Noncommutative Weil conjecture

TL;DR

This paper extends the Weil and Tate conjectures to noncommutative dg categories, proving them in various cases and providing alternative proofs for classical conjectures using noncommutative geometry techniques.

Contribution

It generalizes fundamental conjectures from algebraic geometry to noncommutative dg categories and offers new proofs and applications in this broader setting.

Findings

Proved noncommutative Weil conjecture for twisted schemes and Calabi-Yau dg categories.

Provided an alternative proof of Weil's original conjecture for specific varieties.

Extended L-functions and related conjectures to noncommutative dg categories.

Abstract

In this article, following an insight of Kontsevich, we extend the famous Weil conjecture (as well as the strong form of the Tate conjecture) from the realm of algebraic geometry to the broad noncommutative setting of dg categories. As a first application, we prove the noncommutative Weil conjecture (and the noncommutative strong form of the Tate conjecture) in the following cases: twisted schemes, Calabi-Yau dg categories associated to hypersurfaces, noncommutative gluings of schemes, root stacks, (twisted) global orbifolds, connective dg algebras, and finite-dimensional dg algebras. As a second application, we provide an alternative noncommutative proof of Weil's original conjecture (which avoids the involved tools used by Deligne) in the cases of intersections of two quadrics and linear sections of determinantal varieties. Finally, we extend also the classical theory of L-functions (as well as the corresponding conjectures of Tate and Beilinson) from the realm of algebraic geometry to the broad noncommutative setting of dg categories. Among other applications, this leads to an alternative noncommutative proof of a celebrated convergence result of Serre.