Toroidal compactifications and incompressibility of exceptional congruence covers.

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• Patrick Brosnan (University of Maryland, College Park; University of Maryland, College Park)
• Monday 20 January 2020, 15:00-16:00
• Seminar Room 2, Newton Institute.

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KAH - K-theory, algebraic cycles and motivic homotopy theory

Suppose a finite group G acts faithfully on an irreducible variety X. We say that the G-variety X is compressible if there is a dominant rational morphism from X to a faithful G-variety Y of strictly smaller dimension. Otherwise we say that X is incompressible. In a recent preprint, Farb, Kisin and Wolfson (FKW) have proved the incompressibility of a large class of covers related to the moduli space of principally polarized abelian varieties with level structure. Their arithmetic methods, which use Serre-Tate coordinates in an ingenious way, apply to diverse examples such as moduli spaces of curves and many Shimura varieties of Hodge type. My talk will be about joint work with Fakhruddin and Reichstein, where our goal is to recover some of the results of FKW via the fixed point method from the theory of essential dimension. More specifically, we prove incompressibility for some Shimura varieties by proving the existence of fixed points of finite abelian subgroups of G in smooth compactifications. Our results are weaker than the results of FKW for Hodge type Shimura varieties, because the methods of FKW apply in cases where there is no boundary, while we need a nonempty boundary to find fixed points. However, our method has the advantage of extending to many Shimura varieties which are not of Hodge type, in particular, those associated to groups of type E7. Moreover, by using Pink's extension of the Ash, Mumford, Rapoport and Tai theory of toroidal compactifications to mixed Shimura varieties, we are able to prove incompressibility for congruence covers corresponding to certain universal families: e.g., the universal families of principally polarized abelian varieties.

This talk is part of the Isaac Newton Institute Seminar Series series.

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