Tabuada, Gonçalo (2022) Noncommutative Weil conjecture. Advances in Mathematics, 404A . 108385. doi:10.1016/j.aim.2022.108385 ISSN 0001-8708.

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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.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science, Engineering and Medicine > Science > Mathematics
Library of Congress Subject Headings (LCSH):	Weil conjectures, Noncommutative algebras
Journal or Publication Title:	Advances in Mathematics
Publisher:	Elsevier
ISSN:	0001-8708
Official Date:	6 August 2022
Dates:	Date Event 6 August 2022 Published 26 April 2022 Available 25 March 2022 Accepted 1 September 2020 Submitted
Volume:	404A
Number of Pages:	37
Article Number:	108385
DOI:	10.1016/j.aim.2022.108385
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Restricted or Subscription Access
Copyright Holders:	Elsevier
Date of first compliant deposit:	28 March 2022
Date of first compliant Open Access:	26 April 2023
Related URLs:	Publisher
Open Access Version:	ArXiv

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