# Fermat’s Last Theorem over $\mathbb{Q}(\sqrt{2},\sqrt{3})$

Maleeha Khawaja  School of Mathematics and Statistics, Hicks Building,
University of Sheffield, Sheffield S3 7RH, United Kingdom
<EMAIL_ADDRESS>and  Frazer Jarvis  School of Mathematics and
Statistics, Hicks Building, University of Sheffield, Sheffield S3 7RH, United
Kingdom<EMAIL_ADDRESS>
###### Abstract.

In this paper, we begin the study of the Fermat equation over real biquadratic
fields, taking $\mathbb{Q}(\sqrt{2},\sqrt{3})$ as a first example. To reduce
the existing asymptotic bound for $\mathbb{Q}(\sqrt{2},\sqrt{3})$, we
determine the torsion primes of degree 8 using work of Derickx-Kamienny-Stein-
Stoll [7]. We treat the smaller prime exponents by studying the modular
parametrisation and some smaller integer exponents by studying certain
hyperelliptic curves.

_Dedicated to Iffat (Zaman) Khawaja_

_January 1936 – January 2022_

##  1\. Introduction

Since the groundbreaking work of Wiles [40] on the Fermat equation over
$\mathbb{Q}$, the Fermat equation has been studied extensively over various
number fields. Let $K$ be a number field and let $n\geq 3$ be an integer. The
Fermat equation over $K$ with exponent $n$ is the equation

(1) $x^{n}+y^{n}=z^{n},\qquad x,y,z\in K.$

We say a solution $(a,b,c)$ to (1) over $K$ is trivial if $abc=0$ and non-
trivial otherwise.

In [40], Wiles showed there are no non-trivial solutions to (1) over
$\mathbb{Q}$ for $n\geq 3$. This method of attacking (1) (and Diophantine
equations in general) became known as the _modular approach_. Thereafter,
Jarvis-Meekin [21] extended this work to show there are no non-trivial
solutions to (1) over $\mathbb{Q}(\sqrt{2})$ for $n\geq 4$. This was followed
by work of Freitas-Siksek [13], [14] who established a framework of how to
approach (1) (and, in general, Diophantine equations) over totally real number
fields. In [13], Freitas-Siksek show there are no non-trivial solutions to (1)
over $\mathbb{Q}(\sqrt{d})$ where $d\neq 5,17$ is a square free integer $3\leq
d\leq 23$ for $n\geq 4$. When approaching real quadratic fields of a larger
discriminant, they encountered the obstacle of demonstrating the
irreducibility of certain Galois representations and eliminating the number of
Hilbert newforms that arose as a result of level-lowering. In [29], Michaud-
Jacobs worked around these obstacles by studying quadratic points on certain
modular curves and working directly with Hecke operators. He proved, for most
squarefree $d$ in the range $26\leq d\leq 97$, there are no non-trivial
solutions to (1) over $\mathbb{Q}(\sqrt{d})$ for $n\geq 4$. It is a natural
problem then to study (1) over real biquadratic fields. In a slight divergence
from the modular approach, in [14], Freitas-Siksek initiated the study of
looking at (1) “asymptotically”. As in [14], we say Asymptotic Fermat’s Last
Theorem holds over $K$ if there is a constant $B_{K}$ such that there are no
non-trivial solutions to (1) over $K$ for prime $p>B_{K}$. The authors of [11]
study the solutions to certain $S$-unit equations to show that Asymptotic
Fermat’s Last Theorem holds for several families of number fields – including
some real biquadratic fields. In this paper, we will prove the following
result and discuss some obstacles that arise over more general real
biquadratic fields.

######  Theorem 1.1.

Let $K=\mathbb{Q}(\sqrt{2},\sqrt{3})$. There are no non-trivial solutions to
(1) over $K$ for $n\geq 4$.

In Section 2, we apply and give a brief overview of the modular approach found
in [13] and [14]. In Section 3, we reduce the existing asymptotic bound, for
$\mathbb{Q}(\sqrt{2},\sqrt{3})$, found in the literature. We also compute the
set of torsion primes of degree 8 by extending work of Derickx-Kamienny-Stein-
Stoll [7]. In Section 4, we prove the irreducibility of $\bar{\rho}_{E,p}$ for
a number of small primes $p$, in part, by studying the explicit modular
parameterisation. In Section 5, we treat the smaller integer exponents. To
treat $n=9$ and $n=6$, we study the hyperelliptic curves obtained from the
Fermat curve of degree $n$. To treat $n=4$, we extend work of Mordell [30]. In
Section 6, we give a brief overview of some obstacles that arise when
extending our method to more general real biquadratic fields.

## acknowledgements

We would like to thank Samir Siksek for several useful and enlightening
discussions, particularly with regard to the contents of Section 4 and the
proof of Theorem 5.1. We have also benefitted from helpful correspondence with
Philippe Michaud-Jacobs, Jeremy Rouse and Michael Stoll. We thank the authors,
Yasuhiro Ishitsuka, Tetsushi Ito and Tatsuya Ohshita, of [18] for making us
aware of their work determining all points on the Fermat quartic lying in a
quadratic extension of $\mathbb{Q}(\zeta_{8})$. The first author would like to
thank the University of Sheffield for their financial support via a doctoral
training partnership scholarship111ESPRC grant no EP/T517835/1..

##  2\. The modular approach

Let $K$ be a totally real field (until otherwise specified) and let
$\mathcal{O}_{K}$ denote its ring of integers. Let $p\geq 5$ be a prime.
Suppose $(a,b,c)$ is a non-trivial solution to (1) over $K$ with exponent $p$.
We can associate an elliptic curve $E$ over $K$ (the _Frey curve_) to
$(a,b,c)$:

(2) $E:y^{2}=x(x-a^{p})(x+b^{p}).$

The discriminant of $E$ is given by $\Delta=16(abc)^{2p}$. We denote by
$\bar{\rho}_{E,p}$ the mod $p$ Galois representation associated to $E$.

The following theorem, stated in [14], is formulated from the combination of
the works of Fujiwara [15], Jarvis [19], [20] and Rajaei [34].

######  Theorem 2.1.

(Freitas-Siksek [14, Theorem 7])
Let $K$ be a totally real field and $p\geq 5$ be a prime. Suppose
$K\nsubseteq\mathbb{Q}(\zeta_{p})^{+}$. Let $E$ be an elliptic curve over $K$
with conductor $\mathcal{N}$. Suppose $E$ is modular and $\bar{\rho}_{E,p}$ is
irreducible. Denote by $\Delta_{\mathfrak{q}}$ the discriminant for a local
minimal model222This can be calculated using Tate’s Algorithm [36, Pages
364-368]. of $E$ at a prime ideal $\mathfrak{q}$ of $K$. Suppose the following
conditions are satisfied for all prime ideals $\mathfrak{q}$ that lie over
$p$:

$\displaystyle i)\hskip 5.69054ptE\text{ is semistable at }\mathfrak{q};\hskip
14.22636ptii)\hskip 5.69054ptp\mid v_{\mathfrak{q}}(\Delta_{\mathfrak{q}});$
$\displaystyle iii)\text{ the ramification index satisfies
}e(\mathfrak{q}/p)<p-1.$

Denote by $\mathcal{M}_{p}$ and $\mathcal{N}_{p}$ the following two quantities
associated to $p$:

$\mathcal{M}_{p}:=\prod_{\begin{subarray}{c}\mathfrak{q}\|\mathcal{N},\\\
p|v_{\mathfrak{q}}(\Delta_{\mathfrak{q}})\end{subarray}}\mathfrak{q},\hskip
14.22636pt\mathcal{N}_{p}:=\dfrac{\mathcal{N}}{M_{p}}.$

Then, $\bar{\rho}_{E,p}\sim\bar{\rho}_{\mathfrak{f},\bar{\omega}}$, where
$\mathfrak{f}$ is a Hilbert newform of level $\mathcal{N}_{p}$ and parallel
weight $2$ and $\bar{\omega}$ is a prime ideal of $\mathbb{Q}_{\mathfrak{f}}$
that lies above $p$.

We will apply Theorem 2.1 to the Frey curve (2) in order to contradict the
existence of the putative solution $(a,b,c)$.

Several advances have been made in the direction of establishing the
modularity of elliptic curves over totally real number fields. For example,
the modularity of elliptic curves over real quadratic fields [10] and totally
real cubic fields [8] has been established. Moreover, thanks to the following
result of Box, we now know elliptic curves over most totally real quartic
fields are modular.

######  Theorem 2.2.

(Box [4, Theorem 1.1])
Let $K$ be a totally real quartic field not containing $\sqrt{5}$. Every
elliptic curve over $K$ is modular.

We turn to the question of how to show conditions $i)$ and $ii)$ of Theorem
2.1 are satisfied. Let $\mathcal{H}=Cl(K)/Cl(K)^{2}$ where $Cl(K)$ denotes the
class group of $K$. We can assume, without loss of generality, that any non-
trivial solution $(a,b,c)$ to (1) is integral. By Lemma 3.3 of [13], $a,b,c$
are coprime away from a small set of primes, i.e.,
$gcd(a,b,c)=\mathfrak{m}\cdot\tau^{2}$ for some $\mathfrak{m}\in\mathcal{H}$
and odd prime ideal $\tau\neq\mathfrak{m}$. Consider now the following result
of [14] which addresses conditions $i)$ and $ii)$ above.

######  Lemma 2.1.

(Freitas-Siksek [13, Lemma 3.3])
Let $S$ denote the set of primes above $2$. Let $\mathfrak{q}$ be a prime
ideal of $K$ such that $\mathfrak{q}\notin S\cup\\{\mathfrak{m}\\}$. Then $E$
is semistable at $\mathfrak{q}$ and
$p|v_{\mathfrak{q}}(\Delta_{\mathfrak{q}})$.

Let $K=\mathbb{Q}(\sqrt{2},\sqrt{3})$, $p\geq 5$ prime. Suppose there is a
non-trivial solution $(a,b,c)$ to (1) over $K$ with exponent $p$. Let $E$ be
the Frey curve (2) associated to this solution. By Theorem 2.2, $E$ is
modular. We note that the class number of $K$ is 1 and thus
$\mathfrak{m}=1\cdot\mathcal{O}_{K}$. Suppose $\mathfrak{q}\mid p$ is a prime
ideal of $K$. By Lemma 2.1, assumptions $i)$ and $ii)$ of Theorem 2.1 are
satisfied for $\mathfrak{q}$. In particular, $E$ is semistable away from 2. We
note that $2\mathcal{O}_{K}=\mathfrak{p}^{4}$. We use Tate’s Algorithm [36,
pages 364-368] to see that $E$ is semistable at $\mathfrak{p}$, for $p\geq
19$.

######  Lemma 2.2.

Let $K=\mathbb{Q}(\sqrt{2},\sqrt{3})$. For prime $p\geq 19$, the Frey curve
$E$ is semistable over $K$.

###### Proof.

Let $p\geq 19$ be a prime. Since
$\mathcal{O}_{K}/\mathfrak{p}\cong\mathds{F}_{2}$, we can suppose that

$\mathfrak{p}\mid a,\qquad\mathfrak{p}\nmid b,c.$

After the completion of four cycles of Tate’s algorithm, we arrive at the
model of $E$ with Weierstrass equation

$W:y^{2}=x^{3}+\frac{1}{4}(b^{p}-a^{p})-\frac{1}{16}a^{p}b^{p}$

and discriminant $\Delta_{\mathfrak{p}}=2^{-8}(abc)^{2p}$. Note that, with the
usual notation for elliptic curves,

$\mathfrak{p}\nmid b_{2}=a_{1}^{2}+4a_{2}=b^{p}-a^{p}.$

It then follows from Step 2 of Tate’s algorithm that $E$ is semistable at
$\mathfrak{p}$ and $W$ is the minimal Weierstrass equation for $E$. Thus $E$
is semistable by the remarks preceding the statement of the lemma. ∎

Thus, for $p\geq 19$, to show that the assumptions of Theorem 2.1 are
satisfied it remains to show that $\bar{\rho}_{E,p}$ is irreducible.

##  3\. bounds

Throughout this section, unless otherwise stated, let
$K=\mathbb{Q}(\sqrt{2},\sqrt{3})$. As we saw in the last section, the main
obstacle in proving Fermat’s Last Theorem over $K$ is proving that
$\bar{\rho}_{E,p}$ is irreducible. This is where the bound on the exponent
comes from. In this section, we will reduce the existing asymptotic bound
given in the literature to a smaller one.

In recent work [11], Freitas-Kraus-Siksek lay down a criterion for
establishing Asymptotic Fermat’s Last Theorem for certain number fields and
show this criterion is satisfied for several number fields. In particular,
they show Asymptotic Fermat’s Last Theorem holds for the following family of
real biquadratic fields.

######  Theorem 3.1.

(Freitas-Kraus-Siksek, [11])
The effective Asymptotic Fermat’s Last Theorem holds over
$\mathbb{Q}(\sqrt{2},\sqrt{\ell})$ for prime $\ell\equiv 3$ (mod 8).

As explained in [11], the effectivity of the bound in this case arises from
the modularity of the Frey curve (2). The effective bound can be computed
explicitly from the following result of Freitas-Siksek.

######  Theorem 3.2.

(Freitas-Siksek [12, Theorem 2])
Let $K$ be a totally real Galois field of degree $d_{K}$ and class number
$h_{K}$. Let $B_{K}$ be as defined in [12] and let $p$ be an unramified prime
in $K$ such that $p\nmid B_{K}$ and $p\geq 17$ or $p=11$. Suppose $E$ is an
elliptic curve over $K$. If $E$ is semistable at all primes above $p$ and
$\bar{\rho}_{E,p}$ is reducible then

$p<(1+3^{6dh_{K}})^{2}.$

We emphasise that the constant $B_{K}$ is effectively computable and depends
only on $K$ but omit its definition here. This gives us our first bound.

######  Bound 1.

Let $K=\mathbb{Q}(\sqrt{2},\sqrt{3})$. There are no non-trivial solutions to
(1) over $K$ for primes

$p>(1+3^{24})^{2}\approx 8\times 10^{22}.$

###### Proof.

Suppose there is a non-trivial solution $(a,b,c)$ to (1) over $K$. Let $E$ be
the Frey curve (2) associated to this solution. We recall that, by the
discussion at the end of Section 2, it remains to demonstrate the
irreducibility of $\bar{\rho}_{E,p}$. Since $d_{K}=4$ and $h_{K}=1$, by
Theorem 3.2, $\bar{\rho}_{E,p}$ is irreducible for $p>(1+3^{24})^{2}$. That
is, all the assumptions of Theorem 2.1 are satisfied for $p>(1+3^{24})^{2}$.
We turn to finding the lowered level $\mathcal{N}_{p}$. If $\mathfrak{q}\nmid
2$ then $p\mid v_{\mathfrak{q}}(\Delta_{\mathfrak{q}})$, i.e.,
$\mathfrak{q}\mid\mathcal{M}_{p}$. Recall that
$2\mathcal{O}_{K}=\mathfrak{p}^{4}$, where $\mathfrak{p}$ is a prime ideal.
From the proof of Lemma 2.2, we see that

$v_{\mathfrak{p}}(\Delta_{\mathfrak{p}})=2pv_{\mathfrak{p}}(abc)-32.$

In particular

$p\nmid v_{\mathfrak{p}}(\Delta_{\mathfrak{p}}).$

It follows that $\mathcal{N}_{p}=\mathfrak{p}$. By Theorem 2.1,
$\bar{\rho}_{E,p}\sim\bar{\rho}_{\mathfrak{f},\bar{\omega}}$ where
$\mathfrak{f}$ is a Hilbert newform of parallel weight 2 and level
$\mathfrak{p}$. This space of Hilbert modular forms has dimension 0, i.e., we
have reached a contradiction. ∎

To reduce the bound further, we appeal to the semistability of the Frey curve
(2) and the following result of Kraus.

