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Shimura varieties are a generalisation of (families of) the modular curve(s) defined over the field of complex numbers.  These varieties can be constructed as quotients of certain hermitian symmetric spaces associated to rational reductive groups. Shimura varieties, as per Deligne's definition (and Langlands' choice of nomenclature), form a moduli space of variation of Hodge structures. Conjecturally, all these Hodge structures come from certain motives. For instance, for those of PEL type (or even Hodge type) the Hodge structures are known to come from abelian varieties, and more generally, for those of abelian type, they come from abelian motives.

Shimura, Miyake and Shih proved the existence of canonical models of these complex varieties (over a number field) for a large class of Shimura varieties. But, using the theory of moduli, Deligne, in his 1971 paper Travaux de Shimura, and his 1979 paper Variétés de Shimura: interprétation modulaire, et techniques de construction de modèles canoniques, proved the existence of canonical models for Shimura varieties of abelian type. The latter will be the main focus of this kleine AG (Milne has an English translation of this paper on his website). Canonical models for general Shimura varieties were later shown to exist, by Borovoi and Milne. This makes Shimura varieties a large source of Galois representations, which is fundamental to the Langlands program. Additionally, Langlands was interested in Shimura varieties because he believed he could use them as a test-bed for his conjecture on the automorphicity of all motivic L-functions.

This kleine AG shall be focused on understanding the aforementioned paper by Deligne. It will be held on the 29th of March, 2025, at the following address: Wegelerstraße 10, 53115 Bonn (please note that the venue is not in Paderborn). If you would like to attend the Kleine AG, please fill out the registration form by clicking this link.

The following table is a schedule for the kleine AG. A more detailed program can be found by clicking here: