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

Gandhimohan M. Viswanathan  Department of Physics and National Institute of
Science and Technology of Complex Systems, Universidade Federal do Rio Grande
do Norte, 59078-970 Natal–RN, Brazil

###### 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.

In the last two decades, Mahler measures smyth-book ; zudilin-book  have found
applications across diverse areas of physics, including crystal melting zahabi
, instantons stienstra-instanton , mirror symmetry stienstra-instanton , toric
quiver gauge theories bao2022 ; zahabi , extremal (BPS) black holes zahabi ,
topological string theories bao2022 ; zahabi , networked control systems
chesi2013  and random walks on lattices viswan2017-pre ; zudilin-book ;
guttmann2012 . In statistical mechanics, they play key roles in spanning tree
generating functions, lattice Green functions, and partition functions of
lattice models viswan2017-pre ; guttmann2012 ; wu-of-glasser ; glasser-of-
glasser ; shrock-of-glasser ; glasser-cubic ; stienstra-dimer . Independently
of these advances in physics, mathematicians in the early 1990s discovered a
remarkable connection between Mahler measures and $L$-functions of elliptic
curves boyd1998 ; boyd1981 ; smyth-book ; lalin2007 ; villegas-1999 ; zudilin-
book ; glasser-cubic ; stienstra-dimer ; stienstra-instanton . A natural
question thus arose regarding the possibility of a direct link between
quantities of interest in statistical mechanics on the one hand and
$L$-functions on the other. For example, Stienstra stienstra-dimer  observed
that “there may be some mysterious link between the partition function of a
dimer model and the $L$-function of its spectral curve,” and proceeded to give
a partial answer to this question by finding $L$-functions for several dimer
models. We further investigate this “mysterious link” and show that it
generalizes in an unanticipated and intriguing way to the Ising model —
arguably the most important and widely studied model in statistical mechanics.
Indeed, the findings presented in this work suggest that the connection
extends deeper than previously known or hypothesized viswan2017-pre ;
guttmann2012 ; ising-l-function ; ising-sigma ; ising-elliptic ; viswan2021 ;
viswan-entropy ; baxter .

We begin by expressing the partition function of the Ising model on selected
planar lattices in terms of Mahler measures. This approach yields
hypergeometric formulas for the partition functions. Next, we establish a
connection between the Ising model at special temperatures and the Hasse-Weil
$L$-function of an elliptic curve defined by the projective closure of the
polynomial that appears in the Mahler measure. Finally, at the critical point
of the phase transition, we express the free energy in terms of a Dirichlet
$L$-function. The free energy is a physical quantity, whereas $L$-functions
are number theoretic entities. Why should such dissimilar quantities be
related?

## Mahler measures.

The quantity that would later become known as the Mahler measure was
originally introduced in 1933 by Lehmer lehmer , who was interested in finding
large prime integers zudilin-book . For a monic polynomial, the Mahler measure
is the product of the absolute values of all the roots outside the complex
circle. More generally, let
$p(x)=\sum_{i=0}^{n}a_{i}x^{i}=a_{n}\prod_{i=1}^{n}(x-\alpha_{i})$. Then the
Mahler measure $M(p(x))$ is given by
$M(p(x))=|a_{n}|\prod_{i=1}^{n}\max(1,|\alpha_{i}|).$ The application of
Jensen’s formula leads to the equivalent (and eponymous) analytic definition
due to Mahler mahler1962 , who in the 1960s used it to obtain estimates in
transcendence theory zudilin-book ; mahler-zeta ,
$M(p(x))=\exp\left[\frac{1}{2\pi i}\oint_{|z|=1}{\log|p(z)|}{\mathrm{d}z\over
z}\right].$ The logarithmic Mahler measure is defined as
${\mathrm{m}}(p(x))=\log M(p(x))$. Following convention, ${\mathrm{m}}(p(x))$
is also known as the Mahler measure, with the logarithm implicitly assumed.
Mahler’s analytic definition generalizes to the multivariate case:

