Appears in collection : Arithmetic and Diophantine Geometry, via Ergodic Theory and o-minimality

Let k be a field and X a geometrically connected variety over k. The Tate or degeneracy locus of a l-adic local system on X is the etale counterpart of the Hodge locus of a VHS. While in the last decade tremendous progresses have been made in understanding the latter thanks to, in particular, techniques from o-minimality, much less is known about the former. I will review the main conjectures (and mention briefly some applications) about this locus when k is a number field, and explain what we can currently prove. This should include joint works with Jakob Stix and Akio Tamagawa.