#  Non-integrability of the rational Sasano system of type $A_{4}^{(2)}$

Tsvetana Stoyanova  Department of Mathematics and Informatics, Sofia
University, 5 J. Bourchier Blvd., Sofia 1164, Bulgaria<EMAIL_ADDRESS>
(Date: 22.02.2024)

###### Abstract.

In this paper we prove rigorously that for these values of the parameters for
which the Sasano system of type $A^{(2)}_{4}$ has a particular rational
solution it is not integrable by meromorphic first integrals which are
rational functions in $t$. By an explicit computation we show that the
connected component of the unit element $G^{0}$ of the differential Galois
group of the normal variational equations along a simple particular rational
solution is a direct product of two groups $SL_{2}({\mathbb{C}})$. Thus the
Morales-Ramis theory leads to a non-integrable result. Moreover, using
Bäcklund transformations we extend the obtained particular non-integrable
result to the entire orbit of the parameters.

Key words: Sasano systems, Non-integrability of Hamiltonian systems,
Differential Galois theory, Whittaker equation

2010 Mathematics Subject Classification: 34M55, 37J30, 33C15

##  1\. Introduction

The Sasano systems are higher order Painlevé systems which admit certain
affine Weyl groups of symmetry. They were introduced by Yusuke Sasano in a
series of papers [8, 9, 10, 11, 12, 13, 14] where for a given Weyl group of
symmetry $W(A)$ he found a higher order system of nonlinear ordinary
differential equations for which $W(A)$ acts as its Bäcklund transformations.
The Sasano systems can be considered as higher order analogues of the
classical second-order Panlevé equations. Although they were introduced by
strictly mathematical consideration the Sasano systems turn out to find
applications in mathematical physics through the Drinfeld-Sokolov hierarchies
[2].

One of the essential properties of the Sasano systems is that all of them are
time dependent Hamiltonian systems. The question about their integrability
from the point of view of the Hamiltonian dynamics then naturally arises. In
[19] we have proved that the Sasano system of type $D^{(1)}_{5}$ is not
integrable with values of the parameters
$\alpha_{1}=\alpha_{2}=\alpha_{3}=0,\,\alpha_{0}+\alpha_{5}=0$. In this paper
we study the integrabilty of the Sasano system of type $A^{(4)}_{2}$ given by
[9]

$\displaystyle\dot{x}$ $\displaystyle=$ $\displaystyle
4\,x\,y-2\alpha_{1}+2\,z\,w,$ $\displaystyle\dot{y}$ $\displaystyle=$
$\displaystyle-2\,y^{2}-4\,x-2\,t-w,$ (1.1) $\displaystyle\dot{z}$
$\displaystyle=$ $\displaystyle z^{2}-w+x+2\,y\,z,$ $\displaystyle\dot{w}$
$\displaystyle=$ $\displaystyle-2\,z\,w-\alpha_{0}-2\,y\,w\,,$

where $\alpha_{j},0\leq j\leq 2$ are complex parameters which satisfy the
following relation

$\displaystyle\alpha_{0}+2\alpha_{1}+2\alpha_{2}=1$ (1.2)

The Sasano system (1) admits the affine Weyl group $W(A^{(4)}_{2})$ as a group
of Bäcklund transformations [9]. The system (1) is a time-dependent
Hamiltonian system of $2+1/2$ degrees of freedom [9]

$\displaystyle\frac{dx}{dt}=\frac{\partial H}{\partial
y},\quad\frac{dy}{dt}=-\frac{\partial H}{\partial
x},\quad\frac{dz}{dt}=\frac{\partial H}{\partial
w},\quad\frac{dw}{dt}=-\frac{\partial H}{\partial z}\,,$

with the polynomial Hamiltonian

$\displaystyle H$ $\displaystyle=$ $\displaystyle
2\,x\,y^{2}+2\,x^{2}+2\,t\,x-2\,\alpha_{1}\,y+z^{2}\,w-\frac{w^{2}}{2}+\alpha_{0}\,z+x\,w+2\,y\,z\,w=$
$\displaystyle=$ $\displaystyle
2\,H_{II}(x,y,t;\alpha_{1})+H_{II}^{\textrm{auto}}(z,w,;\alpha_{0})+x\,w+2\,y\,z\,w,.$