######  Lemma 3.1.

(Kraus [25])
Let $K$ be a number field with degree $d_{K}$, narrow class number $h_{K}^{+}$
and group of totally positive units $U_{K}^{+}$. Let $p$ be an unramified
prime in $K$ and $S_{p}$ the set of primes above $p$. Let $E$ be a semistable
elliptic curve over $K$ and suppose $\bar{\rho}_{E,p}$ is reducible. Then
there is a subset $S$ of $S_{p}$ such that the following is true.
(i) If $S$ is non-empty then for every $u\in U_{K}^{+}$

(3) $\prod_{\mathfrak{p}\in S}N_{\mathfrak{p}}(u+\mathfrak{p})=1,$

where $N_{\mathfrak{p}}$ is the morphism
$N_{\mathfrak{p}}:\left(\mathcal{O}_{K}/\mathfrak{p}\right)^{\ast}\rightarrow\mathds{F}_{p}^{\ast}$.
(ii) If $S$ is empty then $E$ has an $L$-rational point of order $p$, where
$L$ is a fixed finite extension of $K$ of degree

$[L:\mathbb{Q}]\leq h_{K}^{+}d_{K}.$

We note that if $\mathfrak{p}\mid p$ then
$(\mathcal{O}_{K}/\mathfrak{p})^{\ast}=\mathds{F}_{p^{n}}^{\ast}$ for some
integer $n\geq 1$. If $u\in\mathds{F}_{p}^{\ast}$ then

$N_{\mathfrak{p}}(u)\coloneqq u^{n}.$

This partial definition of $N_{\mathfrak{p}}$ is sufficient for our purposes.
We use this strategy to obtain our second bound.

######  Bound 2.

Let $K=\mathbb{Q}(\sqrt{2},\sqrt{3})$. There are no non-trivial solutions to
(1) over $K$ for prime

$p>(1+3^{4})^{2}=6724.$

###### Proof.

We adopt the notation of Lemma 3.1. Let $p\geq 19$ be a prime,
$\mathfrak{p}\in S_{p}$. By Lemma 2.2, the Frey curve $E$ is semistable.
Suppose $\bar{\rho}_{E,p}$ is reducible. We note that 2 and 3 are the only
primes that ramify in $K$. Thus, either $p=\mathfrak{p}_{1}\mathfrak{p}_{2}$
or $p=\mathfrak{p}_{1}\mathfrak{p}_{2}\mathfrak{p}_{3}\mathfrak{p}_{4}$. We
shall find units $u$ contradicting (3), so that $S$ must be empty, and showing
that there is therefore a point on the Frey curve $E$ over a number field of
degree at most $8$.

$(a)$ First suppose $p=\mathfrak{p}_{1}\mathfrak{p}_{2}$. Since
$(\mathcal{O}_{K}/\mathfrak{p})^{\ast}=\mathds{F}_{p^{2}}^{\ast}$, we have
$N_{\mathfrak{p}}:\mathds{F}_{p^{2}}^{\ast}\rightarrow\mathds{F}_{p}^{\ast}$.
We consider three cases:

$i)\left(\frac{2}{p}\right)=1;\qquad ii)\left(\frac{3}{p}\right)=1;\qquad
iii)\left(\frac{2}{p}\right)=\left(\frac{3}{p}\right)=-1.$

In each case, we will choose a unit $u\in U_{K}^{+}$ accordingly so that (3)
fails to hold. For ease, we will choose $u$ such that
$u\in\mathds{F}_{p}^{\ast}$.

$i)$ Let $u=3-2\sqrt{2}$. Since $\left(\frac{2}{p}\right)=1$, 2 has a square
root in $\mathds{F}_{p}^{\ast}$ and therefore $u\in\mathds{F}_{p}^{*}$. Thus,
by the definition of $N_{\mathfrak{p}}$ given above:

$N_{\mathfrak{p}}(u)=u^{[\mathds{F}_{p^{2}}^{\ast}:\mathds{F}_{p}^{\ast}]}=u^{2}.$

If $S$ is non-empty then either $\\#S=1$ or $\\#S=2$. If $\\#S=1$ then, by
Lemma 3.1, $N_{\mathfrak{p}}(u)=1$. In other words,

$p\mid(u^{2}-1)=-2^{2}\sqrt{2}u.$

This can only happen for $p=2$ – we recall that $p\neq 2,3$. This leaves the
possibility that $S=\\{\mathfrak{p}_{1},\mathfrak{p}_{2}\\}$. Since we have
discounted the possibility of $N_{\mathfrak{p}}(u)=1$, it follows that
$N_{\mathfrak{p}_{1}}(u)=N_{\mathfrak{p}_{2}}(u)=-1$. In other words

$p\mid(u^{2}+1)=2\cdot 3\cdot u.$

Again, this is not possible as $p\neq 2,3$. This is summarised in the table
below.

$N_{\mathfrak{p}}(u)$ | $u^{2}-1$ | $u^{2}+1$
---|---|---
$u^{2}$ | $-2^{2}\sqrt{2}\cdot u$ | $2\cdot 3\cdot u$

$ii)$ Let $u=2+\sqrt{3}$. Similarly, since $\left(\frac{3}{p}\right)=1$,
$N_{\mathfrak{p}}(u)=u^{2}$. The table below shows that this is possible only
if $p\in\\{2,3\\}$.

$N_{\mathfrak{p}}(u)$ | $u^{2}-1$ | $u^{2}+1$
---|---|---
$u^{2}$ | $2\sqrt{3}\cdot u$ | $2^{2}\cdot u$

$iii)$ Let $u=5-2\sqrt{6}$. Similarly, since $\left(\frac{6}{p}\right)=1$,
$N_{\mathfrak{p}}(u)=u^{2}$. The table below shows that this is possible only
if $p\in\\{2,3,5\\}$.

$N_{\mathfrak{p}}(u)$ | $u^{2}-1$ | $u^{2}+1$
---|---|---
$u^{2}$ | $-2^{2}\sqrt{6}\cdot u$ | $2\cdot 5\cdot u$

$(b)$ Now suppose
$p=\mathfrak{p}_{1}\mathfrak{p}_{2}\mathfrak{p}_{3}\mathfrak{p}_{4}$. Since
$(\mathcal{O}_{K}/\mathfrak{p})^{\ast}=\mathds{F}_{p}^{\ast}$, we have
$N_{\mathfrak{p}}:\mathds{F}_{p}^{\ast}\rightarrow\mathds{F}_{p}^{\ast}$. We
consider the same three cases as in $(a)$:

$i)\left(\frac{2}{p}\right)=1;\qquad ii)\left(\frac{3}{p}\right)=1;\qquad
iii)\left(\frac{2}{p}\right)=\left(\frac{3}{p}\right)=-1.$

In each case, we will consider the same units that we considered in $(a)$.

$i)$ Let $u=3-2\sqrt{2}$. Since $\left(\frac{2}{p}\right)=1$, 2 has a square
root in $\mathds{F}_{p}^{\ast}$ and therefore $u\in\mathds{F}_{p}^{*}$. Thus,

$N_{\mathfrak{p}}(u)=u^{[\mathds{F}_{p}^{\ast}:\mathds{F}_{p}^{\ast}]}=u.$

Suppose $\\#S=m$, where $1\leq m\leq 4$. By Lemma 3.1, $p$ divides $u^{m}-1$.
The table below shows that this is possible only if $p\in\\{2,3,7\\}$. We
recall that $p\geq 19$.

$N_{\mathfrak{p}}(u)$ | $u-1$ | $u^{2}-1$ | $u^{3}-1$ | $u^{4}-1$
---|---|---|---|---
$u$ | $2(1-\sqrt{2})$ | $-4\sqrt{2}\cdot u$ | $-14(7-5\sqrt{2})$ | $24\sqrt{2}(12\sqrt{2}-17)$

$ii)$ Let $u=2+\sqrt{3}$. Similarly, since $\left(\frac{3}{p}\right)=1$ and
$N_{\mathfrak{p}}(u)=u$ then $p$ divides $u^{m}-1$, where $1\leq m\leq 4$. The
table below shows that this is possible only if $p\in\\{2,3,5\\}$.

$N_{\mathfrak{p}}(u)$ | $u-1$ | $u^{2}-1$ | $u^{3}-1$ | $u^{4}-1$
---|---|---|---|---
$u$ | $1+\sqrt{3}$ | $2\sqrt{3}\cdot u$ | $5(3\sqrt{3}+5)$ | $8\sqrt{3}(7+4\sqrt{3})$

$iii)$ Let $u=5-2\sqrt{6}$. Similarly, since $\left(\frac{6}{p}\right)=1$ and
$N_{\mathfrak{p}}(u)=u$ then $p$ divides $u^{m}-1$, where $1\leq m\leq 4$. The
table below shows that this is possible only if $p\in\\{2,3,5,11\\}$.

$N_{\mathfrak{p}}(u)$ | $u-1$ | $u^{2}-1$ | $u^{3}-1$ | $u^{4}-1$
---|---|---|---|---
$u$ | $2\sqrt{2}(\sqrt{2}-\sqrt{3})$ | $-4\sqrt{6}\cdot u$ | $22(-9\sqrt{6}+22)$ | $40(120-49\sqrt{6})$

In summary, we have shown that if $S$ is non-empty then in each case we find a
unit $u$ contradicting (3.1). We conclude that $E$ has an $L$-rational point
of order $p$, and since $d_{K}=4$ and $h_{K}^{+}=2$, it follows that
$[L:\mathbb{Q}]\leq 8$. By Oesterlé’s bound, as relayed in [33, Page 724], it
follows that

$p\leq\left(1+3^{[L:\mathbb{Q}]/2}\right)^{2}\leq\left(1+3^{8/2}\right)^{2}=(1+3^{4})^{2}=6724.$

∎

###  3.1. Torsion primes of degree 8

Let $L$ be the number field as in Lemma 3.1. Recall that $[L:\mathbb{Q}]\leq
d_{K}h_{K}^{+}=8$. Let $d\geq 1$ be an integer. As in [7], let $S(d)$ be the
set of primes $p$ such that there is a number field of degree $d$ and an
elliptic curve over $K$ with a $K$-rational point of order $p$. To reduce
Bound 2, we need to determine the set $S(8)$. Let Primes$(x)$ be the set of
primes less than or equal to $x$. Mazur [27], [28], Kamienny [22] and Parent
[32], [33], have determined $S(1),S(2)$ and $S(3)$, respectively. Building on
these works, Derickx-Kamienny-Stein-Stoll have determined $S(d)$ for $4\leq
d\leq 7$.

######  Theorem 3.3.

(Derickx-Kamienny-Stein-Stoll [7, Theorem 1.2])
For an integer $d$, let $S(d)$, Primes$(x)$ be the sets defined above. Then,

$\begin{split}S(4)&=Primes(17);\\\ S(5)&=Primes(19);\\\
S(6)&=Primes(19)\cup\\{37\\};\\\ S(7)&=Primes(23).\end{split}$

We used the computational techniques and Magma code used to prove Theorem 3.3
to compute the set $S(8)$. We ran the code provided in [38] for $29\leq p\leq
6724$ to arrive at the following result.

######  Theorem 3.4.

For an integer $d$, let $S(d)$, Primes$(x)$ be the sets defined above. Then,

$S(8)=\text{Primes}(23).$

We understand that Derickx-Stoll have independently proven Theorem 3.4 in
unpublished work. In the same work, they have also computed bounds for the
sets $S(9)$ and $S(10)$. We refer the interested reader to [7] for the proof
of Theorem 3.3.

It is clear that Theorem 3.4 enables us to drastically reduce Bound 2.

######  Bound 3.

Let $K=\mathbb{Q}(\sqrt{2},\sqrt{3})$. There are no non-trivial solutions to
(1) over $K$ for prime $p\geq 29$.

##  4\. irreducibility of the mod $p$ galois representation for $13\leq p\leq
23$

We turn our attention to demonstrating the irreducibility of the mod $p$
Galois representation $\bar{\rho}_{E,p}$ for $13\leq p\leq 23$. Since the Frey
curve $E$ has full 2-torsion over $K$, it is sufficient to show that there are
no non-cuspidal $K$-rational points on one of the modular curves
$X_{0}(p),X_{0}(2p)$ or $X_{0}(4p)$. For $p=23$, we found it most convenient
to work with $X_{0}(23)$ and with $X_{0}(2p)$ for $13\leq p\leq 19$. We show
that $X_{0}(23)(K)=X_{0}(23)(\mathbb{Q})$ and
$X_{0}(2p)(K)=X_{0}(2p)(\mathbb{Q})$, for $13\leq p\leq 19$. All points in
$X_{0}(23)(\mathbb{Q})$ and $X_{0}(2p)(\mathbb{Q})$, for $13\leq p\leq 19$,
are cuspidal thus proving the irreducibility of $\bar{\rho}_{E,p}$.

###  4.1. $p=23$

The Mordell-Weil group of the Jacobian $J$ of $X_{0}(23)$ over $K$ is finite
and, further, $J(K)=J(\mathbb{Q})$. Using the Abel-Jacobi map, this allows us
to show that $X_{0}(23)(K)=X_{0}(23)(\mathbb{Q})$.

We are grateful to Philippe Michaud-Rodgers for showing us that we can
determine $X_{0}(46)(K)$ via his Eta Product method (see [29, Section 4.3] for
details of this method).

######  Theorem 4.1.

Let $K=\mathbb{Q}(\sqrt{2},\sqrt{3})$. All $K$-rational points on $X_{0}(23)$
are defined over $\mathbb{Q}$, i.e., $X_{0}(23)(K)=X_{0}(23)(\mathbb{Q})$.

