Zeta Functions of Heisenberg Graphs over Finite Rings

We investigate Ihara-Selberg zeta functions of Cayley graphs for the Heisenberg group over finite rings ℤ/pⁿℤ, where p is a prime. In order to do this, we must compute the Galois group of the covering obtained by reducing coordinates in ℤ/pⁿ⁺¹ℤ modulo pⁿ⁺¹. The Ihara-Selberg zeta functions of the Heisenberg graph mod pⁿ⁺¹ factor as a product of Artin L-functions corresponding to the irreducible representations of the Galois group of the covering. Emphasis is on graphs of degree four. These zeta functions are compared with zeta functions of finite torus graphs which are Cayley graphs for the abelian groups (ℤ/pⁿℤ)^r.