${\mathrm{m}}(p(x_{1},\ldots,x_{d}))=\frac{1}{(2\pi
i)^{d}}\\!\\!\\!\underset{{{|z_{1}|=\ldots=|z_{d}|=1}}}{\\!\\!\\!\\!\oint\\!\cdots\\!\oint}{\\!\\!\\!\\!\\!\\!\log|p(z)|}{\mathrm{d}z_{1}\over
z_{1}}\ldots{\mathrm{d}z_{d}\over z_{d}}.$

A remarkable relationship between Mahler measures and $L$-functions, which
inspired significant subsequent research boyd1981 ; boyd1998 ; smyth-book ;
zudilin-book ; lalin2007 ; villegas-1999 ; deninger ; rogers2010 ; g-singular
; m-singular , was discovered by C. J. Smith (and communicated by Boyd
boyd1981 ) in 1981. $L$-functions are of fundamental importance in number
theory because they link arithmetic properties with complex analysis. The best
known $L$-function is the Riemann zeta function
$\zeta(s)=L(\chi_{0}^{(1)},s)$, where $\chi^{(1)}_{0}$ is the principal
character modulo 1. The Dirichlet $L$-function $L(\chi,s)$ of a Dirichlet
character $\chi(n)$ is defined according to

$L(\chi,s)=\sum_{n=1}^{\infty}{\chi(n)\over n^{s}}\leavevmode\nobreak\ .$

For the Ising model, we will need $L(\chi_{-3},s)$ and $L(\chi_{-4},s)$. The
real odd Dirichlet character $\chi_{-f}$ of conductor $f$ can be written using
the Kronecker symbol as $\chi_{-f}(n)=\left({\leavevmode\nobreak\
-f\leavevmode\nobreak\ \over\leavevmode\nobreak\ n\leavevmode\nobreak\
}\right)$. For example, when $f=3$ we have

$\chi_{-3}(n)=\left\\{\begin{array}[]{ll}0&{\rm if\leavevmode\nobreak\
}n=0\mod 3\\\ 1&{\rm if\leavevmode\nobreak\ }n=1\mod 3\\\ -1&{\rm
if\leavevmode\nobreak\ }n=2\mod 3\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ .\\\ \end{array}\right.$

A related $L$-function is the Hasse-Weil $L$-function $L(E,s)$ of an elliptic
curve $E$ over the rationals $\mathbb{Q}$, defined as follows. Let
$\chi(E,p)=1$ if the $E$ has good reduction at prime $p$ and $\chi(E,p)=0$
otherwise. Let $a_{p}=p+1-|E(\mathbb{F}_{p})|$, where $|E(\mathbb{F}_{p})|$ is
the number of points of $E$ modulo $p$. Then

$L(E,s)=\prod_{{\mathrm{primes}\leavevmode\nobreak\
}p}{L_{p}(E,s)}^{-1}=\sum_{n=1}^{\infty}{a_{n}\over n^{s}}\leavevmode\nobreak\
,$

where $L_{p}(E,s)=1-a_{p}p^{-s}+\chi(E,p)p^{1-2s}\leavevmode\nobreak\ .$ In
the 1990s, Deninger deninger  interpreted $\mathrm{m}(P)$ for a Laurent
polynomial $P$ over $\mathbb{Q}$ as a Deligne period of a mixed motive and
conjectured that $m(1)$ could be expressed in terms of $L(E_{15},2)$. Villegas
villegas-1999  studied $\mathrm{m}(P(k))$ as a function of the parameter $k$
and its relation to modular forms and mirror symmetry. Boyd boyd1998  argued
that given a polynomial $P$ in two variables, there should be relations such
as ${\mathrm{m}}(P)=aL(E,2)/\pi^{2}$, where $a\in\mathbb{Q}$.

