The Poisson summation formula for finite Abelian groups equates the sum over the subgroup H of G to the sum over its dual(H)=H#. This is used when when one sum is much larger than the other.
For a torus, it is a relation between the eigenvalues of the Laplacian and the lengths of closed geodesics.
I think I read this
This is explained in Audrey Terras's books (vol 1 and 2) on Harmonic Analysis.
She also applies the formula to the Ising model in Statistical Mechanics, Boolean switching functions and Macwilliams identities (the last has already been pointed out earlier by Robin Chapman above)
Stein and Shakarchi in their book Fourier Analysis apply the formula to relate Poisson kernels of the disc and the upper half plane, and the heat kernels of R/Z and R
These applications are in addition to those listed on wikipedia