I will discuss the following conjecture: an irreducible ℚ̅ℓ-local system L on a smooth complex algebraic variety S arises in cohomology of a family of varieties over S if and only if L can be extended to an etale local system over some descent of S to a finitely generated subfield of complex numbers. I will describe the motivation for this conjecture coming from relative p-adic Hodge theory, known partial results, and possible approaches (not very successful so far) to formulating a purely p-adic (and thus hopefully more tractable) version of this conjecture. A large part of the talk will be expository, including material based on the ideas of Hélène Esnault, Raju Krishnamoorthy, and Josh Lam.
This video is part of the Institute for Advanced Study‘s Special year seminar 2023.