###### Proof.

We work with the model

$X_{0}(23):y^{2}=x^{6}-8x^{5}+2x^{4}+2x^{3}-11x^{2}+10x-7.$

We denote by $J$ the Jacobian of $X_{0}(23)$. Using Magma, we find that $J$
has rank 0 over $K$ (since the returned upper bound is 0). Note that
$X_{0}(23)$ has good reduction away from the primes 2 and 23, and that 47 and
71 both completely split in $K$. Fix the primes $\mathfrak{p}_{47}$ and
$\mathfrak{p}_{71}$ of $K$ above 47 and 71, respectively. Then,

$\mathcal{O}_{K}/\mathfrak{p}_{47}=\mathbb{F}_{47}\;\text{ and
}\;\mathcal{O}_{K}/\mathfrak{p}_{71}=\mathbb{F}_{71}.$

Furthermore,

$J(\mathbb{F}_{47})\cong\left(\mathbb{Z}/11\mathbb{Z}\right)^{2}\times\left(\mathbb{Z}/19\mathbb{Z}\right)\;\text{
and
}\;J(\mathbb{F}_{71})\cong\left(\mathbb{Z}/11\mathbb{Z}\right)\times\left(\mathbb{Z}/349\mathbb{Z}\right).$

Since $J(K)=J(K)_{tors}$ injects into $J(\mathbb{F}_{47})$ and
$J(\mathbb{F}_{71})$ (see [23, Appendix]), it follows that either $J(K)$ is
trivial or $\\#J(K)=11$. Since $J(\mathbb{Q})\cong\mathbb{Z}/11\mathbb{Z}$,
the latter holds, i.e.,

$J(K)\cong J(\mathbb{Q})\cong\mathbb{Z}/11\mathbb{Z}.$

Let $L=\mathbb{Q}(\sqrt{2})$. Let $\tau$ denote the non-trivial automorphism
of $K/L$, i.e.,

$\tau:\;\sqrt{2}\mapsto\sqrt{2},\;\sqrt{3}\mapsto-\sqrt{3}.$

If $P\in X_{0}(23)(K)$ then $P+P^{\tau}$ is a $L$-rational effective degree 2
divisor on $X_{0}(23)$. We write $\operatorname{Pic}^{0}(X_{0}(23))$ for the
group of rational degree $0$ divisors on $X_{0}(23)/\mathbb{Q}$ modulo linear
equivalence. We recall the standard isomorphism
$J(\mathbb{Q})\cong\operatorname{Pic}^{0}(X_{0}(23))$. Thus,

$[P+P^{\tau}-\infty^{+}-\infty^{-}]\in J(L)=J(\mathbb{Q}).$

We checked that either $P$ is defined over
$\mathbb{Q}(\sqrt{-15}),\mathbb{Q}(\sqrt{-11}),\mathbb{Q}(\sqrt{-7}),\mathbb{Q}(\sqrt{-5})$
or $P\in X_{0}(23)(\mathbb{Q})$ or $P=(x_{P},y_{P})$, where
$x_{P}\in\mathbb{Q}$, i.e., $P$ is defined over a quadratic subfield of $K$.
Bruin-Najman [6] have determined that all quadratic points on $X_{0}(23)$ are
defined over the imaginary quadratic fields listed above. Thus, $P\in
X_{0}(23)(\mathbb{Q})$. (Alternatively in this case one can also check, using
the Chabauty implementation in Magma, that there are no $\mathbb{Q}$-rational
points on the twists of $X_{0}(23)$ over any quadratic subfield of $K$). ∎

We are grateful to Samir Siksek for suggesting the following strategy. To deal
with the exponents $p=19,17,13$, we note that $E^{\prime}(K)=E^{\prime}(L)$
where $L$ is some quadratic subfield of $K$ (which differs for each prime $p$)
and $E^{\prime}$ is the optimal elliptic curve over $\mathbb{Q}$ of conductor
$2p$. By modularity, there is a map

$\pi:X_{0}(2p)\rightarrow E^{\prime}.$

We study the explicit modular parametrisation of $\pi$ and the Mordell-Weil
group of $E$ over $K$ to gain information about points in $X_{0}(2p)(K)$,
since

$\pi(X_{0}(2p))(K)\subseteq E^{\prime}(K)=E^{\prime}(L).$

###  4.2. $p=19$

Using the explicit modular parametrisation of $\pi$, we show that if $P\in
X_{0}(38)(K)$ then either $P\in X_{0}(38)(\mathbb{Q})$ or $P\in
X_{0}(38)(\mathbb{Q}(\sqrt{6}))$ or $P$ induces a $\mathbb{Q}(\sqrt{6})$-point
on the elliptic curve with Cremona label 1728j1. In the second case, it
follows from [31], that $P\in X_{0}(38)(\mathbb{Q})$. In the latter case,
since the Mordell-Weil group of this elliptic curve over
$\mathbb{Q}(\sqrt{6})$ is trivial, it then also follows that $P\in
X_{0}(38)(\mathbb{Q})$.

######  Theorem 4.2.

Let $K=\mathbb{Q}(\sqrt{2},\sqrt{3})$. All $K$-rational points on $X_{0}(38)$
are defined over $\mathbb{Q}$, i.e., $X_{0}(38)(K)=X_{0}(38)(\mathbb{Q})$.

###### Proof.

We work with the following model of $X_{0}(38)$ given by Magma

$x^{2}-wy-xy-xz+yz+z^{2}=0$ $w^{3}-2w^{2}x+3w^{2}y+wxy-
xy^{2}+2wxz-2wyz-2wz^{2}=0.$

There are no points of the form $(w:x:y:z)=(a:b:0:c)\in X_{0}(38)(K)$. Suppose
$P=(a:b:1:c)\in X_{0}(38)(K)$. Let

$E^{\prime}:y^{2}+xy+y=x^{3}+x^{2}+1.$

Then $E^{\prime}$ is the elliptic curve with Cremona label 38b1. Taking the
quotient of $X_{0}(38)$ by the Atkin-Lehner involution $w_{19}$ yields a genus
1 curve which is isomorphic to $E^{\prime}$. We use Magma to compute the
modular parametrisation $\pi$ explicitly

$\pi:X_{0}(38)\rightarrow E^{\prime},\qquad
P\mapsto\left(\dfrac{-3a-1}{a-1},\;h(a,c)\right)$

where $h(a,c)$ is some polynomial expression in $a$ and $c$. We collect some
information about the elliptic curve $E^{\prime}$. Let
$L=\mathbb{Q}(\sqrt{6})$. We find that $E^{\prime}(K)=E^{\prime}(L)$.
Therefore, $\pi(P)\in E^{\prime}(L)$. It follows that

$a\in L.$

If $P\in X_{0}(38)(K)$ then

(4) $b^{2}-a-b-bc+c+c^{2}=0$ (5)
$a^{3}-2a^{2}b+3a^{2}+ab-b+2abc-2ac-2ac^{2}=0$

By taking a linear combination of (4) and (5), we eliminate $c$ to get

(6) $2ab^{2}-(2a^{2}+a+1)b+a^{3}+a^{2}=0.$

If $a=0$ then $P\in\\{(0:0:1:0),(0:0:1:-1)\\}$. Suppose now that $a\neq 0$. It
follows from (6) that

$b=\dfrac{(2a^{2}+a+1)\pm\sqrt{-4a^{4}+4a^{3}-3a^{2}+2a+1}}{2a}.$

Therefore,

$\beta^{2}=-4a^{4}+4a^{3}-3a^{2}+2a+1$

for some $\beta\in K$. Since $a\in L$, either $\beta\in L$ or
$\beta=\sqrt{2}\gamma$, for some $\gamma\in L$. If $\beta\in L$ then $b\in L$
and subsequently $P\in X_{0}(38)(L)$. Ozman-Siksek [31] have determined the
quadratic points on $X_{0}(38)$. In particular, since they find there are no
real quadratic points on $X_{0}(38)$, it must be that $P\in
X_{0}(38)(\mathbb{Q})$. In the second case, $(a,\gamma)\in C(L)$ where

$C:2y^{2}=-4x^{4}+4x^{3}-3x^{2}+2x+1$

is a genus 1 curve. There is an isomorphism

$C\rightarrow
C^{\prime},\qquad(x,y)\mapsto\left(-\dfrac{1}{4x-4},\;\dfrac{-y}{(4x-4)^{2}}\right)$

where $C^{\prime}$ is the elliptic curve with Cremona label 1728j1. Since
$C^{\prime}(L)=C^{\prime}(\mathbb{Q})=\\{\mathcal{O}\\}$, it follows that
$a=1$. From (4) and (6), $P\in\\{(1:1:1:1),(1:1:1:-1)\\}$.

∎

###  4.3. $p=17$

Using the explicit modular parametrisation, we show that if $P\in
X_{0}(34)(K)$ then either $P\in X_{0}(34)(\mathbb{Q}(\sqrt{2}))$ or $P$
induces a $\mathbb{Q}(\sqrt{2})$-rational point on the quadratic twist of $C$
over $\mathbb{Q}(\sqrt{3})$. In the first case, it follows from [31], that
$P\in X_{0}(34)(\mathbb{Q})$. In the second case, we use a reduction argument
to arrive at a contradiction.

######  Theorem 4.3.

Let $K=\mathbb{Q}(\sqrt{2},\sqrt{3})$. All $K$-rational points on $X_{0}(34)$
are defined over $\mathbb{Q}$, i.e., $X_{0}(34)(K)=X_{0}(34)(\mathbb{Q})$.

###### Proof.

We work with the model

(7) $X_{0}(34):x^{4}-y^{4}+x^{3}+3xy^{2}-2x^{2}+x+1=0.$

Making the change of variables $(X,Y)=(x,y^{2})$ yields the curve

$C^{\prime}:X^{4}-Y^{2}+X^{3}+3XY-2X^{2}+X+1=0.$

Since $C^{\prime}$ has genus 1, we can transform it to an elliptic curve

(8) $C^{\prime}\rightarrow
E^{\prime},\qquad(X,Y)\mapsto(2(X^{2}-2X+Y),\;4X(X^{2}-2X+Y))$

where

$E^{\prime}:y^{2}+xy+2y=x^{3}-4x.$

is the elliptic curve with Cremona label 34a1. We deduce the explicit modular
parametrisation

$\pi:X_{0}(34)\longrightarrow
E^{\prime},\qquad(x,y)\longmapsto(2(x^{2}-2x+y^{2}),4x(x^{2}-2x+y^{2})).$

Let $L=\mathbb{Q}(\sqrt{2})$. Note that $E^{\prime}(K)=E^{\prime}(L)$. Suppose
$P=(a,b)\in X_{0}(34)(K)$. Since

$2(a^{2}-2a+b^{2}),\qquad 4a(a^{2}-2a+b^{2})\in L,$

it follows that either $a^{2}-2a+b^{2}=0$ or $a\in L$. Suppose the former is
true, i.e.,

(9) $b^{2}=2a-a^{2}.$

We substitute (9) into (7) to find that

$2a^{3}+a+1=0.$

It follows that $a\not\in K$. Thus, $a\in L$ and hence $b^{2}\in L$. Either
$b\in L$ or $b=\sqrt{3}\beta$ for some $\beta\in L$. If $b\in L$ then
$P=(a,b)\in X_{0}(34)(L)$. Ozman-Siksek [31] have determined the quadratic
points on $X_{0}(34)$. In particular, since they find there are no real
quadratic points on $X_{0}(34)$, it must be that $P\in X_{0}(34)(\mathbb{Q})$.

Suppose $b=\sqrt{3}\beta$ for some $\beta\in L$. Thus, $(a,\beta)$ is an
$L$-rational point on

$C:\;x^{4}-9y^{4}+x^{3}+9xy^{2}-2x^{2}+x+1=0$

where $C$ is the quadratic twist of $X_{0}(34)$ over $\mathbb{Q}(\sqrt{3})$.
Note that 3 is inert in $L$, i.e, the residue field of 3 is the finite field
$\mathds{F}_{9}$. We used Magma to check that $C(\mathds{F}_{9})$ is empty.
Thus, $C(L)$ is empty. ∎

###  4.4. $p=13$

Using the explicit modular parametrisation, we show that if $P\in
X_{0}(26)(K)$ then either $P\in X_{0}(26)(\mathbb{Q}(\sqrt{3}))$ or $P$
induces a point in $C(\mathbb{Q}(\sqrt{3}))$ where $C$ is a certain
hyperelliptic curve of genus 2. In the first case it follows from [6] that
$P\in X_{0}(26)(\mathbb{Q})$. In the second case, we show that
$C(\mathbb{Q}(\sqrt{3}))=\\{\infty\\}$.

######  Theorem 4.4.

Let $K=\mathbb{Q}(\sqrt{2},\sqrt{3})$. All $K$-rational points on $X_{0}(26)$
are defined over $\mathbb{Q}$, i.e., $X_{0}(26)(K)=X_{0}(26)(\mathbb{Q})$.

###### Proof.

We work with the model

(10) $X_{0}(26):y^{2}=x^{6}-8x^{5}+8x^{4}-18x^{3}+8x^{2}-8x+1.$

Let

$E^{\prime}:y^{2}+xy+y=x^{3}-x^{2}-3x+3.$

Then $E^{\prime}$ is the elliptic curve with Cremona label 26b1. Consider the
automorphism $w$ of $X_{0}(26)$ that maps $(x,y,z)$ to $(z,-y,x)$. Taking the
quotient of $X_{0}(26)$ by $w$ yields the elliptic curve $E^{\prime}$. Suppose
$P=(a,b)\in X_{0}(26)(K)$. Using Magma, we find the explicit parametrisation

$\displaystyle\pi:X_{0}(26)$ $\displaystyle\longrightarrow E^{\prime}$
$\displaystyle(a,b)$
$\displaystyle\longmapsto\left(-\dfrac{(a+1)^{2}}{(a-1)^{2}},\;\dfrac{-2b+2a(a-1)}{(a-1)^{3}}\right).$

Let $L=\mathbb{Q}(\sqrt{3})$. Note that $E^{\prime}(K)=E^{\prime}(L)$. It
follows that

(11) $\left(\dfrac{a+1}{a-1}\right)^{2}\in L.$

Thus

$\frac{a+1}{a-1}\in L\;\text{or}\;\frac{a+1}{a-1}\in\;\sqrt{2}\cdot L.$

In the case that $a\in L$, it follows that $b\in L$. In [6, Table 3], Bruin-
Najman have determined all quadratic points on $X_{0}(26)$. In particular,
since they find there are no real quadratic points on $X_{0}(26)$, it must be
that $P\in X_{0}(26)(\mathbb{Q})$. Suppose $\frac{a+1}{a-1}\in\sqrt{2}\cdot
L$, i.e.,

(12) $\dfrac{a+1}{a-1}=\sqrt{2}\alpha,\;\text{ for some }\alpha\in L.$

We note that if $a=1$ then $b^{2}=-16$, i.e., $P\notin K$. Suppose from now
that $a\neq 1$ and note the following identity:

(13)
$\left(\dfrac{a+1}{a-1}\right)^{2}-1=\dfrac{(a+1)^{2}-(a-1)^{2}}{(a-1)^{2}}=\dfrac{4a}{(a-1)^{2}}=\dfrac{4a(a-1)}{(a-1)^{3}}.$

From (11) and (13), we see that

$\dfrac{b}{(a-1)^{3}}\in L.$

Note the following identity

(14)
$16\left(\dfrac{a^{6}-8a^{5}+8a^{4}-18a^{3}+8a^{2}-8a+1}{(a-1)^{6}}\right)\\\
=-4\left(\dfrac{a+1}{a-1}\right)^{6}-3\left(\dfrac{a+1}{a-1}\right)^{4}+10\left(\dfrac{a+1}{a-1}\right)^{2}+13.$

By combining (10), (12) and (14), we obtain

$\left(\frac{4b}{(a-1)^{3}}\right)^{2}=4\left(\dfrac{a+1}{a-1}\right)^{6}-3\left(\dfrac{a+1}{a-1}\right)^{4}+10\left(\dfrac{a+1}{a-1}\right)^{2}+13.$

After making the substitutions $\beta=\frac{4b}{(a-1)^{3}}$ and (12), we
obtain

$\beta^{2}=-32\alpha^{6}-12\alpha^{4}+20\alpha^{2}+13.$

Equivalently,

$(2\beta)^{2}=-2(2\alpha)^{6}-3(2\alpha)^{4}+20(2\alpha)^{2}+52.$

That is, $(2\alpha,2\beta)$ is a $L$-rational point on the curve

$C:y^{2}=-2x^{6}-3x^{4}+20x^{2}+52.$

We find that the fake 2-Selmer set of $C$ over $L$ is empty. Thus, $C(L)$ is
empty (see Bruin-Stoll [5]). ∎

###  4.5. $p=11,7,5$

The Jacobian of the Fermat curve of degree 5, 7 and 11 has finitely many
rational points. Since the divisor obtained from the formal sum of a point and
its Galois conjugates gives a rational divisor, this allows the study of
points on these Fermat curves over fields of low degree. Gross and Rohrlich,
in [16], determine all points on (1) with exponent $p=11$ over fields of
degree at most 5. In particular, they find there are no non-trivial solutions
to (1) over real biquadratic fields for $p=11$. In [39], Tzermias determines
the points on the Fermat septic defined over number fields of degree at most
5. In particular, he finds there are no non-trivial real biquadratic points on
the Fermat septic. In [24], Klassen and Tzermias have classified all points on
the Fermat quintic defined over number fields of degree at most 6. Building on
this work, in [26], Kraus provides an algebraic description of the quartic
points on the Fermat quintic. In particular, he finds that there are no non-
trivial real biquadratic points on the Fermat quintic.

