A Trace-Path Integral Formula over Function Fields

We show that an arithmetic path integral over the $\mathrm{\ell }$-torsion of a Jacobian $J[\mathrm{\ell }]$ is equal to the trace of the Frobenius action on a representation of the Heisenberg group $H(J[\mathrm{\ell }])$, up to an explicitly determined sign. This is an arithmetic analogue of trace–path integral formulae which arise in quantum field theory, where path integrals over a space of sections of a fibration over a circle can be expressed as the trace of the monodromy action on a Hilbert space.