The symbol $H_{II}$ denotes the Hamiltonian associated to the second Painlevé
equation given in the form

$\,H_{II}(x,y,t;\alpha_{1})=x\,y^{2}+x^{2}+t\,x-\alpha_{1}\,y\,,\,$

and the symbol $H_{II}^{\textrm{auto}}$ denotes the autonomous version of
$H_{II}$ given by

$\,H_{II}^{\textrm{auto}}(z,w;\alpha_{0})=z^{2}\,w-\frac{w^{2}}{2}+\alpha_{0}\,z\,.\,$

In this way the Sasano system (1) can be considered as a coupled Painlevé II
equation. The system (1) can be turned into an autonomous Hamiltonian system
with three degrees of freedom by introducing two new dynamical variables : $t$
and a conjugate to it $F$. The new Hamiltonian becomes

$\,\widehat{H}=H+F\,$

and the associated Hamiltonian system is

$\displaystyle\frac{dx}{ds}=\frac{\partial\widehat{H}}{\partial
y},\quad\frac{dy}{ds}=-\frac{\partial\widehat{H}}{\partial
x},\quad\frac{dz}{ds}=\frac{\partial\widehat{H}}{\partial
w},\quad\frac{dw}{ds}=-\frac{\partial\widehat{H}}{\partial z},$ (1.3)
$\displaystyle\frac{dt}{ds}=\frac{\partial\widehat{H}}{\partial
F},\quad\frac{dF}{ds}=-\frac{\partial\widehat{H}}{\partial t}\,.$

The symplectic structure $\Omega$ is canonical in the variables
$(x,y,z,w,t,F)$, that is $\Omega=dy\wedge dx+dw\wedge dz+dF\wedge dt$.

In this paper we strictly prove that for these values of the parameters
$\alpha_{j},j=0,1,2$ for which the Sasano system (1) admits a particular
rational solution the Hamiltonian system (1.3) is not integrable by
meromorphic first integrals which are rational functions in $t$. Recall that
according to the theorem of Liouville-Arnold [1] this means non-existence of
three independent meromorphic first integrals in involution all of them being
rational functions in $t$. As in our works on the non-integrability of the
second order Painlevé equation [3, 16, 17, 18] and of the Sasano system of
type $D^{(1)}_{5}$ [19] in this paper we utilize the Morales-Ramis theory of
non-integrability of Hamiltonian systems. This theory transforms the problem
of integrability of a given complex analytical Hamiltonian system in the sense
of the theorem of Liouville-Arnold to the problem of integrability of a linear
system of ordinary differential equations in the sense of the differential
Galois theory.

From the paper [5] of Matsuda it follows that all the rational solutions of
the Sasano system (1) can be obtained by Bäcklund transformations from the
following simple particular rational solution

$\displaystyle x=w=-\frac{2}{5}\,t,\quad
y=z=0,\quad\alpha_{0}=\frac{2}{5},\quad\alpha_{1}=\frac{1}{5},\quad\alpha_{2}=\frac{1}{10}\,.$
(1.4)

In this paper studying the normal variational equations of (1.3) along the
solution (1.4) we establish a non-integrable result. More precisely, by
successive shearing and similar transformations we reduce the normal
variational equations, which a fourth order linear system, to a system
consisted of two independent second order linear systems. Each of these
systems is equivalent to a non-solvable Whittaker equation. This fact implies
that the connected component $G^{0}$ of the unit element of the differential
Galois group $G$ of the normal variational equations has the form
$G^{0}=SL_{2}({\mathbb{C}})\times SL_{2}({\mathbb{C}})$ which is not Abelian.
Then the key result of this paper is the following theorem

######  Theorem 1.1.

Assume that

$\,\alpha_{0}=\frac{2}{5},\quad\alpha_{1}=\frac{1}{5},\quad\alpha_{2}=\frac{1}{10}\,.\,$

Then the Sasano system of type $A^{(2)}_{4}$ (1) is not integrable by
meromorphic first integrals which are rational functions in $t$.