##  5\. Small Exponents

To complete the proof of Theorem 1.1, it remains to show that there are no
non-trivial solutions to (1) over $K$ for $n=9,6,4$. We note in passing that
there are non-trivial solutions to the Fermat cubic over
$\mathbb{Q}(\sqrt{2})$ (see [21, page 184]).

###  5.1. $n=9$

We are very grateful to Samir Siksek for the lengthy discussions and ideas
that resulted in this proof. We first convert the problem of finding
$K$-points on the Fermat curve of degree 9 to finding the
$\mathbb{Q}(\sqrt{3})$-points on a certain hyperelliptic curve $C$. We then
study the Jacobian of $C$ to show that $C(\mathbb{Q}(\sqrt{3}))=\\{\infty\\}$.

######  Theorem 5.1.

Let $K=\mathbb{Q}(\sqrt{2},\sqrt{3})$. There are no non-trivial solutions to
(1) over $K$ for $n=9$.

We find it convenient to let

$F_{9}\;:\;x^{9}+y^{9}+z^{9}=0.$

That is, $F_{9}$ is the Fermat curve of degree 9. We recall that
$2\mathcal{O}_{K}=\mathfrak{p}^{4}$ and that $K$ has class number $1$. We will
prove that $F_{9}(K)=\\{(1:-1:0),\;(1:0:-1),\;(0:1:-1)\\}$, i.e, $F_{9}(K)$
consists only of trivial solutions. Suppose $(\alpha:\beta:\gamma)\in
F_{9}(K)$. We may suppose that $\alpha$, $\beta$, $\gamma\in\mathcal{O}_{K}$
and that they are coprime. We recall that
$\mathcal{O}_{K}/\mathfrak{p}=\mathbb{F}_{2}$ and

$F_{9}(\mathbb{F}_{2})=\\{(1:1:0),\;(1:0:1),\;(0:1:1)\\}.$

Hence, by permuting $\alpha$, $\beta$, $\gamma$ appropriately, we may suppose
$(\alpha:\beta:\gamma)\equiv(1:1:0)\pmod{\mathfrak{p}}$. Thus

(15) $\mathfrak{p}\mid\gamma,\qquad\mathfrak{p}\nmid\alpha\beta.$

Observe

$\gamma^{18}-(\alpha^{9}-\beta^{9})^{2}\;=\;(\alpha^{9}+\beta^{9})^{2}-(\alpha^{9}-\beta^{9})^{2}\;=\;4(\alpha\beta)^{9}.$

After making the substitutions

(16) $u=\frac{\alpha\beta}{\gamma^{2}},\qquad
v=\frac{\alpha^{9}-\beta^{9}}{\gamma^{9}},$

we see that $Q_{1}=(u,v)\in C_{1}(K)$ where

$C_{1}\;:\;y^{2}=-4x^{9}+1.$

Let

$E_{1}\;:\;y^{2}=4x^{3}+1.$

This is an elliptic curve. Let

$\pi_{1}:C_{1}\rightarrow E_{1},\qquad(x,y)\mapsto(-x^{3},y).$

The elliptic curve $E_{1}$ has minimal Weierstrass model

$E_{1}^{\prime}\;:\;z^{2}+z=x^{3}$

which is obtained from $E_{1}$ by the substitution $y=2z+1$. This has Cremona
label 27a3. In particular $E_{1}^{\prime}$ has good reduction away from $3$.
Let $R_{1}=\pi_{1}(Q_{1})=(-u^{3},v)\in E_{1}(K)$. Then $R_{1}$ corresponds to
the point

$S_{1}=(-u^{3},(v-1)/2)\in E_{1}^{\prime}(K).$

Let $\sigma:K\rightarrow K$ be the automorphism satisfying

$\sigma(\sqrt{2})=-\sqrt{2},\qquad\sigma(\sqrt{3})=\sqrt{3}.$

We note that the fixed field of $\sigma$ is $L=\mathbb{Q}(\sqrt{3})$. Thus
$S_{1}+S_{1}^{\sigma}\in E_{1}^{\prime}(L)$. We checked using Magma that
$E_{1}^{\prime}$ has rank $0$ over $L$, and indeed

(17)
$E_{1}^{\prime}(L)\;=\;\\{\mathcal{O},(0,0),(0,-1)\\}\;\cong\;\mathbb{Z}/3\mathbb{Z}.$

Thus $S_{1}+S_{1}^{\sigma}$ is one of these three points. However,
$\operatorname{ord}_{\mathfrak{p}}(u)<0$ by (15) and (16). It follows that

$S_{1}\equiv\mathcal{O}\pmod{\mathfrak{p}}.$

Hence

$S_{1}^{\sigma}\equiv\mathcal{O}^{\sigma}=\mathcal{O}\pmod{\mathfrak{p}^{\sigma}}.$

However, $\mathfrak{p}$ is a totally ramified prime, so
$\mathfrak{p}^{\sigma}=\mathfrak{p}$. Thus
$S_{1}^{\sigma}\equiv\mathcal{O}\pmod{\mathfrak{p}}$, and

$S_{1}+S_{1}^{\sigma}\equiv\mathcal{O}\pmod{\mathfrak{p}}.$

By (17) we conclude that

$S_{1}+S_{1}^{\sigma}=\mathcal{O}.$

Hence

$R_{1}+R_{1}^{\sigma}=\mathcal{O}.$

Hence

$(-u^{3})^{\sigma}=-u^{3},\qquad v^{\sigma}=-v.$

As the only cube root of $1$ in $K$ is $1$, we have $u^{\sigma}=u$ and so
$u\in L$. Moreover, $v^{2}=-4u^{9}+1\in L$ and $v^{\sigma}=-v$, so
$v=w/\sqrt{2}$ where $w\in L$. Hence $(u,w)\in C(L)$ where

$C\;:\;y^{2}=2(-4x^{9}+1).$

######  Lemma 5.1.

$C(L)=\\{\infty\\}$.

Since $u=\alpha\beta/\gamma^{2}$, Theorem 5.1 follows from Lemma 5.1. We now
prove Lemma 5.1 through studying $J(\mathbb{Q})$ where $J$ is the Jacobian of
$C$.

###### Proof.

Let

$E\;:\;y^{2}=x^{3}+2,$

which is elliptic curve with Cremona label 1728a1. Let

(18) $\pi:C\rightarrow E,\qquad(x,y)\mapsto(-2x^{3},y).$

We find that $E$ has zero torsion and rank $1$ over $\mathbb{Q}$, and that in
fact,

$E(\mathbb{Q})=\mathbb{Z}\cdot(-1,1).$

We write $\operatorname{Pic}^{0}(E)$ for the group of rational degree $0$
divisors on $E/\mathbb{Q}$ modulo linear equivalence, and
$\operatorname{Pic}^{0}(C)$ for the group of rational degree $0$ divisors on
$C/\mathbb{Q}$ modulo linear equivalence. We recall the standard isomorphism
[37, Proposition III.3.4]

(19) $E(\mathbb{Q})\cong\operatorname{Pic}^{0}(E),\qquad P\mapsto[P-\infty],$

where $[D]$ denotes the linear equivalence class of a divisor $D$. Thus

$\operatorname{Pic}^{0}(E)=\mathbb{Z}\cdot\mathcal{Q},\qquad\mathcal{Q}=[(-1,1)-\infty].$

We also recall the standard isomorphism
$J(\mathbb{Q})\cong\operatorname{Pic}^{0}(C)$, and we will represent elements
of the Mordell–Weil group $J(\mathbb{Q})$ as elements of
$\operatorname{Pic}^{0}(C)$. We note that

$J(\mathbb{F}_{5})\cong\mathbb{Z}/6\mathbb{Z}\times\mathbb{Z}/126\mathbb{Z},\qquad
J(\mathbb{F}_{13})\cong\mathbb{Z}/42997\mathbb{Z}.$

As these two groups have coprime orders we conclude that $J$ has trivial
torsion over $\mathbb{Q}$. Moreover, $J$ has $2$-Selmer rank $1$ over
$\mathbb{Q}$, so $J$ has rank at most $1$ over $\mathbb{Q}$. The morphism
$\pi$ in (18) has degree $3$ and induces homomorphisms (see [37, Section
II.3])

$\pi_{*}\;:\;\operatorname{Pic}^{0}(C)\rightarrow\operatorname{Pic}^{0}(E),\qquad\left[\sum
a_{i}P_{i}\right]\mapsto\left[\sum a_{i}\pi(P_{i})\right],$

and

$\pi^{*}\;:\;\operatorname{Pic}^{0}(E)\rightarrow\operatorname{Pic}^{0}(C),\qquad\left[\sum
b_{j}Q_{j}\right]\mapsto\left[\sum
b_{j}\sum_{P\in\pi^{-1}(Q_{j})}e_{\pi}(P)\cdot P\right]$

where $e_{\pi}(P)$ denotes the ramification degree of $\pi$ at $P$.

Let

$\mathcal{P}=\pi^{*}(\mathcal{Q})\;=\;[(1/\sqrt[3]{2},1)+(\omega/\sqrt[3]{2},1)+(\omega^{2}/\sqrt[3]{2},1)-3\infty]\in\operatorname{Pic}^{0}(C)\cong
J(\mathbb{Q}).$

where $\omega$ is a primitive cube root of $1$. The point $\mathcal{P}$ has
infinite order on $J(\mathbb{Q})$. Thus $J$ has rank exactly $1$ over
$\mathbb{Q}$ and no torsion. Therefore
$J(\mathbb{Q})=\mathbb{Z}\cdot\mathcal{P}^{\prime}$, for some
$\mathcal{P}^{\prime}\in J(\mathbb{Q})=\operatorname{Pic}^{0}(C)$. Hence

$\mathcal{P}=k\mathcal{P}^{\prime}$

where $k$ is a non-zero integer. Applying $\pi_{*}$ to both sides we obtain

$k\pi_{*}(\mathcal{P}^{\prime})=\pi_{*}(\mathcal{P})=3\mathcal{Q}.$

However,
$\pi_{*}(\mathcal{P}^{\prime})\in\operatorname{Pic}^{0}(E)=\mathbb{Z}\mathcal{Q}$,
so

$\pi_{*}(\mathcal{P}^{\prime})=\ell\cdot\mathcal{Q}$

for some $\ell\in\mathbb{Z}$. Hence $k\ell=3$, so $k=\pm 1$ or $\pm 3$. We
checked that the image of $\mathcal{P}$ under the composition

$J(\mathbb{Q})\rightarrow J(\mathbb{F}_{5})\rightarrow
J(\mathbb{F}_{5})/3J(\mathbb{F}_{5})$

is non-zero. Thus $k\neq\pm 3$, so $k=\pm 1$, hence

$J(\mathbb{Q})=\operatorname{Pic}^{0}(C)=\mathbb{Z}\cdot\mathcal{P}.$

Suppose $P\in C(L)$. Let $\tau:L\rightarrow L$ be the non-trivial
automorphism. Then $[P+P^{\tau}-2\infty]\in\operatorname{Pic}^{0}(C)$. Thus

$[P+P^{\tau}-2\infty]\;=\;n\cdot\mathcal{P}\;=\;n\cdot\pi^{*}(\mathcal{Q})\;=\;\pi^{*}(n\cdot\mathcal{Q})$

for some integer $n$. We claim that $n=0$. Suppose otherwise, then
$n\cdot\mathcal{Q}\in\operatorname{Pic}^{0}(E)\setminus\\{0\\}$ and by the
isomorphism in (19) we have $n\cdot\mathcal{Q}=[Q-\infty]$ where $Q\in
E(\mathbb{Q})\setminus\\{\mathcal{O}\\}$. Write $Q=(a,b)\in E(\mathbb{Q})$
with $a$, $b\in\mathbb{Q}$. Then

$[P+P^{\tau}-2\infty]\;=\;\pi^{*}([(a,b)-\infty])\;=\;[D-3\infty]$

where

$D=P_{1}+P_{2}+P_{3},\qquad
P_{j}\;=\;\left(-\omega^{j-1}\sqrt[3]{a/2}\,,\,b\right),\qquad j=1,2,3.$

Hence

$D\;\sim\;D^{\prime},\qquad D^{\prime}=P+P^{\tau}+\infty$

where $\sim$ denotes linear equivalence on $C$. Write $\lvert D\rvert$ for the
complete linear system of effective divisors of $C$ linearly equivalent to
$D$. Let $r(D)=\dim\lvert D\rvert$. Note that $D^{\prime}\in\lvert D\rvert$
and $D^{\prime}\neq D$, therefore $r(D)\geq 1$. By Riemann–Roch [2, page 13],

$r(D)-i(D)\;=\;\deg(D)-g=-1,$

where $i(D)\geq 0$ is the so called index of speciality of $D$, and $g=4$ is
the genus of $C$. It follows that $i(D)>0$, and therefore that $D$ is a
special divisor. By Clifford’s theorem [17, Theorem IV.5.4],

$r(D)\;\leq\;\frac{\deg(D)}{2}=\frac{3}{2}.$

Hence $r(D)=1$. Thus the complete linear system $\lvert D\rvert$ is a
$g^{1}_{3}$. As $C$ is hyperelliptic, by [2, page 13], $\lvert
D\rvert=g^{1}_{2}+p$ where $p$ is a fixed base point of the linear system. In
particular, every divisor in $\lvert D\rvert$ is the sum of $p$ and two points
interchanged by the hyperelliptic involution. We apply this to $D$ itself.
Thus two of the points $P_{1}$, $P_{2}$, $P_{3}$ are interchanged by the
hyperelliptic involution. However, they all have the same $y$-coordinate $b$,
so $b=0$. But $(a,b)\in E(\mathbb{Q})$, so $a\in\mathbb{Q}$ and $a^{3}=-2$
giving a contradiction. Hence $n=0$, and so

$P+P^{\tau}\sim 2\infty.$

Thus $P$, $P^{\tau}$ are interchanged by the hyperelliptic involution. We
recall that we want to show that $P=\infty$. Suppose otherwise. Then we can
write $P=(c,d)$ where $c$, $d\in L$ and $c^{\tau}=c$, $d^{\tau}=-d$. Thus
$c\in\mathbb{Q}$, and $d=e/\sqrt{3}$ with $e\in\mathbb{Q}$. Thus
$P^{\prime}=(c,e)\in C^{\prime}(\mathbb{Q})$ where

$C^{\prime}\;:\;y^{2}=6(-4x^{9}+1).$

Let $J^{\prime}$ be the Jacobian of $C^{\prime}$, and

$E^{\prime}\;:\;y^{2}=6(4x^{3}+1).$

We find that $E^{\prime}(\mathbb{Q})=\mathbb{Z}\cdot(1/2,3)$. Let
$\mathcal{Q}^{\prime}=[(1/2,3)-\infty]\in\operatorname{Pic}^{0}(E^{\prime})$,
so $\operatorname{Pic}^{0}(E^{\prime})=\mathbb{Z}\cdot\mathcal{Q}$. Let

$\pi^{\prime}:C^{\prime}\rightarrow E^{\prime},\qquad(x,y)\mapsto(-x^{3},y).$

We find that $J^{\prime}$ has trivial torsion and $2$-Selmer rank $1$, and
following the same steps as before show that
$J^{\prime}(\mathbb{Q})=\operatorname{Pic}^{0}(C)=\mathbb{Z}\cdot\mathcal{P}^{\prime}$,
where $\mathcal{P}^{\prime}=(\pi^{\prime})^{*}(\mathcal{Q})$. Now
$[P^{\prime}-\infty]=n\mathcal{P}^{\prime}$ where $n$ is an integer, and must
be non-zero as $P^{\prime}\neq\infty$. Let $(f,g)=n\cdot(1/2,3)\in
E^{\prime}(\mathbb{Q})\setminus\\{\mathcal{O}\\}$. As before, we find that

$P^{\prime}+2\infty\sim P_{1}^{\prime}+P_{2}^{\prime}+P_{3}^{\prime},\qquad
P_{j}^{\prime}=(-\omega^{j-1}\cdot\sqrt[3]{f}\,,\,g).$

Following the same steps as before, it follows that $g=0$, so $f^{3}=-1/4$
contradicting $f\in\mathbb{Q}$. We can thus conclude that if $P\in C(L)$ then
$P=\infty$. This completes the proof of Lemma 5.1 and therefore Theorem 5.1. ∎

###  5.2. $n=6$

We first use a standard trick to show that a $K$-point on the Fermat curve of
degree 6 induces a $K$-point on a certain hyperelliptic curve $C$. We then use
information about the Mordell-Weil group over $K$ of the elliptic curve
(obtained by taking the quotient of $C$ by an automorphism of $C$) to show
that this $K$-point is defined over a quadratic subfield of $K$. This leads to
the search of $\mathbb{Q}$-rational points on the twists of $C$ over the
quadratic subfields of $K$.

