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Abstract: In this talk, I will talk about the motivic McKay correspondence studied by Batyrev and Denef-Loeser and its generalization to positive and mixed characteristics by myself partly in collaboration with M.M. Wood. The latter relates stringy invariants of singularities to weighted counts of Galois extensions of a local field.
For a finite dimensional linear representation of a finite group, the motivic McKay correspondence says that the motivic stringy invariant of the associated quotient variety is equal to a finite sum of classes of affine spaces in a certain modification of the Grothendieck ring of varieties.
In order to understand wild quotient singularities, which are known to be typical “bad” singularities in positive/mixed characteristics, I started an attempt to generalize the motivic McKay correspondence to positive and mixed characteristics and formulated a conjectural generalization. Here, the finite sum of classes of affine varieties is replaced with a motivic integral over the (conjectural) moduli space of G-covers of formal discs. When the base field (or the residue field if working over a complete DVR) is finite, then the point counting realization of this motivic integral is a weighted count of Galois extensions of the power series field (or the fraction field of the DVR). Such counts were previously studied by number-theorists including Krasner, Serre, Bhargava, Kedlaya and Wood. The point counting version of the McKay correspondence was recently proved by myself. A part of my motivation of this work is a search for a counterexample of resolution of singularities, which has not been successful till now.
If time allows, I will also explain that from this result and a heuristic argument, one can relate Malle’s conjecture on distribution of Galois extensions of number fields and Manin’s conjecture on distribution of rational points on Fano varieties.