For our purposes related to the Ising model, we will need the Mahler measures
of two extensively researched Laurent polynomials in the variables
$x,y\in\mathbb{C}$, specifically lalin2007 ; guttmann2012 ; g-singular ,

$\displaystyle P_{1}(x,y)$ $\displaystyle=k+x+1/x+y+1/y\leavevmode\nobreak\ ,$
$\displaystyle P_{2}(x,y)$
$\displaystyle=(x+1)(y+1)({x}+y)-kxy\leavevmode\nobreak\ .$

The projective closures of the equations $P_{1}(x,y)=0$ and $P_{2}(x,y)=0$
define elliptic curves for most values of the free parameter $k\in\mathbb{C}$
m-singular ; g-singular . The exceptions are singular cubic curves when
$k\in\\{0,\pm 4\\}$ for $P_{1}=0$, and similarly $k\in\\{-1,0,8\\}$ for
$P_{2}=0$. The following Mahler measures are conventionally known guttmann2012
as $m$ and $g$:

$\displaystyle m(k)$ $\displaystyle={\mathrm{m}}(P_{1})\leavevmode\nobreak\ ,$
(1) $\displaystyle g(k)$
$\displaystyle={\mathrm{m}}(P_{2})\leavevmode\nobreak\ .$ (2)

Plots are shown in Supplementary Material S1. We will also need below, for the
triangular and honeycomb lattices, the following less well known Mahler
measure:

$t(k)={\mathrm{m}}(k+x+1/x+y+1/y+xy+1/xy)\leavevmode\nobreak\ .$ (3)

There are many known formulas relating $m(k)$ and $g(k)$ to $L$-functions of
elliptic curves for special values of $k$. For our purposes here related to
the Ising model, it will suffice to cite the following examples:

$\displaystyle m(8)$
$\displaystyle={24\over\pi^{2}}L(E_{24},2)\leavevmode\nobreak\ ,$ (4)
$\displaystyle g(-4)$
$\displaystyle={18\over\pi^{2}}L(E_{36},2)\leavevmode\nobreak\ .$ (5)

Here, $E_{24}$ and $E_{36}$ are the elliptic curves of conductor $24$ and $36$
corresponding to $P_{1}=0$ and $P_{2}=0$, respectively. These are Eqs. (27)
and (17) listed by Rogers rogers2010  (or (2.19) and (2.09) in the arXiv
version). Many such formulas have been proven lalin2007 ; g-singular ;
m-singular ; rogers2010 .

Relationships represented by equations such as (4) and (5) are considered deep
because the Mahler measure on the left-hand side is a function only of the
polynomial — not of the elliptic curve that it defines boyd1998 . In contrast,
the $L$-function $L(E,s)$ depends on the elliptic curve itself. See refs.
smyth-book ; zudilin-book ; boyd1998  for details.

We will see below that the critical temperatures of the Ising model on our
chosen lattices are related to the singularities in $m(k)$ and $g(k)$ at $k=4$
and $k=8$ respectively. It is known guttmann2012  that $m(4)$ and $g(8)$ can
be expressed in terms of Dirichlet $L$-functions,

$\displaystyle m(4)$
$\displaystyle={4\over\pi}L(\chi_{-4},2)={4G\over\pi}\leavevmode\nobreak\ ,$
(6) $\displaystyle g(8)$ $\displaystyle={15\sqrt{3}\over
4\pi}L(\chi_{-3},2)\leavevmode\nobreak\ ,$ (7)

where $G$ is the Catalan constant.

Formulas such as (4)–(7) are known only for special values of
$k\in\mathbb{Q}$, whereas hypergeometric formulas are known for arbitrary
$k\in\mathbb{C}$. Let ${}_{p}F_{q}$ denote, as usual, the generalized
hypergeometric function:

${}_{p}F_{q}\left[\begin{array}[]{c}{a_{1},a_{2},\dots,a_{p}}\\\
{b_{1},b_{2},\dots,b_{q}}\end{array};x\right]=\sum_{n=0}^{\infty}{(a_{1})_{n}(a_{2})_{n}\dots(a_{p})_{n}\over(b_{1})_{n}(b_{2})_{n}\dots(b_{q})_{n}}\leavevmode\nobreak\
{x^{n}\over{n!}}.$

The Pochhammer symbol $(x)_{n}={\Gamma(x+n)/\Gamma(x)}$ is the rising
factorial. One can express $m(k)$ and, for $|k|$ sufficiently large, $g(k)$
as,