######  Theorem 5.2.

Let $K=\mathbb{Q}(\sqrt{2},\sqrt{3})$. There are no non-trivial solutions to
(1) over $K$ for $n=6$.

###### Proof.

We find it convenient to let

$F_{6}:x^{6}+y^{6}=z^{6}.$

That is, $F_{6}$ is the Fermat curve of degree 6. We will prove that
$F_{6}(K)=\\{(0:-1:1),(-1:0:1),(0:1:1),(1:0:1)\\}$, i.e., $F_{6}(K)$ consists
only of trivial solutions. Suppose $(\alpha:\beta:\gamma)\in F_{6}(K)$.
Similar to the proof of Theorem 5.1, observe

$\gamma^{12}-(\alpha^{6}-\beta^{6})^{2}=(\alpha^{6}+\beta^{6})^{2}-(\alpha^{6}-\beta^{6})^{2}=4(\alpha\beta)^{6}.$

Let

$a=\dfrac{\alpha\beta}{\gamma^{2}},\qquad
b=\dfrac{\alpha^{6}-\beta^{6}}{\gamma^{6}}.$

Then $P=(a,b)\in C(K)$ where

$C:y^{2}=-4x^{6}+1.$

Let

$E:y^{2}=x^{3}-4.$

This is the elliptic curve with Cremona label 432b1. Let

$\pi:C\rightarrow
E,\qquad(x,y)\mapsto\left(\dfrac{1}{x^{2}},\;\dfrac{y}{x^{3}}\right).$

We checked using Magma that $E$ has rank 1 over $K$ (and $\mathbb{Q}$) and
that

$E(K)=E(\mathbb{Q})\cong\mathbb{Z}.$

Since $\pi(P)\in E(\mathbb{Q})$. It follows that $a^{2}\in\mathbb{Q}$ and
hence $b^{2}\in\mathbb{Q}$. Either $a\in\mathbb{Q}$ and hence $b\in\mathbb{Q}$
or

$a=\frac{a^{\prime}}{\sqrt{d}},\qquad
b=\frac{b^{\prime}}{\sqrt{d}},\qquad\text{for }\;d\in\\{2,3,6\\},\qquad
a^{\prime},b^{\prime}\in\mathbb{Q}.$

If $a,b\in\mathbb{Q}$ then $P\in C(\mathbb{Q})$. The Jacobian of $C$ has rank
1 over $\mathbb{Q}$. Using the Chabauty implementation in Magma, we find that

$C(\mathbb{Q})=\\{(0,\;\pm 1)\\}.$

If $a=0$ then it’s clear that $(\alpha:\beta:\gamma)$ is a trivial solution.

In the latter case, we remark that $a,b$ are necessarily defined over the same
quadratic subfield of $K$ since

$\dfrac{b}{a}\in\mathbb{Q}.$

Thus, $(a^{\prime},\;b^{\prime}d)\in C_{d}(\mathbb{Q})$ where

$C_{d}:\;y^{2}=-4x^{6}+d^{3},$

where $d\in\\{2,3,6\\}$. Using Magma, we found that

$C_{3}(\mathbb{Q})=C_{6}(\mathbb{Q})=\emptyset$

since the fake 2-Selmer set of $C_{d}$ over $\mathbb{Q}$ is empty (see Bruin-
Stoll [5]) if $d=3$ or $d=6$. It remains to determine $C_{2}(\mathbb{Q})$. We
will work with the model

$C_{2}:\;y^{2}=-x^{6}+2.$

We note that the curve $C_{2}$ has genus 2 and the rank of the Jacobian of
$C_{2}$ over $\mathbb{Q}$ is 2. Thus, we are unable to determine
$C_{2}(\mathbb{Q})$ using Chabauty. Instead, we used Bruin’s elliptic curve
Chabauty method to do so. Let $F=\mathbb{Q}(\sqrt[3]{2})$ and
$\epsilon=1+\sqrt[3]{2}+\sqrt[3]{2}^{2}\in F$. Note the factorisation

$C_{2}:\;y^{2}=-x^{6}+2=-(x^{2}-\sqrt[3]{2})(x^{4}+\sqrt[3]{2}x^{2}+\sqrt[3]{2}^{2}).$

Using this method, finding $C_{2}(\mathbb{Q})$ amounts to finding all points
$Q=(x_{Q},y_{Q})\in E_{2}(F)$ where

$E_{2}:\;y^{2}=x^{3}+(\epsilon\cdot\sqrt[3]{2})x^{2}+(\epsilon^{2}\cdot\sqrt[3]{2}^{2})x$

such that

$\frac{x_{Q}}{\epsilon}\in\mathbb{Q}.$

We used the Magma implementation of Bruin’s elliptic curve Chabauty method to
find that

$\frac{x_{Q}}{\epsilon}\in\\{0,1\\}.$

It immediately follows that $C_{2}(\mathbb{Q})$ consists only of the four
points

$C_{2}(\mathbb{Q})=\\{(\pm 1,\;\pm 1)\\}.$

Thus, $(a^{\prime},\;b^{\prime})\in\\{(\pm 1,\;\pm 1)\\}$ and if $P=(a,b)\in
C(K)$ then $P\in\\{(\pm\;1/{\sqrt{2}},\;\pm\;1/\sqrt{2})\\}$. Since

$b=\dfrac{\alpha^{6}-\beta^{6}}{\gamma^{6}},$

it is clear that $(\alpha:\beta:\gamma)$ is a trivial solution. ∎

###  5.3. $n=4$

Points on the Fermat quartic over quadratic fields were first studied by
Aigner [1]; somewhat later, Faddeev [9] gave another proof which also
classified the points over cubic fields, using quite intricate algebraic
geometry, and Mordell [30] then found an elementary proof of the same result.
Mordell starts with the knowledge that there are no non-trivial points on the
Fermat quartic over $\mathbb{Q}$, and studies points over all quadratic
extensions. We shall generalise his method, observing that we can also
classify points over quadratic extensions of certain quadratic fields. More
precisely, if $L$ is any field for which there are no points on the Fermat
quartic, and if the two elliptic curves with Cremona labels 32a1 and 64a1 have
rank $0$ over $L$, then we give a procedure to write down all the points on
the Fermat quartic over quadratic extensions of $L$.

We note that we require only that the elliptic curves with Cremona labels 32a1
and 64a1 have rank $0$ over $L$. This is generally false – for example, over
$L=\mathbb{Q}(\sqrt{3})$, the elliptic curve with Cremona label 64a1 has rank
$1$ (the other has rank $0$), and over $L=\mathbb{Q}(\sqrt{6})$, the elliptic
curve with Cremona label 32a1 has rank $1$ (and the other has rank $0$).

In an earlier version of this paper, we previously conjectured that there are
no points on the Fermat quartic over any real biquadratic field. We thank
Pedro José Cazorla Garcia for pointing out to us that the point
$(\sqrt{3},2,\sqrt{5})$ lies on the Fermat quartic over
$\mathbb{Q}(\sqrt{3},\sqrt{5})$.

After the completion of this work, we were made aware that Ishitsuka-Ito-
Ohshita [18, Theorem 7.3] have previously determined all points on the Fermat
quartic lying in a quadratic extension of $\mathbb{Q}(\zeta_{8})$. Since
$\mathbb{Q}(\sqrt{2})$ is contained in $\mathbb{Q}(\zeta_{8})$, this is indeed
stronger than the statement of Theorem 5.3. We thank the authors Yasuhiro
Ishitsuka, Tetsushi Ito and Tatsuya Ohshita for making us aware of this.

We note that the authors of the aforementioned work study the Jacobian of the
Fermat quartic over $\mathbb{Q}(\zeta_{8})$ and that the proof of Theorem 5.3,
extending work of Mordell [30], makes use of a different strategy.

######  Theorem 5.3.

The points on the Fermat quartic lying in quadratic extensions of
$\mathbb{Q}(\sqrt{2})$ lie in one of $\mathbb{Q}(\sqrt{2},i)$,
$\mathbb{Q}(\sqrt{2},\sqrt{-7})$, $\mathbb{Q}(\sqrt[4]{2})$ or
$\mathbb{Q}(\sqrt[4]{2}i)$.

###### Proof.

Let $K$ be a quadratic extension of $L=\mathbb{Q}(\sqrt{2})$. We look for
solutions to $x^{4}+y^{4}=1$ in $K$. Let $t=\frac{1-x^{2}}{y^{2}}$, so that
$x^{2}+ty^{2}=1$. This gives a parametrisation

$x^{2}=\frac{1-t^{2}}{1+t^{2}},\qquad y^{2}=\frac{2t}{1+t^{2}}.$

We observe that if $x,y\in K$ then $x^{2},y^{2}\in K$ and therefore so is $t$.

Suppose first that $t\in L$. Then $x^{2},y^{2}\in L$. In order for $x$ and $y$
to lie in the same quadratic extension $K$ of $L$, either $x\in L$, $y\in L$
or $x/y\in L$. This means that one of

$\frac{1-t^{2}}{1+t^{2}},\qquad\frac{2t}{1+t^{2}}\qquad\text{ or
}\qquad\frac{2t}{1-t^{2}}$

is a square in $L$. Equivalently, $(1-t^{2})(1+t^{2})$, $2t(1+t^{2})$ or
$2t(1-t^{2})$ is a square in $L$. These correspond to $L$-rational points of
one of the curves

  * •

$u^{2}=(1-t^{2})(1+t^{2})$;

  * •

$u^{2}=2t(1+t^{2})$;

  * •

$u^{2}=2t(1-t^{2})$.

Both of the first two possibilities are isomorphic to $y^{2}=x^{3}+4x$ (the
elliptic curve with Cremona label 32a1) via the maps
$(t,u)\mapsto\left(\frac{2t+2}{1-t},\frac{u}{(1-t)^{2}}\right)$ and
$(t,u)\mapsto(2t,2u)$ respectively, and the third to $y^{2}=x^{3}-4x$ (the
elliptic curve with Cremona label 64a1) via $(t,u)\mapsto(-2t,2u)$. We checked
that both of these curves have rank $0$ over $\mathbb{Q}(\sqrt{2})$.

The elliptic curve $y^{2}=x^{3}+4x$ has no additional points over $L$ and the
$\mathbb{Q}$-points are $(0,0)$ and $(2,\pm 4)$ and $\mathcal{O}$. Using the
explicit isomorphisms between the original forms and the Weierstrass forms, we
see that these points correspond on the first curve to $t=\pm 1$ and $t=0$,
and on the second to $t=0$, $t=1$ and $t=\infty$. These values of $t$ give
points $(x^{2},y^{2})=(1,0)$, $(-1,0)$, $(0,1)$, $(0,-1)$, giving points over
$\mathbb{Q}$ or $\mathbb{Q}(i)$.

The curve $y^{2}=x^{3}-4x$ has $\mathbb{Q}$-points $(-2,0)$, $(0,0)$, $(2,0)$,
and $\mathcal{O}$ but there are also four further points over $L$, given by
$(2+2\sqrt{2},\pm(4+4\sqrt{2})$ and $(2-2\sqrt{2},\pm(4-4\sqrt{2})$. The
rational points correspond to $t=\pm 1$ and $t=0$, and the point at infinity
to $t=\infty$, as before. The new points correspond to $t=-1\pm\sqrt{2}$, and
these give $(x^{2},y^{2})=(1/\sqrt{2},1/\sqrt{2})$ and
$(x^{2},y^{2})=(-1/\sqrt{2},-1/\sqrt{2})$, corresponding to solutions in the
quadratic extensions of $\mathbb{Q}(\sqrt{2})$ obtained by adjoining square
roots of $\sqrt{2}$ and $-\sqrt{2}$, i.e., adjoining $\sqrt[4]{2}$ or
$\sqrt[4]{2}i$. In particular, we recover the solutions

$1^{4}+1^{4}=\sqrt[4]{2}^{4},\qquad
1^{4}+1^{4}=\left(\sqrt{-\sqrt{2}}\right)^{4},$

and similar points got by negating one or more of the terms.

The case $t\in K$ but $t\notin L$ is slightly more involved. We write
$F(t)=t^{2}+\beta t+\gamma$ for the minimal polynomial of $t$ over $L$, so
$\beta,\gamma\in L$. We put $A=(1+t^{2})xy$ and $B=(1+t^{2})y$, so that

$\displaystyle A^{2}$ $\displaystyle=$ $\displaystyle 2t(1-t^{2})$
$\displaystyle B^{2}$ $\displaystyle=$ $\displaystyle 2t(1+t^{2}).$

As $A^{2},B^{2}\in K$ and $K=L(t)$, we can write

$\displaystyle A$ $\displaystyle=$ $\displaystyle\lambda+\mu t$ $\displaystyle
B$ $\displaystyle=$ $\displaystyle\lambda^{\prime}+\mu^{\prime}t$

for $\lambda$, $\mu$, $\lambda^{\prime}$, $\mu^{\prime}$ in $L$. Comparing the
first equations in these displays,

$(\lambda+\mu t)^{2}=2t(1-t^{2}).$

In particular, the equation

$(\lambda+\mu z)^{2}-2z(1-z^{2})=0$

has a root $z=t$. As the equation is defined over $L$, the LHS is divisible by
the minimal polynomial $F(z)$, and, as this is a cubic, we have

(20) $(\lambda+\mu z)^{2}-2z(1-z^{2})=F(z)(\rho+\sigma z),$

a factorisation over $L$ (so $\rho$ and $\sigma$ both belong to $L$ also).

Then $z=-\rho/\sigma$ is a solution to the LHS of (20) defined over $L$. In
particular, we have a solution with $z\in L$ to

$Y^{2}=2z(1-z^{2})=-2z^{3}+2z,$

where $Y=\lambda+\mu z\in L$. Thus we get an $L$-point on the elliptic curve

$Y^{2}=-2X^{3}+2X.$

We recall that this is the elliptic curve with Cremona label 64a1 the
$L$-rational solutions correspond to $z=\pm 1$, $z=0$ and $z=-1\pm\sqrt{2}$.

In exactly the same way, looking at $B^{2}$, we will get a solution over $L$
to

(21)
$(\lambda^{\prime}+\mu^{\prime}z)^{2}-2z(1+z^{2})=F(z)(\rho^{\prime}+\sigma^{\prime}z),$

and therefore a solution over $L$ to $Y^{2}=2z(1+z^{2})$ which is isomorphic
to $Y^{2}=X^{3}+4X$ (the elliptic curve with Cremona label 32a1). The only
$z$-values are $z=0$ and $z=1$.

