The $p$-cyclic McKay correspondence via motivic integration

Abstract

We study the McKay correspondence for representations of the cyclic group of order $p$ in characteristic $p$. The main tool is the motivic integration generalized to quotient stacks associated to representations. Our version of the change of variables formula leads to an explicit computation of the stringy invariant of the quotient variety. A consequence is that a crepant resolution of the quotient variety (if any) has topological Euler characteristic $p$ like in the tame case. Also, we link a crepant resolution with a count of Artin-Schreier extensions of the power series field with respect to weights determined by ramification jumps and the representation.

44 pages, v3: The term "strongly Kawamata log terminal" has been changed to "stringily Kawamata log terminal," as it is more consistent with the definition

References in corpus (2)

• Log Crepant Birational Maps and Derived Categories
• Pure subrings of regular local rings, endomorphism rings and Frobenius morphisms

Cited by in corpus (21)

• The wild McKay correspondence and $p$-adic measures
• Mass formulas for local Galois representations and quotient singularities I: a comparison of counting functions
• Toward motivic integration over wild Deligne-Mumford stacks
• Wilder McKay correspondences
• Mass formulas for local Galois representations and quotient singularities II: dualities and resolution of singularities
• Vanishing theorems for threefolds in characteristic $p>5$
• Modular quotient varieties and singularities by the cyclic group of order $2p$
• Motivic Zeta Functions on $\mathds{Q}$-Gorenstein Varieties
• The Wild McKay Correspondence for Cyclic Groups of Prime Power Order
• Densities of rational points and number fields
• Manin's conjecture vs. Malle's conjecture
• Crepant resolutions of quotient varieties in positive characteristics and their Euler characteristics
• Notes on the motivic McKay correspondence for the group scheme ${\alpha }_p$
• Motivic integration over wild Deligne-Mumford stacks
• Discrepancies of $p$-cyclic quotient varieties
• On Convergence of Stringy Motives of Wild $p^n$-Cyclic Quotient Singularities
• The v-function in the wild McKay correspondence is not determined by the ramification filtration
• Crepant resolution of ${\mathbb{A}}^4/A_4$ in characteristic 2
• Moduli of formal torsors
• On the local étale fundamental group of KLT threefold singularities
• Euler characteristic of crepant resolutions of specific modular quotient singularities