quadratic form theory.

Findings

Extended Davenport and Hasse's lifting theorem to prime power products.

Constructed multiplicative $f$-ic forms on complete intersections.

Identified algebraic torus structures within the varieties.

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 $W$.