We now want to consider all these cases, as in Mordell. We’ll write
$(z_{1},z_{2})$ for the situation where the equation (20) is solved by $z_{1}$
and equation (21) is solved by $z_{2}$.

#### 1\. $(0,0)$

This is exactly the same as Mordell’s case (I).

#### 2\. $(-1,1)$

This is Mordell’s case (VI). If $z_{1}=-1$ is a root of the LHS of (20) then
$\lambda+\mu=0$ and, since $-1$ must then be a root of the RHS of (21), it
follows that $\rho+\sigma=0$. Similarly, if $z_{1}=1$ is a root of the LHS of
(21) then $\lambda^{\prime}+\mu^{\prime}=2$,
$\rho^{\prime}+\sigma^{\prime}=0$. Equation (20) is

$\lambda^{2}(1+z)-2z(1-z)=\rho F(z)$

(after dividing out by $1-z$). We can rewrite the LHS as
$(2-\mu^{\prime}+\mu^{\prime}z)^{2}-2z-2z^{3}=\rho^{\prime}(1-z)F(z)$. Thus,
after dividing through by $1-z$, we get

$2(z^{2}+z+2)-4\mu^{\prime}+\mu^{\prime 2}(1-z)=\rho^{\prime}F(z).$

These have the same coefficients of $z^{2}$, so $\rho=\rho^{\prime}$.
Comparing constant terms and $z$ terms:

$\lambda^{2}=(2-\mu^{\prime})^{2},\qquad\lambda^{2}-2=2-\mu^{\prime 2},$

so either $(\lambda,\mu^{\prime})=(0,2)$ or $(\lambda,\mu^{\prime})=(\pm
2,0)$. In the first case, the first equation becomes

$-2z(1-z)=\rho.F(z),$

but this contradicts the fact that $F(z)$ is irreducible. In the second case,

$\rho F(z)=4(1+z)-2z(1-z)=2(z^{2}+z+2),$

so $F(z)=z^{2}+z+2$. Thus, $t=\frac{-1\pm\sqrt{-7}}{2}$ and $K=L(\sqrt{-7})$.

We perform a similar analysis in the remaining cases and give a brief summary
in each case, omitting the details of all our calculations for brevity.

#### 3\. $(1,0)$

This is Mordell’s case (II). We find that $\lambda^{2}=-2\in L$ – this is a
contradiction, since $\lambda\in L$.

#### 4\. $(-1,0)$

This is Mordell’s case (III). We find that $t=0$ – this is a contradiction,
since $t\not\in L$.

#### 5\. $(-1+\sqrt{2},0)$

We find that $\lambda^{2}=2(1-\sqrt{2})<0$ – this is a contradiction, since
$\lambda\in L$.

#### 6\. $(-1-\sqrt{2},0)$

We find that $\lambda^{2}=2(1+\sqrt{2})$ – this is a contradiction, since
$\lambda\in L$.

#### 7\. $(0,1)$

This is Mordell’s case (IV). We find that $(\mu^{\prime}-2)^{2}=-2$ – this is
a contradiction, since $\mu^{\prime}\in L$.

#### 8\. $(1,1)$

This is Mordell’s case (V). We find that
$\lambda+\mu=\rho+\sigma=\rho^{\prime}+\sigma^{\prime}=0$ and
$\lambda^{\prime}+\mu^{\prime}=2$. The equations (20) and (21) then become

$\displaystyle\lambda^{2}(1-z)^{2}-2z(1-z^{2})$ $\displaystyle=$
$\displaystyle\rho(1-z)F(z)$
$\displaystyle(2-\mu^{\prime}+\mu^{\prime}z)^{2}-2z-2z^{3}$ $\displaystyle=$
$\displaystyle\rho^{\prime}(1-z)F(z).$

By inspecting the $z^{3}$ and $z^{2}$ terms, it follows that
$\rho^{\prime}=\rho$ and $\lambda=\mu^{\prime}=0$. Inspecting the constant
term then yields $-4=0$.

#### 9\. $(-1+\sqrt{2},1)$

We find that $(\mu^{\prime}-2)^{2}=-\lambda^{2}(1+\sqrt{2})$ – this is a
contradiction, since $\lambda,\mu^{\prime}\in L$.

#### 10\. $(-1+\sqrt{2},1)$

We find that $\lambda^{2}=(\mu^{\prime}-2)^{2}(1+\sqrt{2})$ – this is a
contradiction, since $\lambda,\mu\in L$. ∎

This completes the proof of Theorem 1.1.

##  6\. A note on more general real biquadratic fields

We are currently studying the extent to which the above methods extend to more
general real biquadratic fields. For example, we find that the Frey curve (2)
is semistable over $\mathbb{Q}(\sqrt{2},\sqrt{11})$ and the space of newforms
of level $\mathcal{N}_{\mathfrak{p}}$ has dimension 0. Applying the above
method then yields the following result.

######  Theorem 6.1.

Let $K=\mathbb{Q}(\sqrt{2},\sqrt{11})$. Let $p\geq 29$ be a prime, $p\neq
197$. There are no non-trivial solutions to (1) over $K$ with exponent $p$.

We note that, in the statement of Theorem 6.1, the exclusion of $p=197$ arises
during the application of Lemma 3.1 when proving the irreducibility of
$\bar{\rho}_{E,p}$ (we were not able to prove the irreducibility of
$\bar{\rho}_{E,\;197}$ by other means). For real biquadratic fields of a
similar nature, one should expect the exclusion of a small set of primes to
arise in a similar way.

## References

  * [1] A. Aigner  “Über die Möglichkeit von $x^{4}+y^{4}=z^{4}$ in quadratischen Körpern”  In _Jber. Deutsch. Math.-Verein._ 43, 1934, pp. 226–229
  * [2] E. Arbarello, M. Cornalba, P.. Griffiths and J. Harris  “ _Geometry of algebraic curves. Vol. I_ ”  Springer-Verlag, New York, 1985
  * [3] W. Bosma, J. Cannon and C. Playoust  “The Magma algebra system. I. The user language”  In _J. Symbolic Comput._ 24, 1997, pp. 235–265
  * [4] J. Box  “Elliptic curves over totally real quartic fields not containing $\sqrt{5}$ are modular”  In _Trans. Amer. Math. Soc._ 375, 2022, pp. 3129–3172  USED:␣ARXIV: 2103.13975
  * [5] N. Bruin and M. Stoll  “Two-cover descent on hyperelliptic curves”  In _Math. Comp._ 78, 2009, pp. 2347–2370
  * [6] P. Bruin and F. Najman  “Hyperelliptic modular curves and isogenies of elliptic curves over quadratic fields”  In _LMS J. Comput. Math._ 18 Wiley, 2015, pp. 578–602
  * [7] M. Derickx, S. Kamienny, W. Stein and M. Stoll  “ _Torsion points on elliptic curves over number fields of small degree_ ”, 2021  arXiv:1707.00364
  * [8] M. Derickx, F. Najman and S. Siksek  “Elliptic curves over totally real cubic fields are modular”  In _Algebra & Number Theory_ 14 Mathematical Sciences Publishers, 2020, pp. 1791–1800
  * [9] D. Faddeev  “Group of divisor classes on the curve defined by the equation $x^{4}+y^{4}=1$”  In _Soviet Math. Dokl._ 1, 1960, pp. 1149–1151
  * [10] N. Freitas, B.. Hung and S. Siksek  “Elliptic curves over real quadratic fields are modular”  In _Invent. Math._ 201, 2015, pp. 159–206
  * [11] N. Freitas, A. Kraus and S. Siksek  “Class field theory, Diophantine analysis and the asymptotic Fermat’s Last Theorem”  In _Adv. Math._ 363, 2020, pp. 106964
  * [12] N. Freitas and S. Siksek  “Criteria for Irreducibility of mod $p$ Representations of Frey Curves”  In _J. Théor. Nombres Bordeaux_ 27, 2015, pp. 67–76
  * [13] N. Freitas and S. Siksek  “Fermat’s Last Theorem over some small real quadratic fields”  In _Algebra & Number Theory_ 9, 2014, pp. 875–895
  * [14] N. Freitas and S. Siksek  “The asymptotic Fermat’s last theorem for five-sixths of real quadratic fields”  In _Compos. Math._ 151, 2015, pp. 1395–1415
  * [15] K. Fujiwara  “ _Level optimization in the totally real case_ ”, 2006  arXiv:math/0602586
  * [16] B.. Gross and D.. Rohrlich  “Some results on the Mordell-Weil Group of the Jacobian of the Fermat curve.”  In _Invent. Math._ 44, 1978, pp. 201–224
  * [17] R. Hartshorne  “ _Algebraic geometry_ ”, Graduate Texts in Mathematics, No. 52  Springer-Verlag, New York-Heidelberg, 1977
  * [18] Y. Ishitsuka, T. Ito and T. Ohshita  “Explicit calculation of the mod 4 Galois representation associated with the Fermat quartic”  In _Int. J. Number Theory_ 16, 2019, pp. 881–905
  * [19] F. Jarvis  “Level lowering for modular mod $l$ Galois representations over totally real fields”  In _Math. Ann._ 313, 1999, pp. 141–160
  * [20] F. Jarvis  “Mazur’s Principle for totally real fields of odd degree”  In _Compos. Math._ 116, 1999, pp. 39–79
  * [21] F. Jarvis and P. Meekin  “The Fermat equation over $\mathbb{Q}(\sqrt{2})$”  In _J. Number Theory_ 109, 2004, pp. 182–196
  * [22] S. Kamienny  “Torsion points on elliptic curves and $q$-coefficients of modular forms”  In _Invent. Math._ 109, 1992, pp. 221–229
  * [23] N.. Katz  “Galois Properties of Torsion Points on Abelian Varieties.”  In _Invent. Math._ 62, 1981, pp. 481–502
  * [24] M. Klassen and P. Tzermias  “Algebraic points of low degree on the Fermat quintic”  In _Acta Arith._ 82, 1997, pp. 393–401
  * [25] A. Kraus  “Courbes elliptiques semi-stables sur les corps de nombres”  In _Int. J. Number Theory_ 03, 2011, pp. 611–633
  * [26] A. Kraus  “Quartic points on the Fermat quintic”  In _Ann. Math. Blaise Pascal_ 25, 2018, pp. 199–205
  * [27] B. Mazur  “Modular curves and the Eisenstein ideal”  In _Inst. Hautes Études Sci. Publ. Math._ 47, 1977, pp. 33–186
  * [28] B. Mazur and D. Goldfeld  “Rational isogenies of prime degree”  In _Invent. Math._ 44, 1978, pp. 129–162
  * [29] P. Michaud-Jacobs  “Fermat’s Last Theorem and modular curves over real quadratic fields”  In _Acta Arith._ 203, 2022, pp. 319–351
  * [30] L. Mordell  “The Diophantine equation $x^{4}+y^{4}=1$ in algebraic number fields”  In _Acta Arith._ 14, 1967, pp. 347–355
  * [31] E. Ozman and S. Siksek  “Quadratic points on modular curves”  In _Math. Comp._ 88, 2019, pp. 2461–2484
  * [32] P. Parent  “No 17-torsion on elliptic curves over cubic number fields”  In _J. Theor. Nombres Bordeaux_ 15, 2003, pp. 831–838
  * [33] P. Parent  “Torsion des courbes elliptiques sur les corps cubiques”  In _Ann. Inst. Fourier_ 50, 2000, pp. 723–749
  * [34] A. Rajaei  “On the levels of mod $l$ Hilbert modular forms”  In _J. Reine Angew. Math._ 537, 2001, pp. 33–65
  * [35] S. Roman  “ _Field Theory_ ”, Graduate Texts in Mathematics  Springer, 2006
  * [36] J.. Silverman  “ _Advanced Topics in the Arithmetic of Elliptic Curves_ ”  Springer-Verlag New York, 1994
  * [37] J.. Silverman  “ _The Arithmetic of Elliptic Curves_ ”  Springer, 2009
  * [38] M. Stoll  “ _Magma code for the paper[7]_”, 2020  URL: http://www.mathe2.uni-bayreuth.de/stoll/magma/#DKSS
  * [39] P. Tzermias  “Algebraic points of low degree on the Fermat curve of degree seven”  In _Manuscripta Math._ 97, 1998, pp. 483–488
  * [40] A. Wiles  “Modular elliptic curves and Fermat’s Last Theorem”  In _Ann. of Math._ 141 Annals of Mathematics, 1995, pp. 443–551


# Freezing density scaling of transport coefficients in the Weeks-Chandler-
Andersen fluid

S. A. Khrapak<EMAIL_ADDRESS>Joint Institute for High Temperatures,
Russian Academy of Sciences, 125412 Moscow, Russia  A. G. Khrapak  Joint
Institute for High Temperatures, Russian Academy of Sciences, 125412 Moscow,
Russia

###### Abstract

It is shown that the transport coefficients (self-diffusion, shear viscosity,
and thermal conductivity) of the Weeks-Chandler-Anderson (WCA) fluid along
isotherms exhibit a freezing density scaling (FDS). The functional form of
this FDS is essentially the same or closely related to those in the Lennard-
Jones fluid, hard-sphere fluid, and some liquefied noble gases. This proves
that this FDS represents a quasi-universal corresponding state principle for
simple classical fluids with steep interactions. Some related aspects such as
Stokes-Einstein relation without a hydrodynamic diameter and gas-to-liquid
dynamical crossover are briefly discussed. Simple fitting formula for the
transport coefficients of the dense WCA fluid are suggested.

##  I Introduction

Understanding transport properties of fluids is crucial in fields such as
chemical physics and engineering, condensed matter, materials science, and
environmental science. Despite certain progress Frenkel (1955); Balucani and
Zoppi (1994); March and Tosi (2002); Hansen and McDonald (2006) our
understanding remains rather limited and fragmented. It is unlikely that a
general theory can be developed, allowing to describe the actual transport
properties of fluids with varying molecular composition, interactions,
location on the phase diagram and other conditions. Even for simple fluids
such a task seems unrealistic. For this reason, various scaling relationships
and empirical correlations still play a very important role Rosenfeld (1977);
Dzugutov (1996); Dyre (2018); Bell  _et al._ (2019); Khrapak (2024a).

It has been recently discovered that appropriately reduced self-diffusion,
shear viscosity, and thermal conductivity coefficients of the Lennard-Jones
(LJ) fluid along isotherms exhibit quasi-universal scaling on the density
divided by its value at the freezing point Khrapak and Khrapak (2021a, 2022a,
2022b); Heyes  _et al._ (2023). This freezing density scaling (FDS) implies
that the transport coefficients are functions of a single variable
$\rho/\rho_{\rm fr}$, where $\rho$ is the number density and $\rho_{\rm fr}$
is its value at the freezing point. Since the dependence of $\rho_{\rm fr}$ on
temperature is relatively well known for the LJ system, FDS represents a very
convenient corresponding states principle to estimate transport properties of
the LJ fluid.

Originally considered as a useful empirical observation Khrapak and Khrapak
(2021a), the FDS has been later discussed in the context of quasi-universal
excess entropy scaling and isomorph theory Khrapak and Khrapak (2022a, b);
Heyes  _et al._ (2023). It occurs, however, that FDS holds even at quite low
densities where neither the original form of the excess entropy scaling
(predicting exponential dependence of transport coefficients on the excess
entropy Rosenfeld (1977, 1999)) nor the isomorph theory are expected to work.
The applicability domain of the FDS approach covers almost the entire gas-
liquid-fluid region in the LJ system phase diagram (probably excluding very
low densities, deep gas-liquid coexistence region, and the vicinity of the
critical point), including metastable regions of superheated and supercooled
liquids and supersaturated vapour Khrapak and Khrapak (2023a).

