Motives and the Langlands program

Project Information

MotLang

Grant agreement ID: 101040935

DOI

10.3030/101040935

EC signature date 14 March 2022

Start date 1 April 2022

End date 31 March 2027

Funded under

• European Research Council (ERC)

Total cost

€ 1 409 163,00

EU contribution

€ 1 409 163,00

Coordinated by

TECHNISCHE UNIVERSITAT DARMSTADT
Germany

Periodic Reporting for period 1 - MotLang (Motives and the Langlands program)

Reporting period: 2022-04-01 to 2024-09-30

Summary of the context and overall objectives of the project

The project uses algebraic cycles in the form of motivic sheaves to address fundamental questions about the Langlands program for function fields. This program conjectures a decomposition of the space of automorphic forms by motivic Langlands parameters. A breakthrough of V. Lafforgue for function fields with generalizations by Xue, Zhu, Drinfeld and Gaitsgory led to tremendous progress in the construction of l-adic Langlands parameters. This should be viewed as the l-adic realization of the conjectured motivic parametrization. Independently, recent advances for motivic sheaves, as envisioned by Grothendieck and realized by Voevodsky, Ayoub, Cisinski--Déglise, provide powerful techniques to handle algebraic cycles. Despite the immense progress in both areas, hardly any advances have been made specifying the motivic nature of Langlands original prediction. The project aims to formulate and implement motivic Langlands parameterizations. It is divided into three subprojects: (1) Satake equivalence, (2) Drinfeld's lemma and (3) Langlands parametrizations. Subprojects (1) and (2) aim to lay the necessary foundations. Their results are combined in Subprojects (3) towards the overall goal of the project.

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

In the reporting period the activities were focused in reaching the objectives of Subprojects (1) and (2). In Subproject (1), achievements of team members include extensions of the Satake equivalence for motivic sheaves to include constant term functors, twisted groups and versions for the Beilinson--Drinfeld Grassmannian, a classification of normal Schubert varieties in the affine Grassmannian and the Arkhipov-Bezrukavnikov equivalence for twisted groups. In Subproject (2), achievements include the development of the theory of constructible sheaves with condensed coefficients and building on this the categorical Drinfeld's lemma for constructible Weil sheaves as well as Frobenius rigidity for cellular objects in the stable homotopy category providing first instances of a motivic version of Drinfeld's lemma.

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)

The results are expected to be useful in addressing the fundamental problem of constructing motivic Langlands parametrization for function fields and beyond. This would specify and possibly address a long-standing open problem in the areas still lacking a precise formulation.