A motivic circle method

Project Information

MotivicCircleMethod

Grant agreement ID: 893012

DOI

10.3030/893012

Project terminated on 31 January 2022

EC signature date 16 March 2020

Start date 1 July 2020

End date 24 October 2022

Funded under

• EXCELLENT SCIENCE - Marie Skłodowska-Curie Actions

Total cost

€ 186 167,04

EU contribution

€ 186 167,04

Coordinated by

INSTITUTE OF SCIENCE AND TECHNOLOGY AUSTRIA
Austria

Periodic Reporting for period 1 - MotivicCircleMethod (A motivic circle method)

Reporting period: 2020-07-01 to 2022-06-30

Summary of the context and overall objectives of the project

The Hardy–Littlewood circle method is a well-known technique of analytic number theory that has been used to solve several important number theory problems such as the ternary Goldbach conjecture. It has also benefitted other fields, including quantum computing and algebraic geometry. The aim of the project is to adapt the method to the so-called motivic setting, giving it a more geometric flavour. The proposed applications concern spaces of rational curves on hypersurfaces. These spaces are of great interest in theoretical physics and understanding their structure can shed light on several open questions in algebraic geometry and number theory related to homological stability, weak approximation and equidistribution. Moreover, in the case of cubic fourfolds the motivic circle method is expected to provide a gateway to a series of heuristics on the relationship of these fourfolds with K3 surfaces, and thus possibly allow a new angle of attack for the major open question of the rationality of cubic fourfolds.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

We have achieved a satisfying set up of the motivic circle method, and constructed a new weight function on the Grothendieck ring of varieties with exponentials. Moreover, we have adapted the usual geometry of numbers argument appearing in the classical circle method to the motivic setting, and have used it to prove a general bound on motivic exponential sums. Using the latter, we have obtained bounds on the minor arc contribution, which asymptotically should allow to extract geometric invariants of moduli spaces of rational curves on hypersurfaces. On the other hand, we have managed to analyse the major arcs contribution, identifying in particular a singular series written as a motivic Euler product. Results will be disseminated at the ICMS Workshop in April and at other conferences and seminars throughout the year 2022.

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

Our work shows without doubt that a motivic version of the circle method is possible and will have numerous geometric applications. We have constructed most of the ingredients for making the motivic circle method work, including a new weight function for bounding motivic exponential sums. The first new results on moduli spaces of rational curves are now within our reach, and should be obtained in the months after the termination of the fellowship. The fellowship reached its intended impact, since it led the laureate to a permanent research position at the CNRS. This position will be an excellent place for disseminating the results of the project.

MSCA Fellow Margaret Bilu