Since the Bäcklund transformations of the Sasano system of type $A^{(2)}_{4}$
are canonical transformations which are rational in all of the dynamical
variables [9] we can extend the key result of Theorem 1.1 to the main theorem
of this paper

######  Theorem 1.2.

The Sasano system of type $A^{(2)}_{4}$ is not integrable by meromorphic first
integrals which are rational functions in $t$.

This paper is organized as follows. In the next section we briefly recall the
main theorem of the Morales-Ramis theory of non-integrability of Hamiltonian
systems. In Section 3 we prove the key Theorem 1.1. In the last section using
the Bäcklund transformations of the Sasano system (1) we establish the main
result of this paper.

##  2\. Preliminaries

In this section we briefly recall the Morales-Ramis theory on non-
integrability of Hamiltonian systems following [6, 7].

Consider a complex analytical Hamiltonian system

$\displaystyle\dot{x}=X_{H}(x)$ (2.5)

with a Hamiltonian $H:M\rightarrow{\mathbb{C}}$ on a complex symplectic
manifold $M$ of dimension $2n$. Recall that by the theorem of Liouville-Arnold
[1] the Hamiltonian system (2.5) is completely integrable if there exist $n$
functionally independent first integrals $f_{1}=H,f_{2},f_{3},\ldots$ and in
involution. Let $x(t)$ be a non-equilibrium particular solution of (2.5).
denote by $\Gamma$ the phase curve corresponding to this solution. The
variational equations $(\textrm{VE})$ along $\Gamma$ have the form

$\,\dot{\xi}=\frac{\partial X_{H}(x(t))}{\partial x}\,\xi,\quad\xi\in
T_{\Gamma}M\,.\,$

We always can reduce the variational equations using the Hamiltonian in the
following sense. Consider the normal bundle of $\Gamma$ on the level variety
$M_{h}=\\{x\,|\,H(x)=h\\}$. The projection of the variational equations on
this bundle is called the normal variational equations $(\textrm{NVE})$. The
dimension of the $(\textrm{NVE})$ is $2n-2$. The solutions of the
$(\textrm{NVE})$ define a Picard-Vessiot extension $L$ of the basic
differential field $K$ of coefficients of the $(\textrm{NVE})$. This in its
turn defines a differential Galois group $G=\textrm{Gal}(L/K)$. Then the main
theorem of the Morales-Ramis theory states [6, 7]

######  Theorem 2.1.

(Morales-Ruiz and Ramis) If the Hamiltonian system (2.5) is completely
integrable by meromorphic first integrals in a neighborhood of $\Gamma$, not
necessarily independent on $\Gamma$ itself, then the connected component
$(G)^{0}$ of the unit element of the differential Galois group
$G=\textrm{Gal}(L/K)$ is Abelian.

Theorem 2.1 provides a necessary condition for completely integrability and a
sufficient condition for non-integrability of a given Hamiltonian system. That
is, if the connected component of the unit element of the Galois group $G$ is
not Abelian, then the corresponding Hamiltonian system (2.5) is not integrable
by meromorphic first integrals.

##  3\. Non-integrability for
$(\alpha_{0},\alpha_{1},\alpha_{2})=(2/5,1/5,1/10)$

In this section we will prove that when
$(\alpha_{0},\alpha_{1},\alpha_{2})=(2/5,1/5,1/10)$ the extended autonomous
Sasano system (1.3) is not integrable by meromorphic first integrals which are
rational functions in $t$. To obtain this result we will study the variational
equations along the basic rational solutions

$\displaystyle x=-\frac{2}{5}\,s,\quad y=0,\quad z=0,\quad
w=-\frac{2}{5}\,s,\quad t=s,\quad F=\frac{2}{5}\,s^{2}\,.$ (3.6)

Sinse $t=s$ from here on we use $t$ as an independent variable instead of $s$.