$\displaystyle m(k)$ $\displaystyle=\Re\left(\log k-{2\over
k^{2}}\leavevmode\nobreak\ {{}_{4}}F_{3}\left[\begin{array}[]{c}{1,1,{3\over
2},{3\over 2}}\\\ {2,2,2}\end{array};{16\over
k^{2}}\right]\right)\leavevmode\nobreak\ ,$ (10) $\displaystyle g(k)$
$\displaystyle=\frac{1}{3}\Re\bigg{\\{}\log\left((4+k)(k-2)^{4}\over
k^{2}\right)$
$\displaystyle\quad\quad-{2k^{2}\over(4+k)^{3}}\leavevmode\nobreak\
_{4}F_{3}\left[\begin{array}[]{c}{1,1,{4\over 3},{5\over 3}}\\\
{2,2,2}\end{array};{27k^{2}\over(4+k)^{3}}\right]$ (13)
$\displaystyle\quad\quad-{8k\over(k-2)^{3}}\leavevmode\nobreak\
_{4}F_{3}\left[\begin{array}[]{c}{1,1,{4\over 3},{5\over 3}}\\\
{2,2,2}\end{array};{27k\over(k-2)^{3}}\right]\bigg{\\}}\leavevmode\nobreak\ .$
(16)

Eq. (10) for $m(k)$ is well known zudilin-book ; guttmann2012 ; villegas-1999
; lalin2007 . It can be obtained by using $\log|z|=\Re(\log z)$, where
$\Re(z)$ is the real part of $z$, expanding the logarithm in the Mercator
series, applying the binomial theorem, and then integrating term by term. The
formula for $g(k)$ can be approached using $q$-series lalin2007 .
Specifically, Eq. (16) can be proven using a modular expansion stienstra-dimer
, see for instance Eq.(2-37) of lalin2007  or Eq. (17) of guttmann2012 . We
did not find in the literature glasser-of-glasser  a hypergeometric formula
for $t(k)$, which we will need below to treat the honeycomb and triangular
lattices. We attempted to calculate $t(k)$ using the Mercator series, however
the resulting expression involves multiple sums that resisted simplification.
See Supplementary Material S2 for the full derivation. In contrast to the
straightforward calculation of $m(k)$, detailed in Supplementary Material S3
for comparison, the brute force method described above is unable to yield the
hypergeometric formula for $t(k)$. As a result, we reach an impasse.

A different, more creative, approach is needed for $t(k)$. The method below
was inspired by ref. guttmann2012 . It is easy to check explicitly that at
each step below any change of variables leaves the Mahler measure invariant:

$\displaystyle t(k)$
$\displaystyle=\mathrm{m}(k+xy+1/(xy)+x/y+y/x+x^{2}+1/x^{2})$
$\displaystyle=\mathrm{m}(k-2+(x+1/x)(y+1/y+x+1/x))$
$\displaystyle={\mathrm{m}}\left(k-2+\left(x^{2}+1/{x^{2}}\right)\left(xy+1/{xy}\right)\left(x/y+y/x\right)\right)$
$\displaystyle={\mathrm{m}}\left(k-2+\left(x^{2}+1/{x^{2}}\right)\left(y+1/{y}\right)\left({x^{2}}/y+y/{x^{2}}\right)\right)$
$\displaystyle={\mathrm{m}}\left((x^{4}+1)(y^{2}+1)({x^{4}}+y^{2})-(2-k)x^{4}y^{2}\right)$
$\displaystyle={\mathrm{m}}\left((x+1)(y+1)({x}+y)-(2-k)xy\right).$ (17)

Finally, compare with (2) and $P_{2}$ to arrive at a key result that opens the
way for further calculations:

$t(k)=g(2-k)\leavevmode\nobreak\ .$ (18)

See Supplementary Material S4 for independent checks.

