Objective This proposal belongs to the field of arithmetic geometry and lies at the interface of number theory, algebraic geometry and the theory of automorphic forms. More precisely, I aim to use algebraic cycles in the form of motivic sheaves to address fundamental questions about the Langlands program for function fields. This aims at motivic Langlands parametrizations. The Langlands program predicts 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. The recent work of Fargues--Scholze for non-Archimedean local fields uses similar techniques in another context. 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. My project aims to develop new tools to make motivic sheaves amenable for applications in the Langlands program for function fields. In my ongoing joint work with Scholbach, we have successfully implemented key constructions such as IC-Chow groups of shtuka spaces and the motivic Satake equivalence (Math. Ann.) bypassing the use of standard conjectures on algebraic cycles. As in my joint work with Haines, this in particular requires to study the geometry of infinite dimensional varieties such as the affine Grassmannian. Based on these results, I will attack the longstanding open problem on the relation of automorphic forms and motives in the function field case. Fields of science natural sciencesmathematicspure mathematicsarithmeticsnatural sciencesmathematicspure mathematicsgeometrynatural sciencesmathematicspure mathematicsalgebraalgebraic geometry Programme(s) HORIZON.1.1 - European Research Council (ERC) Main Programme Topic(s) ERC-2021-STG - ERC STARTING GRANTS Call for proposal ERC-2021-STG See other projects for this call Funding Scheme HORIZON-AG - HORIZON Action Grant Budget-Based Coordinator TECHNISCHE UNIVERSITAT DARMSTADT Net EU contribution € 1 409 163,00 Address Karolinenplatz 5 64289 Darmstadt Germany See on map Region Hessen Darmstadt Darmstadt, Kreisfreie Stadt Activity type Higher or Secondary Education Establishments Links Contact the organisation Opens in new window Website Opens in new window Participation in EU R&I programmes Opens in new window HORIZON collaboration network Opens in new window Other funding € 0,00
