What are some topics of advanced number theory every young geometers should know? (soft question)

Question

By "advanced number theory", I mean topics like arithmetic/Diophantine geometry, modular/automorphic forms and Shimura varieties. I'm interested in derived/non-commutative algebraic geometry, some topology and higher category theory as well as string theory related things. According to a professor teaching my K-theory class, I only have to know (for topology) elliptic curves and algebraic number theory, both of which I have learned but nothing beyond them. However, as I'm trying to learn topics like geometric Langlands, which are closely related to modular/automorphic forms and other number theory topics, I've been increasingly aware of the influence of number theory in my field.