For the normal variational equations $(\textrm{NVE})$ of the system (1.3)
along the particular solution (3.6) we obtain the system

$\displaystyle\dot{x}_{1}$ $\displaystyle=$
$\displaystyle-\frac{8}{5}\,t\,y_{1}-\frac{4}{5}\,t\,z_{1},$
$\displaystyle\dot{y}_{1}$ $\displaystyle=$ $\displaystyle-4\,x_{1}-w_{1},$
$\displaystyle\dot{z}_{1}$ $\displaystyle=$ $\displaystyle x_{1}-w_{1},$
$\displaystyle\dot{w}_{1}$ $\displaystyle=$
$\displaystyle\frac{4}{5}\,t\,y_{1}+\frac{4}{5}\,t\,z_{1}\,.$

The $(\textrm{NVE})$ have one singular point over
${\mathbb{C}}{\mathbb{P}}^{1}$ at $t=\infty$ which is an irregular
singularity. The origin is one ordinary point for it.

The so obtained $(\textrm{NVE})$ can be written in the form

$\displaystyle t^{-1}\,\dot{X}(t)=A(t)\,X(t)\,,$ (3.7)

where $X(t)$ is the vector-column $X(t)=(x_{1}(t),y_{1}(t),z_{1}(t),w_{1}(t))$
and $A(t)$ is the holomophic matrix at $t=\infty$

$\displaystyle
A(t)=\left(\begin{array}[]{rrrr}0&-\frac{8}{5}&-\frac{4}{5}&0\\\\[0.6458pt]
-\frac{4}{t}&0&0&-\frac{1}{t}\\\\[0.6458pt]
\frac{1}{t}&0&0&-\frac{1}{t}\\\\[0.6458pt]
0&\frac{4}{5}&\frac{4}{5}&0\end{array}\right)\,.$ (3.12)

Denote by $M_{n}(t)$ the set of all square matrices of order $n$ whose
elements are functions in $t$.

######  Theorem 3.1.

There exist a matrix $S(t)\in M_{4}(t)$ whose elements are algebraic functions
in $t$ and an algebraic function $f(\tau)$ in $\tau$, such that the
transformation

$\displaystyle X(t)=S(t)\,V(t),\qquad t=f(\tau)$ (3.13)

reduces the system (3.7) to the system

$\displaystyle\tau^{-5}\,V^{\prime}(\tau)=Q(\tau)\,V(\tau),\quad^{\prime}=\frac{d}{d\tau}\,.$
(3.14)

The matrix $Q(\tau)$ is a holomorphic matrix at $\tau=\infty$ which has the
form

$\,Q(\tau)=\left(\begin{array}[]{cc}Q_{1}(\tau)&0\\\
0&Q_{2}(\tau)\end{array}\right)\,$

with $Q_{j}(\tau)\in M_{2}(\tau),j=1,2$.

###### Proof.

In order to simplify the system (3.7) we apply to it shearing transformations
at the irregular singularity $t=\infty$. For details about such
transformations the reader can look at the book of Wasow [20].

All the eigenvalues of matrix $A(\infty)$ are equal to zero. The
transformation $X(t)=T_{1}\,V_{1}(t)$ with

$\,T_{1}=\left(\begin{array}[]{crcr}4&0&0&0\\\ 0&-5&0&-5\\\ 0&5&0&10\\\
0&0&4&0\end{array}\right),\qquad
T_{1}^{-1}=\left(\begin{array}[]{crrc}\frac{1}{4}&0&0&0\\\\[0.6458pt]
0&-\frac{2}{5}&-\frac{1}{5}&0\\\\[0.6458pt] 0&0&0&\frac{1}{4}\\\\[0.6458pt]
0&\frac{1}{5}&\frac{1}{5}&0\end{array}\right)\,$

takes the system (3.7) into the system

$\displaystyle
t^{-1}\,\dot{V}_{1}(t)=T_{1}^{-1}\,A(t)\,T(t)=B(t)\,V_{1}(t)=\left[B_{0}+B_{1}\,t^{-1}\right]$
(3.15)

with

$\,B_{0}=\left(\begin{array}[]{cccc}0&1&0&0\\\ 0&0&0&0\\\ 0&0&0&1\\\
0&0&0&0\end{array}\right),\qquad
B_{1}=\left(\begin{array}[]{rcrc}0&0&0&0\\\\[0.6458pt]
\frac{28}{5}&0&\frac{12}{5}&0\\\\[0.6458pt] 0&0&0&0\\\\[0.6458pt]
-\frac{12}{5}&0&-\frac{8}{5}&0\end{array}\right)\,.\,$