We will use the above relation between Mahler measures to reformulate the
Ising model on the triangular and honeycomb lattices. The Mahler measure
corresponding to the partition function of the Ising model on the square
lattice was investigated in the 2010s guttmann2012 ; viswan2017-pre , leading
to hypergeometric formulas hucht2011 ; viswan2021 ; viswan2017-pre . However,
even for the square lattice, no connection to $L$-functions via Mahler
measures seems to have been previously recognized. Notably, Mahler measures
were not used in some of the first studies of the links between the Ising
model, elliptic curves and $L$-functions. In 2001, correlations of the Ising
model on the square lattice were studied via $L$-functions of finite graphs
ising-l-function . A few years later it shown that linear differential
operators associated with the susceptibility of the Ising model are associated
with elliptic curves, with an interpretation of complex multiplication for
elliptic curves as (complex) fixed points of generators of the exact
renormalization group ising-sigma ; ising-elliptic . Below we present results
that significantly advance beyond the known results.

## The Ising model

We next use Mahler measures to establish a direct (and unforeseen) link
between the free energy of the Ising model, elliptic curves and $L$-functions.
To this end, we begin by defining the Hamiltonian of the ferromagnetic Ising
model with $N$ classical spins with isotropic nearest-neighbor interactions
and zero external magnetic field as
$H_{N}=-J\sum_{\mbox{$\left\langle\,{ij}\,\right\rangle$}}\sigma_{i}\sigma_{j}$.
Here $J$ is the isotropic coupling, $\sigma_{i}=\pm 1$ and
$\left\langle\,{ij}\,\right\rangle$ the set of all pairs $(i,j)$ of nearest-
neighbor “spins” on the chosen lattice. We restrict our attention below to the
square, triangular and honeycomb lattices, denoted where necessary by the
symbols $\square,\leavevmode\nobreak\ \triangle$ and ⎔ respectively. Let
$\beta=1/(k_{\mbox{\tiny B}}T)$ denote the temperature parameter and $K=\beta
J$ the reduced temperature. The canonical partition function is given by
$Z_{N}(K)=\sum\exp[-\beta H_{N}]$, where the sum is over all possible states.
The per-site free energy $f(K)$ is given by $-\beta f=\log Z$, where $Z$ is
the per-site partition function $Z=\lim_{N\to\infty}Z_{N}^{1/N}$.

The partition functions in the infinite square, triangular and hexagonal
lattices are given by

$\displaystyle\log{Z_{\square}\over 2}$
$\displaystyle=\frac{1}{2\pi^{2}}\iint_{0}^{\pi}\log[\cosh^{2}2K$
$\displaystyle-\sinh 2K(\cos x+\cos y)]\leavevmode\nobreak\
\mathrm{d}x\mathrm{d}y\leavevmode\nobreak\ ,$ (19)
$\displaystyle\log{Z_{\triangle}\over 2}$
$\displaystyle=\frac{1}{8\pi^{2}}\iint_{0}^{2\pi}\ln[\cosh^{3}2K+\sinh^{3}2K$
$\displaystyle-\sinh 2K\leavevmode\nobreak\ (\cos x+\cos
y+\cos(x+y))]\leavevmode\nobreak\ \mathrm{d}x\mathrm{d}y\leavevmode\nobreak\
,$ (20) $\displaystyle\log{Z_{\hexagon}\over 2}$
$\displaystyle=\frac{1}{16\pi^{2}}\iint_{0}^{2\pi}\ln\tfrac{1}{2}[\cosh^{3}2K+1$
$\displaystyle-\sinh^{2}2K\leavevmode\nobreak\ (\cos x+\cos
y+\cos(x+y))]\,\mathrm{d}x\mathrm{d}y\,.$ (21)

Here (19) is Onsager’s solution onsager , while the triangular and honeycomb
lattices were solved in hexagon . The critical temperatures are
$K_{c}^{\square}=\tfrac{1}{2}\log\left(\sqrt{2}+1\right)$,
$K_{c}^{\triangle}=\tfrac{1}{4}\log(3)$ and
$K^{\hexagon}_{c}=\tfrac{1}{2}\log\left(\sqrt{3}+2\right)$.

