(Homological Projective Duality)-invariance of the Tate, Beilinson and Riemann conjectures

Projektinformationen

HPD-inv of TBR

ID Finanzhilfevereinbarung: 892994

DOI

10.3030/892994

Projekt abgeschlossen

EK-Unterschriftsdatum 17 April 2020

Startdatum 1 September 2020

Enddatum 31 August 2022

Finanziert unter

• EXCELLENT SCIENCE - Marie Skłodowska-Curie Actions

Gesamtkosten

€ 224 933,76

EU-Beitrag

€ 224 933,76

Koordiniert durch

UNIVERSITY OF WARWICK
United Kingdom

Periodic Reporting for period 1 - HPD-inv of TBR ((Homological Projective Duality)-invariance of the Tate, Beilinson and Riemann conjectures)

Berichtszeitraum: 2020-09-01 bis 2022-08-31

Zusammenfassung vom Kontext und den Gesamtzielen des Projekts

Three of the most important conjectures in mathematics - the Tate conjecture, the Beilinson conjecture and the generalized Riemann hypothesis - concern the location and order of the zeros/poles of the L-functions. These conjectures play a central role in mathematics. For example, in the particular case of an elliptic curve, the Beilinson conjecture reduces to the Birch and Swinnerton-Dyer conjecture, and in the particular case of a point, the generalized Riemann hypothesis reduces to the famous Riemann hypothesis. These are two of the seven Millennium Prize Problems. The overall objective of this project is to prove that the aforementioned conjectures of Tate, Beilinson, and Riemann, are invariant under homological projective duality.

Arbeit, die ab Beginn des Projekts bis zum Ende des durch den Bericht erfassten Berichtszeitraums geleistet wurde, und die wichtigsten bis dahin erzielten Ergebnisse

The researcher proved that the generalized Riemann hypothesis is invariant under homological projective duality in the sense of Kuznetsov. This result was published in the article - Noncommutative Riemann hypothesis, Proceedings of the American Mathematical Society 150 (2022), 2385-2404 - and was presented at several conferences, workshops and seminars. In the case where the base field is a totally imaginary number field, the researcher proved moreover that the Tate and Beilinson conjectures are also invariant under homological projective duality in the sense of Kuznetsov. In order to achieve the aforementioned results the researcher develop, among other tools, a general theory of noncommutative L-functions which is of independent interest.

Fortschritte, die über den aktuellen Stand der Technik hinausgehen und voraussichtliche potenzielle Auswirkungen (einschließlich der bis dato erzielten sozioökonomischen Auswirkungen und weiter gefassten gesellschaftlichen Auswirkungen des Projekts)

The results obtained in this project deepen the current understanding of the Tate, Beilinson and Riemann conjectures. Moreover, they opened the door to many other research directions and applications. Furthermore, it is expected that the results obtained in this project will influence the mathematical community to learn (more) about noncommutative L-functions and to adopt them as part of their toolkit.

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