Motivic Gauss and Jacobi sums

TL;DR

The paper lifts classical Gauss and Jacobi sum relations to the realm of motives by constructing motivic counterparts for these sums via Artin–Schreier and Fermat varieties. It proves key isomorphisms between Artin–Schreier and Fermat motives, and identifies motivic Gauss/Jacobi elements as correspondences that realize Frobenius endomorphisms on these motives. The results yield motivic incarnations of Davenport–Hasse formulas,Base-change and multiplication formulas, and illuminate how motivic invertibility and Frobenius actions encode Weil-number relations. Assuming Beilinson and Tate conjectures, the work deduces structural results about the Picard group of invertible Chow motives and connects these to the group of Weil numbers generated by Jacobi sums, with implications for gamma/beta-type phenomena in the motivic setting.

Abstract

We study the Gauss and Jacobi sums from a viewpoint of motives. We exhibit isomorphisms between Chow motives arising from the Artin-Schreier curve and the Fermat varieties over a finite field, that can be regarded as (and yield a new proof of) classically known relations among Gauss and Jacobi sums such as Davenport-Hasse's multiplication formula. As a key step, we define motivic analogues of the Gauss and Jacobi sums as algebraic correspondences, and show that they represent the Frobenius endomorphisms of such motives. This generalizes Coleman's result for curves. These results are applied to investigate the group of invertible Chow motives with coefficients in a cyclotomic field.

Theorems & Definitions (81)

• Theorem 1.1
• Theorem 1.2
• Remark 2.1
• Lemma 2.2
• Lemma 3.1
• proof
• Example 3.2
• Lemma 3.3
• proof
• Proposition 3.4