Noncommutative Riemann hypothesis

TL;DR

This paper extends the generalized Riemann hypothesis to noncommutative schemes using noncommutative l-adic cohomology, demonstrating invariance under derived equivalences and proving new cases.

Contribution

It introduces a noncommutative version of the generalized Riemann hypothesis and shows its invariance under derived equivalences and homological dualities.

Findings

Invariance of the hypothesis under derived equivalences

Proof of the hypothesis in new noncommutative cases

Extension of classical conjecture to noncommutative geometry

Abstract

In this note, making use of noncommutative $l$-adic cohomology, we extend the generalized Riemann hypothesis from the realm of algebraic geometry to the broad setting of geometric noncommutative schemes in the sense of Orlov. As a first application, we prove that the generalized Riemann hypothesis is invariant under derived equivalences and homological projective duality. As a second application, we prove the noncommutative generalized Riemann hypothesis in some new cases.