Mahler measures, elliptic curves and $L$-functions for the free energy of the Ising model

Abstract

This work establishes links between the Ising model and elliptic curves via Mahler measures. First, we reformulate the partition function of the Ising model on the square, triangular and honeycomb lattices in terms of the Mahler measure of a Laurent polynomial whose variety's projective closure defines an elliptic curve. Next, we obtain hypergeometric formulas for the partition functions on the triangular and honeycomb lattices and review the known series for the square lattice. Finally, at specific temperatures we express the free energy in terms of a Hasse-Weil $L$-function of an elliptic curve. At the critical point of the phase transition on all three lattices, we obtain the free energy more simply in terms of a Dirichlet $L$-function. These findings suggest that the connection between statistical mechanics and analytic number theory may run deeper than previously believed.