G-Zips and Tautological Rings of Shimura Varieties

Abstract: I will discuss a conjecture describing the tautological ring of a Shimura variety and present recent progress towards the conjecture. This is the subject of joint work with W. Goldring. The tautological ring is the subring of the Chow ring generated by all Chern classes of all automorphic vector bundles. It has been completely described for the Siegel modular variety by work of van der Geer. An important feature is that more is known in positive characteristic than in characteristic zero. This is because the mod p special fibre of the Shimura variety admits a "zip period map" into the stack of G-Zips, which is a mod p analogue of the classical period map in Hodge theory. I will explain how the theory of G-Zips can be used to deduce significant information about the tautological ring.

number theory

Audience: researchers in the topic

London number theory seminar

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