Some big conjectures in AG/NT:
- The Ash Conjecture : Galois representations coming from torsion cohomology classes of locally symmetric spaces.
- Deligne’s weight monodromy conjecture : Something to do with complete intersections.
What’s the deal with prismatic cohomology?
- Something coming from Bhatt: an integral version of p-adic Hodge theory which is supposed to link a bunch of other cohomology theories.