Remarkably, the functional form of the FDS scaling in the LJ fluid is similar
(although not identical) to that in the hard-sphere (HS) fluid Khrapak and
Khrapak (2021a). It also applies reasonably well to liquefied noble gases
Khrapak and Khrapak (2022b), where the LJ potential can serve only as a rough
approximation. Thus, it can be expected that FDS is not a special property of
the LJ fluid, but applies to a wider class of fluids. The purpose of this
paper is to validate this hypothesis using the Weeks-Chandler-Anderson (WCA)
model. In particular, we demonstrate that the reduced self-diffusion,
viscosity, and thermal conductivity coefficients of the WCA fluid exhibit the
same FDS scaling as the LJ fluid. This finding proves that the FDS approach is
a very useful practical tool to estimate the transport coefficients of various
simple fluids in a rather wide parameter regime.

##  II Formulation

We consider a classical system of $N$ particles of mass $m$ confined to the
volume $V$ (so that $\rho=N/V$) and interacting via the WCA repulsive
potential Weeks  _et al._ (1971):

$\phi(r)=4\epsilon\left[\left(\frac{\sigma}{r}\right)^{12}-\left(\frac{\sigma}{r}\right)^{6}\right]+\epsilon\quad\quad{\rm
for}\quad r<r_{c}$ (1)

and zero otherwise. Here $\epsilon$ and $\sigma$ are the energy and length
scales, and $r_{c}=2^{1/6}\sigma$ is the cut-off distance. The conventional
reduced units for temperature and density are $T^{*}=k_{\rm B}T/\epsilon$ and
$\rho^{*}=\rho\sigma^{3}$, respectively.

The WCA potential corresponds to the LJ potential, truncated and shifted at
the location of its minimum. It was originally used as a reference repulsive
potential when separating the LJ potential into the repulsive and attractive
parts Weeks  _et al._ (1971). Nowadays, the WCA system is often used as one of
the models of simple fluids de Kuijper  _et al._ (1990); Heyes and Okumura
(2006); Nasrabad (2008); Ahmed and Sadus (2009); Khrapak  _et al._ (2011);
Bell (2019); Attia  _et al._ (2022); Dyre and Pedersen (2023).

Since the WCA potential is purely repulsive, WCA system does not exhibit gas-
liquid phase transition and gas-liquid coexistence region. There are no
critical or triple points in this model. Its phase diagram is thus very
different from that of the LJ system. Similarly to the hard-sphere system only
the fluid-solid phase transition is present. Thus, identical values of
$\rho/\rho_{\rm fr}$ can be associated with very different phase states for
the LJ and WCA systems.

The transport coefficients of the WCA fluid are taken from Ref. Nasrabad
(2008), where equilibrium molecular dynamics simulations have been used to
compute thermodynamic, structural and dynamical quantities. The data cover
relatively wide region of the phase diagram: Five isotherms with $T^{*}=0.5$,
$1.0$, $2.0$, $4.0$, $6.0$ and the density in the range $0.3\leq\rho^{*}\leq
1.0$. The tabulated data are found in the Supplementary Material of Ref.
Nasrabad (2008). The transport coefficients of the LJ and WCA fluids are
usually reduced using $\sigma$ as a unit of distance, $\sqrt{\epsilon/m}$ as a
unit of velocity, and $\sigma/\sqrt{\epsilon/m}$ as a unit of time. To compare
transport properties in different systems it is more convenient to use a
universal macroscopic normalization (sometimes referred to as Rosenfeld’s
normalization Rosenfeld (1999)) with mean interparticle separation
$\Delta=\rho^{-1/3}$ as a unit of distance, thermal velocity $v_{\rm
T}=\sqrt{k_{\rm B}T/m}$ as a unit of velocity, and $\Delta/v_{\rm T}$ as a
unit of time. The reduced self-diffusion ($D$), viscosity ($\eta$), and
thermal conductivity ($\lambda$) coefficients are

$D_{\rm R}=D\frac{\rho^{1/3}}{v_{\rm T}},\quad\eta_{\rm
R}=\eta\frac{\rho^{-2/3}}{mv_{\rm T}},\quad\lambda_{\rm
R}=\lambda\frac{\rho^{-2/3}}{v_{\rm T}},$ (2)

Here the subscript $R$ denotes Rosenfeld normalization. We have recalculated
the original data presented in Ref. Nasrabad (2008) to get $D_{\rm R}$,
$\eta_{\rm R}$, and $\lambda_{\rm R}$.

The data on the fluid-solid coexistence of the WCA model can be found for
example in Refs. Ahmed and Sadus (2009); Mirzaeinia  _et al._ (2017); Dyre and
Pedersen (2023); Attia  _et al._ (2022). In a recent publication Attia  _et
al._ (2022), the points on the fluid and solid boundaries of the fluid-solid
coexistence have been determined by numerical integration of the Clausius-
Clapeyron relation in a very wide range of $T^{*}$. For the temperature range
$0.5\lesssim T^{*}\lesssim 6$, relevant for the present investigation, we use
a fourth order polynomial to fit the tabulated data for the liquidus. This
yields

$\displaystyle\rho^{*}_{\rm fr}$ $\displaystyle\simeq
0.77622+0.2346T^{*}-0.05393(T^{*})^{2}$
$\displaystyle+0.00789(T^{*})^{3}-4.56519\times 10^{-4}(T^{*})^{4}.$

This expression is used to evaluate the freezing density and the FDS scaling
parameter ${\mathcal{R}}=\rho/\rho_{\rm fr}$.

##  III Results

###  III.1 Stokes-Einstein relation

First, let us verify that the Stokes-Einstein (SE) relation between the self-
diffusion and viscosity coefficients is satisfied in the WCA fluid. We chose
the form known as SE relation without a hydrodynamic diameter Costigliola  _et
al._ (2019)

$D\eta\left(\frac{\Delta}{k_{\rm B}T}\right)\equiv D_{\rm R}\eta_{\rm
R}=\alpha_{\rm SE},$ (3)

where $\alpha_{\rm SE}$ is a numerical coefficient. In this version of the SE
relation the effective hydrodynamic diameter is essentially replaced by the
mean interparticle separation. There exist different theoretical approaches
explaining why it should be approximately so Li and Chang (1955); Zwanzig
(1983); Balucani  _et al._ (1990). The constant $\alpha_{\rm SE}$ is weakly
system-dependent Khrapak (2019); Khrapak and Khrapak (2021b). For simple
fluids, the general tendency is that it increases from $\alpha_{\rm SE}\simeq
0.14$ for plasma-related systems with soft (Coulomb-like) interparticle
interaction to $\alpha_{\rm SE}\simeq 0.17$ for extremely steep hard-sphere
interactions Khrapak and Khrapak (2021b).

Numerous confirmations of the applicability of the SE relation without the
hydrodynamic diameter to dense simple fluids have been reported, including
charged particles in the plasma-related context Daligault  _et al._ (2014);
Khrapak (2018, 2021); Khrapak and Khrapak (2021b), soft (inverse power)
repulsive particle fluid Heyes and Brańka (2007), the LJ fluid Costigliola
_et al._ (2019); Ohtori and Ishii (2015); Ohtori  _et al._ (2017); Khrapak and
Khrapak (2021b), and the HS fluid Ohtori  _et al._ (2018); Pieprzyk  _et al._
(2019); Khrapak and Khrapak (2021b). Several important non-spherical molecular
liquids have been examined using numerical simulations in Ref. Ohtori  _et
al._ (2020) and applicability of the SE relation (3) has also been confirmed.
This indicates that the applicability regime is apparently wider than simple
atomic systems with spherically symmetric interactions. Among recent
demonstrations of the applicability of SE relation in the form of Eq. (3) to
real liquids we mention iron at conditions of planetary cores Li  _et al._
(2021), dense supercritical methane (at least for the most state points
investigated) Ranieri  _et al._ (2021); Khrapak (2022a), silicon melt at high
temperatures Luo  _et al._ (2022), and water Khrapak and Khrapak (2023b);
Khrapak (2024b).

In view of previous investigations of the SE relation in the WCA fluid
Cappelezzo  _et al._ (2007); Ohtori  _et al._ (2018), there should be no doubt
that it applies in this case. The main questions are related to the
applicability regime and the value of the constant $\alpha_{\rm SE}$.

Figure 1: (Color online) Stokes-Einstein coefficient $\alpha_{\rm SE}$ in the
WCA fluid as a function of the FDS scaling parameter
${\mathcal{R}}=\rho/\rho_{\rm fr}$. The symbols correspond to the numerical
results from Ref. Nasrabad (2008). As the fluid-solid phase transition is
approached (${\mathcal{R}}=1$) the coefficients saturates at a constant value
$\simeq 0.15$, shown by the dashed line. The dotted curve corresponds to
dilute gas asymptote $\sim{\mathcal{R}}^{-4/3}$. The vertical axis is reversed
to highlight the accuracy of the SE relation at high densities.  Figure 2:
(Color online) Reduced self-diffusion (a), viscosity (b), and thermal
conductivity (c) coefficients of the WCA fluid versus the FDS scaling
parameter ${\mathcal{R}}=\rho/\rho_{\rm fr}$. Results for five isotherms:
$T^{*}=0.5$, $1.0$, $2.0$, $4.0$, $6.0$ denoted by symbols correspond to the
numerical results from Ref. Nasrabad (2008). The dashed curves correspond to
the LJ model FDS evaluated using the modified excess entropy scaling of Ref.
Bell  _et al._ (2019) for an isotherm $T^{*}=2.0$. The solid curves are the
exponential fits of Eqs. (4)-(6) applicable in the dense fluid regime
${\mathcal{R}}\gtrsim 0.6$.

The dependence of $\alpha_{\rm SE}$ on the FDS scaling parameter
${\mathcal{R}}$ is shown in Fig. 1. We observe that at low densities
$\alpha_{\rm SE}$ first decreases as density increases (note reversed
direction of the vertical axis in order to highlight the accuracy of the SE
relation at high densities). In the dilute gaseous regime the transport
coefficients scale approximately as Khrapak and Khrapak (2021a): $D_{\rm
R}\sim\eta_{\rm R}\sim\lambda_{\rm R}\sim(\rho^{*})^{-2/3}$. This implied
$\alpha_{\rm SE}\sim(\rho^{*})^{-4/3}$. If we further neglect a relatively
weak temperature dependence of the freezing density in the range of
temperatures investigated, we arrive at the scaling $\alpha_{\rm
SE}\sim{\mathcal{R}}^{-4/3}$, shown by the dotted line in Fig. 1. This is only
a rather crude approximation, since the freezing density actually increases
from $\rho_{\rm fr}\simeq 0.88$ at $T^{*}=0.5$ to $\rho_{\rm fr}\simeq 1.36$
at $T^{*}=6.0$ Attia  _et al._ (2022). Nevertheless, we observe in Fig. 1 that
the numerical data do approach this asymptote as the density decreases.

As the density increases and the freezing line is approached, the data are
saturating around a constant value $\alpha_{\rm SE}\simeq 0.15$, shown by a
horizontal dashed line in Fig. 1. The data points approach this asymptote at
${\mathcal{R}}\gtrsim 0.6$. This is where the SE relation is applicable and
this is where vibrational paradigm of atomic dynamics in liquids holds Khrapak
(2024a). The picture is very similar to that in the LJ fluid, see Fig. 1 from
Ref. Khrapak and Khrapak (2021b). Thus, the SE relation without the
hydrodynamic diameter works in the dense WCA fluid just as it does in the LJ
fluid.

###  III.2 Freezing density scaling

The reduced self-diffusion ($D_{\rm R}$), viscosity ($\eta_{R}$), and thermal
conductivity ($\lambda_{\rm R}$) coefficients as functions of the FDS scaling
parameter ${\mathcal{R}}=\rho/\rho_{\rm fr}$ are shown in Fig. 2. Symbols
correspond to five different isotherms studied in Ref. Nasrabad (2008). The
dashed curves correspond to the freezing density scaling in the LJ fluid, as
calculated using the model of Ref. Bell  _et al._ (2019). We observe that FDS
of transport coefficients in the WCA fluid essentially coincides with that in
the LJ fluid. Taking into account that similar FDS operates in the HS fluid
Khrapak and Khrapak (2021a) and in noble gases Khrapak and Khrapak (2022b)
indicates that this form of FDS is quasi-universal for sufficiently steep
interaction potentials.

It is observed in Fig. 2 that the self-diffusion coefficient decreases
monotonically with density, while the shear viscosity and thermal conductivity
coefficients increase towards the fluid-solid phase transition. Since in the
low density regime the transport coefficients scale as $D_{\rm R}\sim\eta_{\rm
R}\sim\lambda_{\rm R}\sim(\rho^{*})^{-2/3}$, minima in the reduced viscosity
and thermal conductivity coefficients should be expected, just as in other
classical fluids Khrapak and Khrapak (2022b, c). These minima correspond to a
crossover between different mechanisms of momentum and energy transport and
hence to a gas-liquid dynamical crossover (Frenkel line on the phase diagram),
which is a topic of considerable recent interest and debate Brazhkin  _et al._
(2012, 2013, 2018); Bryk  _et al._ (2018); Bell  _et al._ (2020); Khrapak
(2022b). However, since the available numerical results start from the density
$\rho^{*}=0.3$, the minima are not really observed in Fig. 2 and we do not
elaborate on this point further.

The FDS that applies to steep interaction potentials such as LJ, WCA, HS
cannot be truly universal. For extremely soft interaction potentials such as
Coulomb and screened Coulomb (Yukawa) potentials, which are of interest in the
plasma-related context Baus and Hansen (1980); Fortov  _et al._ (2004, 2005);
Ivlev  _et al._ (2012), the dependence of the transport coefficients on the
density is qualitatively similar. The minima in shear viscosity and thermal
conductivity occur at $\Gamma/\Gamma_{\rm fr}\simeq 0.05$ Huang  _et al._
(2023); Khrapak (2023), where $\Gamma=Q^{2}/ak_{\rm B}T$ is the Coulomb
coupling parameter, defined as the ratio of the Coulomb energy of interaction
at a mean interparticle separation to the kinetic energy, expressed using
system temperature. Here $Q$ is the electrical charge,
$a=(4\pi\rho/3)^{-1/3}\propto\rho^{-1/3}$ is the Wigner-Seitz radius, and
$\Gamma_{\rm fr}$ corresponds to the coupling parameter at the fluid-solid
phase transition (freezing). Rewritten in terms of the relative density, the
condition for the minima reads $\rho/\rho_{\rm fr}=(\Gamma/\Gamma_{\rm
fr})^{3}\simeq 0.000125$, which is drastically different from $\rho/\rho_{\rm
fr}\simeq 0.3$ in the LJ and HS fluids Khrapak and Khrapak (2021a, 2022c).
Thus, the shapes of FDS must be different for soft and steep interactions.
Where exactly the transition between soft and steep regimes occurs requires
further investigation and might be subject of future work.

