The field of L-theory, the algebraic K-theory of quadratic forms, lies at the intersection of algebraic topology and of number theory. My impression is that it is an underpopulated discipline partially because it requires background in fields which most graduate students would think of as being disjoint. I think it is both deep and interesting.

A typical problem would be the calculation of a high-dimensional cobordism group (topological problem). You would show this to be isomorphic to a polynomial extension over the integers, and the actual computation would be to calculate the corresponding L-groups for the corresponding polynomial extensions over the rationals (number theory), and then localize to pass to results over the integers.

As a reference, I would recommend any book by Andrew Ranicki (High Dimensional Knot Theory is very nice, for example). See also this book review.