NTS ABSTRACTSpring2024

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Jan 25

Jason Kountouridis

The monodromy of simple surface singularities in mixed characteristic

Given a smooth proper surface X over a $p$-adic field, it is a generally open problem in arithmetic geometry to relate the mod $p$ reduction of X with the monodromy action of inertia on the $\ell$-adic cohomology of X, the latter viewed as a Galois representation ($\ell \neq p$). In this talk, I will focus on the case of X degenerating to a surface with simple singularities, also known as rational double points. This class of singularities is linked to simple simply-laced Lie algebras, and in turn this allows for a concrete description of the associated local monodromy. Along the way we will see how Springer theory enters the picture, and we will discuss a mixed-characteristic version of some classical results of Artin, Brieskorn and Slodowy on simultaneous resolutions.