A main thoroughfare of representation theory into arithmetic geometry is
through geometric objects called Shimura varieties, which are spaces
attached to a reductive algebraic group over $\textbf{Q}$. These varieties
are often moduli spaces of abelian varieties, which admit solutions that can
be defined over a ring, like $\textbf{Z}$, and not just a field. Number
theorists are very interested in the reduction mod p of these varieties, as
they encode lots of rich algebraic and arithmetic data, but studying these
reductions purely geometrically is a seemingly intractable problem because
the mod p reductions are very singular and are related to hard combinatorial
questions.

However, due to the rich group-theoretic structure underlying Shimura
varieties, we can study the local geometry of the variety. These "local
models" turn out to be certain degenerations of Grassmannians linked to a
simple combinatorial gadget, and the arithmetic theory allows us to
prescribe local coordinates on these that map to the singularities of the
mod-p Shimura variety. This is enough to do things like count points on such
varieties, or determine precisely how bad the singularities are over the
finite field, even if we can’t understand the global geometry. It turns out
that these particular degenerations are ones that appear in the theory of
limit linear series for algebraic curves, and so give these degenerations a
concrete moduli-theoretic interpretation, which turns out to be very useful.
This interpretation also links these degenerations to toric and tropical
geometry.