We next rewrite these partition function in terms of Mahler measures. For the
square lattice, we will use $m(k)$. For the triangular and honeycomb lattices,
we will use $t(k)$ and later apply (18). Noting $Z(0)=2$ for all three
lattices, for nonzero real $K\in\mathbb{R}\backslash\\{0\\}$, we get

$\displaystyle\\!\log{Z_{\square}\over
2}=\\!\frac{1}{2}\left(\log|\tfrac{1}{2}\sinh 2K|+m(2\cosh 2K\coth
2K)\right)\leavevmode\nobreak\ ,$ $\displaystyle\\!\log{Z_{\triangle}\over
2}=\\!\frac{1}{2}\\!\left(\\!\log\\!|\tfrac{1}{2}\sinh
2K|\\!+\\!t\\!\left(\\!\\!-{\\!2\cosh^{3}\\!2K\\!\\!+\\!2\sinh^{3}\\!2K\over\sinh
2K}\right)\\!\\!\right)\\!,$ $\displaystyle\\!\log{Z_{\hexagon}\over
2^{1/2}}=\\!\frac{1}{4}\\!\left(\\!2\log|\sinh
2K|\\!+\\!t\left(\\!-{2\cosh^{3}2K+2\over\sinh^{2}2K}\right)\\!\right)\leavevmode\nobreak\
.$ (22)

Let

$\displaystyle k_{\square}$ $\displaystyle=2\cosh 2K\coth
2K\leavevmode\nobreak\ ,$ (23) $\displaystyle k_{\triangle}$
$\displaystyle=2+{2\cosh^{3}2K+2\sinh^{3}2K\over\sinh 2K}$
$\displaystyle=2\cosh^{2}(2K)(\coth(2K)+1)\leavevmode\nobreak\ ,$ (24)
$\displaystyle k_{\hexagon}$
$\displaystyle=2+{2\cosh^{3}2K+2\over\sinh^{2}2K}=\cosh^{2}(2K)\text{csch}^{2}(K)\leavevmode\nobreak\
.$ (25)

Then from (22) and applying (18) we get

$\displaystyle\log{Z_{\square}\over 2}=\frac{1}{2}\left(\log\tfrac{1}{2}\sinh
2K+m(k_{\square})\right)\leavevmode\nobreak\ ,$
$\displaystyle\log{Z_{\triangle}\over
2}=\frac{1}{2}\left(\log\tfrac{1}{2}\sinh
2K+g(k_{\triangle})\right)\leavevmode\nobreak\ ,$
$\displaystyle\log{Z_{\hexagon}\over 2^{1/2}}=\frac{1}{4}\left(2\log\sinh
2K+g(k_{\hexagon})\right)\leavevmode\nobreak\ .$ (26)

Hypergeometric formulas follow from (10) and (16):

$\displaystyle\ln Z_{\square}=\ln(2\cosh 2K)-{1\over
k_{\square}^{2}}\leavevmode\nobreak\
{{}_{4}}F_{3}\left[\begin{array}[]{c}{1,1,{3\over 2},{3\over 2}}\\\
{2,2,2}\end{array};{16\over k_{\square}^{2}}\right]\leavevmode\nobreak\ ,$
(29) $\displaystyle\ln{Z_{\triangle}\over 2^{1/2}}=\tfrac{1}{2}\log\sinh
2K+\tfrac{1}{6}\log\left((4+k_{\triangle})(k_{\triangle}-2)^{4}\over
k_{\triangle}^{2}\right)$ $\displaystyle\quad-{2k_{\triangle}^{2}\over
2(4+k_{\triangle})^{3}}\leavevmode\nobreak\
_{4}F_{3}\left[\begin{array}[]{c}{1,1,{4\over 3},{5\over 3}}\\\
{2,2,2}\end{array};{27k_{\triangle}^{2}\over(4+k_{\triangle})^{3}}\right]$
(32) $\displaystyle\quad-{8k_{\triangle}\over
2(k_{\triangle}-2)^{3}}\leavevmode\nobreak\
_{4}F_{3}\left[\begin{array}[]{c}{1,1,{4\over 3},{5\over 3}}\\\
{2,2,2}\end{array};{27k_{\triangle}\over(k_{\triangle}-2)^{3}}\right]\leavevmode\nobreak\
,$ (35) $\displaystyle\log{Z_{\hexagon}\over
2^{1/2}}=\tfrac{1}{2}\log\sinh^{2}K+\tfrac{1}{12}\log\left((4+k_{\hexagon})(k_{\hexagon}-2)^{4}\over
k_{\hexagon}^{2}\right)$ $\displaystyle\quad-{k_{\hexagon}^{2}\over
2(4+k_{\hexagon})^{3}}\leavevmode\nobreak\
_{4}F_{3}\left[\begin{array}[]{c}{1,1,{4\over 3},{5\over 3}}\\\
{2,2,2}\end{array};{27k_{\hexagon}^{2}\over(4+k_{\hexagon})^{3}}\right]$ (38)
$\displaystyle\quad-{2k_{\hexagon}\over(k_{\hexagon}-2)^{3}}\leavevmode\nobreak\
_{4}F_{3}\left[\begin{array}[]{c}{1,1,{4\over 3},{5\over 3}}\\\
{2,2,2}\end{array};{27k_{\hexagon}\over(k_{\hexagon}-2)^{3}}\right]\leavevmode\nobreak\
.$ (41)

