A motivic circle method

Projektinformationen

MotivicCircleMethod

ID Finanzhilfevereinbarung: 893012

DOI

10.3030/893012

Das Projekt endete am 31 Januar 2022

EK-Unterschriftsdatum 16 März 2020

Startdatum 1 Juli 2020

Enddatum 24 Oktober 2022

Finanziert unter

• EXCELLENT SCIENCE - Marie Skłodowska-Curie Actions

Gesamtkosten

€ 186 167,04

EU-Beitrag

€ 186 167,04

Koordiniert durch

INSTITUTE OF SCIENCE AND TECHNOLOGY AUSTRIA
Austria

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

Berichtszeitraum: 2020-07-01 bis 2022-06-30

Zusammenfassung vom Kontext und den Gesamtzielen des Projekts

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.

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

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.

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)

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