In a topological quantum field theory, path integrals can often be expressed instead as the trace of a monodromy action on a Hilbert space.

In this talk I will discuss an arithmetic analogue of this phenomena for function fields, where the phase space is replaced with the \ell-torsion points of the Jacobian of a curve over a finite field, the path integral is replaced with a sum over the points of J[\ell], and the monodromy is instead replaced with the Frobenius action. Time permitting, I will also briefly outline the proof of the arithmetic trace-path integral formula.