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

Technical Summary: Multiplicative f-ic Forms on Algebraic Varieties Arising from Thaine's Generalized Jacobi Sums

Problem Statement
This paper addresses two interconnected themes: the arithmetic properties of generalized Jacobi sums and their geometric realization as multiplicative forms on algebraic varieties.

1. Arithmetic: Classical Jacobi sums and their associated multiplication matrices are well-understood in the context of prime powers ($p^r$). However, extending these structures to products of distinct prime powers ($p_1 p_2$) and establishing multiplicative identities that bridge the gap between classical and generalized settings remains a challenge. Specifically, the authors seek to generalize Davenport and Hasse's lifting theorem from prime-power settings to products of prime powers using Thaine's framework of $d$-compositions.
2. Geometric: While Pfister's theory of multiplicative quadratic forms over fields is classical, the construction of multiplicative forms of higher degree ($f$-ic forms, where $f \ge 2$) on algebraic varieties is less developed. The paper aims to construct such forms on complete intersections and investigate the algebraic group structure (specifically, algebraic tori) that arises on dense open subsets of these varieties.

Methodology
The authors employ a synthesis of algebraic number theory, cyclotomy, and algebraic geometry:

• Thaine's Framework: The study utilizes F. Thaine's generalized Jacobi sums $E_p(a, b)$ and generalized multiplication matrices $H_p$ for elements $p \in K^\times$. These objects are defined axiomatically to satisfy properties analogous to classical Jacobi sums but are applicable over general fields.
• $d$-Composition: A central tool is Thaine's $d$-composition operation ($A \overset{d}{*} B$) on matrices. The authors analyze how this operation combines generalized Jacobi sums and multiplication matrices associated with $p_1$ and $p_2$ to yield objects associated with the product $p_1 p_2$.
• Fourier Analysis: In Section 3.3, the multiplicative identities are interpreted through a "twisted double finite Fourier transform," relating cyclotomic numbers and Jacobi sums via convolution theorems on finite abelian groups.
• Geometric Construction: The arithmetic identities are translated into geometric operations. By defining algebraic varieties $V$ as complete intersections of hypersurfaces derived from correlation sums of coefficients in cyclotomic extensions, the authors construct bilinear maps $\phi_d: V \times V \to V$.
• Norm Relations: The construction relies on relative norm relations in cyclotomic extensions $L/K$. Specifically, the multiplicative forms are derived from the norm map $N_{L/M}$ where $M$ is an intermediate field, linking the algebraic structure of the variety to the field-theoretic properties of the coefficients.

Key Contributions and Results

1. Generalized Multiplicative Identities (Theorem 1.1):
   The authors prove that generalized Jacobi sums for a product $p_1 p_2$ can be constructed via the $d$-composition of sums for $p_1$ and $p_2$. Specifically, if $E_{p_1}$ and $E_{p_2}$ are generalized Jacobi sums, then $E_{p_1 p_2, d}(a, b) := -\sigma_{-d}(E_{p_1}(a, b))E_{p_2}(a, b)$ is a generalized Jacobi sum for $p_1 p_2$. This extends the classical theory where Jacobi sums are intrinsically tied to prime powers, showing that Thaine's generalized sums are multiplicative with respect to products.

2. Extension of Davenport-Hasse Lifting (Theorem 1.3):
   The paper establishes an equivalence between the lifting of Jacobi sums (Davenport-Hasse) and the lifting of multiplication matrices of Gaussian periods. It proves that the $n$-fold product of a multiplication matrix under $(-1)$-composition corresponds to the lifting of Jacobi sums from $\mathbb{F}_{p^r}$ to $\mathbb{F}_{p^{nr}}$.

3. Construction of Multiplicative Quadratic Forms (Theorem 1.4):
   For an odd prime $l$, the authors construct a regular quadratic form $q(x)$ on an algebraic variety $V \subset K^{l-1}$ defined by the intersection of $\frac{l-1}{2}-1$ quadrics. They demonstrate that $q(x)$ is multiplicative on $V$, meaning $q(x)q(y) = q(\phi(x, y))$ for a specific bilinear map $\phi$. This generalizes Hoshi's earlier work on multiplicative quadratic forms.

4. Construction of Multiplicative $f$-ic Forms (Theorem 1.5):
   Generalizing to degree $f \ge 2$, the paper constructs multiplicative $f$-ic forms $h(x)$ on varieties $V$ defined as complete intersections of $e-1$ hypersurfaces of degree $f$ (where $l = ef + 1$). The form $h(x)$ is shown to be regular and multiplicative with respect to the composition law derived from the $d$-composition of coefficients.

5. Algebraic Torus Structure (Theorem 1.6):
   A significant geometric result is the identification of a dense open subset $W \subset V$ (where the norm of the associated element is non-zero) as an algebraic $K$-torus. The composition law $x \overset{-1}{*} y$ restricts to a commutative group law on $W$, making $W$ isomorphic to a norm-defined subtorus of the Weil restriction $R_{L/K}(\mathbb{G}_m)$. This places Pfister's theory of multiplicative quadratic forms within a broader context of multiplicative $f$-ic forms on varieties carrying torus structures.

6. Explicit Lifts of Diophantine Solutions (Section 4.3):
   The authors apply these results to provide explicit formulas for lifting solutions of Diophantine systems associated with Jacobi sums. For prime powers $p_1, p_2$, solutions for $p_1 p_2$ are constructed from solutions for $p_1$ and $p_2$ using the bilinear maps derived from the multiplicative forms. This recovers and generalizes known results for $l=3, 5, 7$ (e.g., relating to Gauss's Disquisitiones Arithmeticae and systems studied by Katre, Rajwade, and van Wamelen).

Significance and Claims
The paper claims to place the theory of multiplicative quadratic forms over fields (Pfister's theory) into a broader setting of multiplicative $f$-ic forms on affine algebraic varieties.

• Arithmetic Significance: It extends the scope of Davenport-Hasse lifting and Jacobi sum identities from prime powers to products of prime powers, utilizing Thaine's $d$-composition to unify these cases.
• Geometric Significance: It demonstrates that the "invertible part" of the coefficient variety obtained by geometrizing Jacobi sum relations naturally carries the structure of an algebraic torus. This provides a geometric mechanism for generating new solutions to Diophantine equations via group laws on these varieties.
• Methodological Contribution: The work bridges the gap between formal arithmetic operations (generalized Jacobi sums) and geometric structures (multiplicative forms on complete intersections), showing that the algebraic torus appearing in the construction is not an independent object but the invertible locus of the variety defined by the generalized relations.

The authors explicitly state that their results extend the study of multiplicative forms initiated by Hoshi [Hos03] from quadratic examples to higher-degree forms and clarify the group-theoretic structure underlying these constructions. They do not claim to solve general Diophantine problems but rather provide a systematic framework for constructing multiplicative forms and lifting solutions in specific cyclotomic contexts.