Finally, a good representation of the reduced transport coefficients of the
dense WCA fluid can be achieved with simple exponential functions of the form

$\displaystyle D_{\rm R}\simeq 0.3e^{-2.4{\mathcal{R}}^{2.7}},$ (4)
$\displaystyle\eta_{\rm R}=0.5e^{2.4{\mathcal{R}}^{2.7}},$ (5)
$\displaystyle\lambda_{\rm R}\simeq 1.2e^{2.15{\mathcal{R}}^{1.1}}.$ (6)

These fits are shown by the solid lines in Fig. 2. The agreement with
numerical results at $\rho/\rho_{\rm fr}\gtrsim 0.6$ is convincing (but there
is still potential for improvement when new data become available). They
should not be applied for lower densities, because they are not consistent
with the behavior of transport coefficients in this regime. Note that the fits
for the self-diffusion and viscosity coefficients satisfy the SE relation at
high densities, $D_{\rm R}\eta_{\rm R}\simeq 0.15$. Formulas (4) - (6) are
purely empirical fits convenient for practical application. Nevertheless,
there is a certain similarity with the excess entropy scaling proposed by
Rosenfeld Rosenfeld (1977, 1999), where the reduced transport coefficients are
described as exponential functions of excess entropy. Actually, the FDS
scaling parameter ${\mathcal{R}}$ and the excess entropy are intimately
related, as illustrated in Fig. 2 of Ref. Khrapak and Khrapak (2022a) for the
LJ fluid. These relations have not been used in constructing Eqs. (4)-(6).
Other functional forms for the dependencies $\eta_{\rm R}({\mathcal{R}})$ and
$D_{\rm R}({\mathcal{R}})$ have been recently proposed in Ref. Heyes  _et al._
(2023).

##  IV Conclusion

Unlike transport properties of dilute gases and solids, transport properties
of liquids are much less understood. This is why various approximate
relationships and empirical correlations still play important role in
describing their transport properties.

This paper has been focused on the freezing density scaling, proposed
originally to describe the self-diffusion, viscosity, and thermal conductivity
coefficients of the LJ fluid. It has been demonstrated here that the same or
very similar FDS applies to the WCA fluid as well. Given the fact that the
phase diagrams of the LJ and WCA systems are completely different and taking
into account the previous observation that LJ-type FDS is similar to that in
the fluid of hard spheres and some liquefied noble gases, our results provide
strong indication about quasi-universal character of this FDS. Therefore, we
confirm the usefulness of the FDS approach to estimate transport coefficients
of dense fluids with steep interatomic interactions. This can be of interest
for researchers in chemical physics, (soft) condensed matter, physics of
fluids, materials science and beyond.

The authors have no conflicts of interest to disclose.

Data sharing is not applicable to this article as no new data were created or
analyzed in this study.

## References

  * Frenkel (1955) Y. Frenkel,  _Kinetic theory of liquids_ (Dover, New York, NY, 1955).
  * Balucani and Zoppi (1994) U. Balucani and M. Zoppi,  _Dynamics of the Liquid State_ (Clarendon Press, Oxford, 1994).
  * March and Tosi (2002) N. H. March and M. P. Tosi,  _Introduction to Liquid State Physics_ (World Scientific Pub Co Inc, 2002).
  * Hansen and McDonald (2006) J.-P. Hansen and I. R. McDonald,  _Theory of Simple Liquids -_ (Elsevier, Amsterdam, 2006).
  * Rosenfeld (1977) Y. Rosenfeld, “Relation between the transport coefficients and the internal entropy of simple systems,” Phys. Rev. A 15, 2545–2549 (1977).
  * Dzugutov (1996) M. Dzugutov, “A universal scaling law for atomic diffusion in condensed matter,” Nature 381, 137–139 (1996).
  * Dyre (2018) J. C. Dyre, “Perspective: Excess-entropy scaling,” J. Chem. Phys. 149, 210901 (2018).
  * Bell  _et al._ (2019) I. H. Bell, R. Messerly, M. Thol, L. Costigliola, and J. C. Dyre, “Modified entropy scaling of the transport properties of the Lennard-Jones fluid,” J. Phys. Chem. B 123, 6345–6363 (2019).
  * Khrapak (2024a) S.A. Khrapak, “Elementary vibrational model for transport properties of dense fluids,” Phys. Rep. 1050, 1–29 (2024a).
  * Khrapak and Khrapak (2021a) S. A. Khrapak and A. G. Khrapak, “Transport properties of Lennard-Jones fluids: Freezing density scaling along isotherms,” Phys. Rev. E 103, 042122 (2021a).
  * Khrapak and Khrapak (2022a) S. A. Khrapak and A. G. Khrapak, “Freezing temperature and density scaling of transport coefficients,” J. Phys. Chem. Lett. 13, 2674–2678 (2022a).
  * Khrapak and Khrapak (2022b) S. A. Khrapak and A. G. Khrapak, “Freezing density scaling of fluid transport properties: Application to liquefied noble gases,” J. Chem. Phys. 157, 014501 (2022b).
  * Heyes  _et al._ (2023) D. M. Heyes, D. Dini, S. Pieprzyk, and A. C. Brańka, “Departures from perfect isomorph behavior in Lennard-Jones fluids and solids,” J. Chem. Phys. 158, 134502 (2023).
  * Rosenfeld (1999) Y. Rosenfeld, “A quasi-universal scaling law for atomic transport in simple fluids,” J. Phys.: Condens. Matter 11, 5415–5427 (1999).
  * Khrapak and Khrapak (2023a) S. A. Khrapak and A. G. Khrapak, “Vibrational model for thermal conductivity of Lennard-Jones fluids: Applicability domain and accuracy level,” Phy. Rev. E 108, 064129 (2023a).
  * Weeks  _et al._ (1971) J. D. Weeks, D. Chandler, and H. C. Andersen, “Role of repulsive forces in determining the equilibrium structure of simple liquids,” J. Chem. Phys. 54, 5237–5247 (1971).
  * de Kuijper  _et al._ (1990) A. de Kuijper, J. A. Schouten, and J. P. J. Michels, “The melting line of the Weeks-Chandler-Anderson Lennard-Jones reference system,” J. Chem. Phys. 93, 3515 (1990).
  * Heyes and Okumura (2006) D. M. Heyes and H. Okumura, “Equation of state and structural properties of the Weeks-Chandler-Andersen fluid,” J. Chem. Phys. 124, 164507 (2006).
  * Nasrabad (2008) A. Nasrabad, “Thermodynamic and transport properties of the Weeks-Chandler-Andersen fluid: Theory and computer simulation,” J. Chem. Phys. 129, 244508 (2008).
  * Ahmed and Sadus (2009) A. Ahmed and R. J. Sadus, “Phase diagram of the Weeks-Chandler-Andersen potential from very low to high temperatures and pressures,” Phys. Rev. E 80, 061101 (2009).
  * Khrapak  _et al._ (2011) S. A. Khrapak, M. Chaudhuri, and G. E. Morfill, “Communication: Universality of the melting curves for a wide range of interaction potentials,” J. Chem. Phys. 134, 241101 (2011).
  * Bell (2019) I. H. Bell, “Probing the link between residual entropy and viscosity of molecular fluids and model potentials,” PNAS 116, 4070–4079 (2019).
  * Attia  _et al._ (2022) E. Attia, J. C. Dyre, and U. R. Pedersen, “Comparing four hard-sphere approximations for the low-temperature WCA melting line,” J. Chem. Phys. 157, 034502 (2022).
  * Dyre and Pedersen (2023) J. C. Dyre and U. R. Pedersen, “Comparing zero-parameter theories for the WCA and harmonic-repulsive melting lines,” J. Chem. Phys. 158, 164504 (2023).
  * Mirzaeinia  _et al._ (2017) A. Mirzaeinia, F. Feyzi, and S. M. Hashemianzadeh, “Equation of state and Helmholtz free energy for the atomic system of the repulsive Lennard-Jones particles,” J. Chem. Phys. 147, 214503 (2017).
  * Costigliola  _et al._ (2019) L. Costigliola, D. M. Heyes, T. B. Schrøder, and J. C. Dyre, “Revisiting the Stokes-Einstein relation without a hydrodynamic diameter,” J. Chem. Phys. 150, 021101 (2019).
  * Li and Chang (1955) J. C. M. Li and P. Chang, “Self-diffusion coefficient and viscosity in liquids,” J. Chem. Phys. 23, 518–520 (1955).
  * Zwanzig (1983) R. Zwanzig, “On the relation between self-diffusion and viscosity of liquids,” J. Chem. Phys. 79, 4507–4508 (1983).
  * Balucani  _et al._ (1990) U. Balucani, R. Vallauri, and T. Gaskell, “Generalized Stokes-Einstein relation,” Berichte der Bunsengesellschaft für physikalische Chemie 94, 261–264 (1990).
  * Khrapak (2019) S. Khrapak, “Stokes–Einstein relation in simple fluids revisited,” Mol. Phys. 118, e1643045 (2019).
  * Khrapak and Khrapak (2021b) S. A. Khrapak and A. G. Khrapak, “Excess entropy and Stokes-Einstein relation in simple fluids,” Phys. Rev. E 104, 044110 (2021b).
  * Daligault  _et al._ (2014) J. Daligault, K. Rasmussen, and S. D. Baalrud, “Determination of the shear viscosity of the one-component plasma,” Phys. Rev. E 90, 033105 (2014).
  * Khrapak (2018) S. Khrapak, “Practical formula for the shear viscosity of Yukawa fluids,” AIP Adv. 8, 105226 (2018).
  * Khrapak (2021) S. A. Khrapak, “Self-diffusion in simple liquids as a random walk process,” Molecules 26, 7499 (2021).
  * Heyes and Brańka (2007) D. M. Heyes and A. C. Brańka, “Physical properties of soft repulsive particle fluids,” Phys. Chem. Chem. Phys. 9, 5570 (2007).
  * Ohtori and Ishii (2015) N. Ohtori and Y. Ishii, “Explicit expression for the Stokes-Einstein relation for pure Lennard-Jones liquids,” Phys. Rev. E 91, 012111 (2015).
  * Ohtori  _et al._ (2017) N. Ohtori, S. Miyamoto, and Y. Ishii, “Breakdown of the Stokes-Einstein relation in pure Lennard-Jones fluids: From gas to liquid via supercritical states,” Phys. Rev. E 95, 052122 (2017).
  * Ohtori  _et al._ (2018) N. Ohtori, H. Uchiyama, and Y. Ishii, “The Stokes-Einstein relation for simple fluids: From hard-sphere to Lennard-Jones via WCA potentials,” J. Chem. Phys. 149, 214501 (2018).
  * Pieprzyk  _et al._ (2019) S. Pieprzyk, M. N. Bannerman, A. C. Brańka, M. Chudak, and D. M. Heyes, “Thermodynamic and dynamical properties of the hard sphere system revisited by molecular dynamics simulation,” Phys. Chem. Chem. Phys. 21, 6886–6899 (2019).
  * Ohtori  _et al._ (2020) N. Ohtori, Y. Kondo, K. Shintani, T. Murakami, T. Nobuta, and Y. Ishii, “The Stokes-Einstein relation for non-spherical molecular liquids,” Chem. Lett. 49, 379–382 (2020).
  * Li  _et al._ (2021) Q. Li, T. Sun, Y. Zhang, J.-W. Xian, and L. Vocadlo, “Atomic transport properties of liquid iron at conditions of planetary cores,” J. Chem. Phys. 155, 194505 (2021).
  * Ranieri  _et al._ (2021) U. Ranieri, S. Klotz, R. Gaal, M. M. Koza, and L. E. Bove, “Diffusion in dense supercritical methane from quasi-elastic neutron scattering measurements,” Nature Commun. 12, 1958 (2021).
  * Khrapak (2022a) S.A. Khrapak, “Diffusion, viscosity, and Stokes-Einstein relation in dense supercritical methane,” J. Mol. Liq. 354, 118840 (2022a).
  * Luo  _et al._ (2022) J. Luo, C. Zhou, Q. Li, Y. Lin, and L. Liu, “Atomic transport properties of silicon melt at high temperature,” J. Crystal Growth 590, 126701 (2022).
  * Khrapak and Khrapak (2023b) S. A. Khrapak and A. G. Khrapak, “Stokes–Einstein relation without hydrodynamic diameter in the TIP4P/Ice water model,” J. Chem. Phys. 158, 206101 (2023b).
  * Khrapak (2024b) S. A. Khrapak, “System size dependence of the diffusion coefficients in MD simulations: A simple correction formula for pure dense fluids,” J. Phys. Chem. B 128, 287 (2024b).
  * Cappelezzo  _et al._ (2007) M. Cappelezzo, C. A. Capellari, S. H. Pezzin, and L. A. F. Coelho, “Stokes-Einstein relation for pure simple fluids,” J. Chem. Phys. 126, 224516 (2007).
  * Khrapak and Khrapak (2022c) S. A. Khrapak and A. G. Khrapak, “Minima of shear viscosity and thermal conductivity coefficients of classical fluids,” Phys. Fluids 34, 027102 (2022c).
  * Brazhkin  _et al._ (2012) V. V. Brazhkin, Yu. D. Fomin, A. G. Lyapin, V. N. Ryzhov, and K. Trachenko, “Two liquid states of matter: A dynamic line on a phase diagram,” Phys. Rev. E 85, 031203 (2012).
  * Brazhkin  _et al._ (2013) V. V. Brazhkin, Yu. D. Fomin, A. G. Lyapin, V. N. Ryzhov, E. N. Tsiok, and K. Trachenko, “Liquid-gas” transition in the supercritical region: fundamental changes in the particle dynamics,” Phys. Rev. Lett. 111, 145901 (2013).
  * Brazhkin  _et al._ (2018) V. V. Brazhkin, C. Prescher, Yu. D. Fomin, E. N. Tsiok, A. G. Lyapin, V. N. Ryzhov, V. B. Prakapenka, J. Stefanski, K. Trachenko, and A. Sapelkin, “Comment on Behavior of supercritical fluids across the Frenkel line,” J. Phys. Chem. B 122, 6124–6128 (2018).
  * Bryk  _et al._ (2018) T. Bryk, F. A. Gorelli, I. Mryglod, G. Ruocco, M. Santoro, and T. Scopigno, “Reply to comment on behavior of supercritical fluids across the Frenkel line,” J. Phys. Chem. B 122, 6120–6123 (2018).
  * Bell  _et al._ (2020) I. H. Bell, G. Galliero, S. Delage-Santacreu, and L. Costigliola, “An entropy scaling demarcation of gas- and liquid-like fluid behaviors,” J. Chem. Phys. 152, 191102 (2020).
  * Khrapak (2022b) S. A. Khrapak, “Gas-liquid crossover in the Lennard-Jones system,” J. Chem. Phys. 156, 116101 (2022b).
  * Baus and Hansen (1980) M Baus and J. P. Hansen, “Statistical mechanics of simple Coulomb systems,” Phys. Rep. 59, 1–94 (1980).
  * Fortov  _et al._ (2004) V. E. Fortov, A. G. Khrapak, S. A. Khrapak, V. I. Molotkov, and O. F. Petrov, “Dusty plasmas,” Phys.-Usp. 47, 447 – 492 (2004).
  * Fortov  _et al._ (2005) V. E. Fortov, A. V. Ivlev, S. A. Khrapak, A. G. Khrapak, and G. E. Morfill, “Complex (dusty) plasmas: Current status, open issues, perspectives,” Phys. Rep. 421, 1–103 (2005).
  * Ivlev  _et al._ (2012) A. Ivlev, H. Löwen, G. Morfill, and C. P. Royall,  _Complex Plasmas and Colloidal Dispersions: Particle-Resolved Studies of Classical Liquids and Solids_ (World Scientific, 2012).
  * Huang  _et al._ (2023) D. Huang, M. Baggioli, Sh. Lu, Z. Ma, and Y. Feng, “Revealing the supercritical dynamics of dusty plasmas and their liquidlike to gaslike dynamical crossover,” Physical Review Research 5, 013149 (2023).
  * Khrapak (2023) S. A. Khrapak, “Vibrational model of heat transfer in strongly coupled Yukawa fluids (dusty plasma liquids),” Plasma Phys. Rep. 49, 15–22 (2023).