  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.