See Figure 1. At sufficiently negative temperatures, the hypergeometric
expression for $Z_{\triangle}$ fails because $|k_{\triangle}|$ becomes too
small (see Supplementary Material S1). Series (29) is not new, as discussed
below.

Figure 1: Comparison of $-\beta f=\log Z$ for the square, triangular and
honeycomb lattices calculated via numerical integration and via the
hypergeometric formulas.

Next, we obtain a connection with $L$-functions of elliptic curves, using (4)
and (5) as illustrative examples. Let
$K^{\prime}_{\square},\leavevmode\nobreak\ K^{\prime}_{\triangle}$ and
$K^{\prime}_{\hexagon}$ be the temperatures corresponding to $k_{\square}=8$
and $k_{\triangle}=k_{\hexagon}=-4$. The choice $k_{\hexagon}=-4$ does not
correspond to temperatures $K_{\hexagon}\in\mathbb{R}$. Since our goal is to
show the connection with $L$-functions, we can obtain the required complex
valued temperature $K_{\hexagon}\in\mathbb{C}$ via analytic continuation. Only
one of several complex-valued temperatures is shown. For the square lattice,
there are 2 real-valued temperatures corresponding to $k_{\square}=8$, only
one of which we show here for illustration purposes. For the triangular
lattice, the reduced temperature $K=\beta J$ is negative, which can be
achieved for $\beta>0$ by choosing $J<0$. The values thus obtained are

$\displaystyle K^{\prime}_{\square}$
$\displaystyle=\frac{1}{2}\log\left(\sqrt{6}-\sqrt{2\sqrt{6}+5}+2\right)\approx
0.132\leavevmode\nobreak\ ,$ $\displaystyle K^{\prime}_{\triangle}$
$\displaystyle=\frac{1}{4}\log\left(2\sqrt{3}-3\right)\approx-0.19\leavevmode\nobreak\
,$ $\displaystyle K^{\prime}_{\hexagon}$
$\displaystyle=\log\left(-\sqrt{\sqrt{3}-i\sqrt{2\sqrt{3}-3}-1}\right)\approx
2.77i\leavevmode\nobreak\ .$

Substituting these and (4) and (5), into (26), we obtain a direct connection
between the free energy and $L$-functions of elliptic curves:

$\displaystyle-\beta f_{\square}(K^{\prime}_{\square})$
$\displaystyle=\frac{1}{2}\left(\log|\tfrac{1}{2}\sinh
2K_{\square}|+{24\over\pi^{2}}L(E_{24},2)\right)$
$\displaystyle\quad\quad+\log 2\leavevmode\nobreak\ ,$ $\displaystyle-\beta
f_{\triangle}(K^{\prime}_{\triangle})$
$\displaystyle=\frac{1}{2}\left(\log|\tfrac{1}{2}\sinh
2K_{\triangle}|+{18\over\pi^{2}}L(E_{36},2)\right)$
$\displaystyle\quad\quad+\log 2\leavevmode\nobreak\ ,$ $\displaystyle-\beta
f_{\hexagon}(K^{\prime}_{\hexagon})$
$\displaystyle=\frac{1}{4}\left(2\log|\sinh
2K_{\hexagon}|+{18\over\pi^{2}}L(E_{36},2)\right)$
$\displaystyle\quad\quad+\tfrac{1}{2}\log 2\leavevmode\nobreak\ .$ (42)

