Local models of moduli spaces of shtukas

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Arithmetic Algebraic Geometry aims at a better understanding of integer solutions of systems of polynomial equations with integer coefficients. One approach to the study of the solution space X is by studying the solutions modulo a fixed prime number p. The number of these solutions is used to define a function in one complex variable ζ(X, s) where s ∈ C, Re(s) >> 0. These ζ-functions admit conjecturally a meromorphic continuation to the whole complex plane. Favorable properties of these functions carry information on the solution space X, and are closely related with its geometry. By an idea of Langlands, one should be able to express these ζ-functions for certain classes of polynomial equations, the Shimura varieties X, explicitly in terms of automorphic L-functions - thereby proving many favorable properties of these functions. The methods led for example to a proof of Fermat’s Last Theorem by Wiles. In this sense, the following project lies at the interface of Algebraic Number theory, Algebraic Geometry and the theory of Automorphic Forms. Concretely, the project focuses on the local geometry of certain moduli spaces of Drinfeld shtukas, the function field analogs of Shimura varieties, at places of bad reduction and their applications to the local factors of their ζ-functions.Moduli spaces of shtukas were introduced by Drinfeld for the general linear group to study Landlands’ correspondence for function fields. They were generalized by Varshavsky to reductive groups and by Arasteh Rad and Hartl to general smooth affine groups. In a first step of the project, Beilinson-Drinfeld local models are used to study the local geometry of these moduli spaces for Bruhat-Tits groups. In the Shimura case, these models were used by Rapoport in order to apply the Landlands-Kottwitz method for parahoric level, and first result on the local geometry were obtained by Görtz. The novelty of the project is the study of deeper level structures. In a second step of the project, the function given by the semisimple trace of Frobenius on the sheaf of nearby cycles of these models is studied. This function is explicitly determined by the Kottwitz conjecture for parahoric level, and can be used in the Langlands-Kottwitz method. In future projects, these results shall on the one hand contribute to the calculation of local factors of ζ-functions for moduli spaces of shtukas, and on the other hand to a better understanding of analogies with the case of Shimura varieties.