Noncommutative Weil conjecture

Abstract

In this article, following an insight of Kontsevich, we extend the 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. Moreover, we establish a functional equation for the noncommutative Hasse-Weil zeta functions, compute the l-adic and p-adic absolute values of the eigenvalues of the cyclotomic Frobenius, and provide a complete description of the category of noncommutative numerical motives in terms of Weil q-numbers. As a first application, we prove the noncommutative Weil conjecture and the noncommutative strong form of the Tate conjecture in several cases: twisted schemes, Calabi-Yau dg categories associated to hypersurfaces, noncommutative gluings of schemes, root stacks, (twisted) global orbifolds, and finite-dimensional dg algebras. As a second application, we provide an alternative noncommutative proof of the Weil conjecture and of the strong form of the Tate conjecture in the particular cases of intersections of two quadrics and linear sections of determinantal varieties.