We finally present our last result, unveiling a connection that might be even
deeper. The positive critical temperatures are given by the singularities at
$k_{\triangle}=k_{\hexagon}=8$ and $k_{\square}=4$. Using (6) and (7) in (26)
and after simplification, we arrive at the critical free energy as Dirichlet
$L$-functions $L(\chi,s)$ at $s=2$:

$\displaystyle-\beta f_{\square}(K_{c}^{\square})$
$\displaystyle=\frac{1}{2}\log 2\leavevmode\nobreak\ +\leavevmode\nobreak\
{2\over\pi}\leavevmode\nobreak\ L(\chi_{-4},2)\leavevmode\nobreak\ ,$
$\displaystyle-\beta f_{\triangle}(K_{c}^{\triangle})$
$\displaystyle=\frac{1}{4}\log{4\over 3}+{15\sqrt{3}\over
8\pi}L(\chi_{-3},2)\leavevmode\nobreak\ ,$ $\displaystyle-\beta
f_{\hexagon}(K_{c}^{\hexagon})$
$\displaystyle=\frac{1}{4}\log{12}+{15\sqrt{3}\over
16\pi}L(\chi_{-3},2)\leavevmode\nobreak\ .$ (43)

Onsager’s critical free energy for the square lattice (see Eq. (121) in
onsager ) is usually written in terms of $G$, whereas in (43) we have
highlighted instead the $L$-function.

## Concluding remarks.

We have shown that Mahler measures can serve as a bridge connecting the Ising
model with elliptic curves and $L$-functions. Eqs. (42) and (43) unveil
unexpected relationships for the free energy. We emphasize that the latter is
a physical quantity, while $L$-functions originate in number theory. They are
fundamentally different — yet intriguingly related. As Poincaré once remarked
poincare , the most valuable relationships are those that “reveal unsuspected
relations between other facts, long since known, but wrongly believed to be
unrelated to each other.” We hope that the findings presented above motivate
others to further explore these and similar relationships.

Other advances include the identity (18) for Mahler measures, as well as the
reformulation (26) in terms of Mahler measures, and similarly the
Hypergeometric formulas (35) and (41) for the triangular and honeycomb
lattices. In contrast, (29) has been known for more than a decade hucht2011 ;
viswan2021 ; viswan2017-pre . It is noteworthy that Glasser and Onsager had
derived a (different) hypergeometric formula in the 1970s that they never
published viswan2021 .

Finally, we discuss the main limitations of this work and related open
problems: (i) We have given only one example of a Hasse-Weil $L$-function for
each of the three lattices, while the list in ref. rogers2010  yields several
others. (ii) Only three lattices have been studied. Can we generalize to other
planar lattices? The $Z$-invariant Ising model baxter ; yang2002  might be a
promising candidate for further study. (iii) What of nonplanar lattices?
Intriguingly, while spanning tree constants on planar lattices are related to
$L$-functions $L(\cdot,s)$ evaluated at $s=2$, it is known guttmann2012  that
on the simple cubic lattice the spanning tree constant can be expressed in
terms of an $L$-function evaluated at $s=3$. Might the free energy of the
critical Ising model on the simple cubic lattice similarly be given by an
$L$-function at $s=3$ ? Such fascinating questions are currently unanswerable
because the 3D Ising model remains unsolved viswan-entropy ; perk1 ; perk2 ;
perk3 ; perk4 . Exploring fresh ideas will be essential for advancing our
understanding of these formidable challenges.

We thank J. H. H. Perk for discussions via e-mail over a period of several
years. This work was supported by CNPq (Grant No. 302414/2022-3).

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