Over the last five decades (at least), the structures and ideas of quantum field theory have been enormously useful in algebraic geometry and topology, generating new directions of research, new techniques, new theories, and new problems of substantial depth. This talk will describe an extension of this influence to include arithmetic geometry. An interplay of ideas similar to those previously observed appear to be on the horizon, leading, in particular, to a fruitful interpretation of fundamental objects in number theory from the point of view of quantum field theory.  No prior knowledge of arithmetic geometry will be assumed.