Periodic Reporting for period 1 - HPD-inv of TBR ((Homological Projective Duality)-invariance of the Tate, Beilinson and Riemann conjectures)
Reporting period: 2020-09-01 to 2022-08-31
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.
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.
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.