Height zeta functions package arithmetic counting problems into analytic objects. In their simplest form, they count algebraic numbers, polynomials, or rational points according to height. In this talk I will describe a complementary viewpoint: certain height zeta functions can be interpreted as partition functions for an arithmetic electrostatic system.

The basic observation is that the familiar invariants of polynomials acquire physical interpretations: Mahler measure behaves like an external potential, discriminants encode internal energy and resultants encode pairwise interactions between polynomials. Thus a polynomial, or more generally a Galois orbit, may be viewed as a charged object with internal structure. This analogy turns height zeta functions into arithmetic analogues of statistical-mechanical partition functions.

Our goal is to see how heights, discriminants, resultants, and configurations of conjugate algebraic numbers fit naturally into an electrostatic framework.