Characteristic classes of p-adic local systems

Abstract: Given an étale $\mathbb{Z}_p$-local system of rank $n$ on an algebraic variety $X$, continuous cohomology classes of the group $GL_n(\mathbb{Z}_p)$ give rise to classes in (absolute) étale cohomology of the variety. These characteristic classes can be thought of as p-adic analogs of Chern-Simons characteristic classes of vector bundles with a flat connection.

On a smooth projective variety over complex numbers, Chern-Simons classes of all flat bundles are torsion in degrees $>1$ by a theorem of Reznikov. Likewise, $p$-adic characteristic classes on smooth varieties over an algebraically closed field of characteristic zero vanish (at least for $p$ large as compared to the rank of the local system) in degrees $>1$. But for varieties over non-closed fields the characteristic classes of $p$-adic local systems turn out to often be non-zero even rationally. When $X$ is defined over a $p$-adic field, characteristic classes of a $p$-adic local system on it can be partially expressed in terms of Hodge-theoretic invariants of the local system. This relation is established through considering an analog of Chern classes for vector bundles on the pro-étale site of $X$.

This is joint work with Lue Pan.

algebraic geometrynumber theory

Audience: researchers in the topic

MIT number theory seminar

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