The first shearing transformation $V_{1}(t)=S_{1}(t)\,V_{2}(t)$ with
$S_{1}(t)={\mathrm{diag}}(1,t^{-g},t^{-2g},t^{-3g})$ takes the system (3.15)
into the system

$\displaystyle
t^{-1}\,\dot{V}_{2}(t)=\left[S^{-1}(t)\,B(t)\,S(t)-t^{-1}\,S^{-1}(t)\,\dot{S}(t)\right]\,V_{2}(t)=P(t)\,V_{2}(t)\,,$
(3.16)

where

$\,P(t)=\left(\begin{array}[]{cccc}0&t^{-g}&0&0\\\\[0.6458pt]
\frac{28}{5}\,t^{g-1}&g\,t^{-2}&\frac{12}{5}\,t^{-g-1}&0\\\\[0.6458pt]
0&0&2g\,t^{-2}&t^{-g}\\\\[0.6458pt]
-\frac{12}{5}\,t^{3g-1}&0&-\frac{8}{5}\,t^{g-1}&3g\,t^{-2}\end{array}\right)\,.\,$

With $g=1/4$ the system (3.16) becomes

$\,t^{-1}\,\dot{V}_{2}(t)=\left(\begin{array}[]{cccc}0&t^{-1/4}&0&0\\\\[0.6458pt]
\frac{28}{5}\,t^{-3/4}&\frac{1}{4}\,t^{-2}&\frac{12}{5}\,t^{-5/4}&0\\\\[0.6458pt]
0&0&\frac{1}{2}\,t^{-2}&t^{-1/4}\\\\[0.6458pt]
-\frac{12}{5}\,t^{-1/4}&0&-\frac{8}{5}\,t^{-3/4}&\frac{3}{4}\,t^{-2}\end{array}\right)\,V_{2}(t)\,.\,$

Now setting $t=\alpha\,\tau^{4}$ where $\alpha$ will be chosen late and
multiplying by $4\,\alpha^{2}\,\tau$ we get

$\displaystyle\tau^{-6}\,V_{2}^{\prime}(\tau)=P(\tau)\,V_{2}(\tau),\qquad^{\prime}=\frac{d}{d\tau}\,,$
(3.17)

where

$\,P(\tau)=\left(\begin{array}[]{cccc}0&4\,\alpha^{7/4}&0&0\\\\[0.6458pt]
\frac{4.28}{5}\,\alpha^{5/4}\,\tau^{-2}&\tau^{-7}&\frac{48}{5}\,\alpha^{3/4}\,\tau^{-4}&0\\\\[0.6458pt]
0&0&2\,\tau^{-7}&4\,\alpha^{7/4}\\\\[0.6458pt]
-\frac{48}{5}\,\alpha^{7/4}&0&-\frac{32}{5}\,\alpha^{5/4}\,\tau^{-2}&3\,\tau^{-7}\end{array}\right)\,.\,$

Again all the eigenvalues of the leading matrix $P(\infty)$ are equal to zero.
But this time the matrix $P(\infty)$ is similar to a Jordan block of order
$4$. The similar transformation $V_{2}(\tau)=T_{2}\,V_{3}(\tau)$ with

$\,T_{2}=\left(\begin{array}[]{cccc}0&0&-\frac{5}{48}\,\alpha^{-7/4}&0\\\\[0.6458pt]
0&0&0&-\frac{5}{4.48}\,\alpha^{-7/2}\\\\[0.6458pt]
4\,\alpha^{7/4}&0&0&0\\\\[0.6458pt] 0&1&0&0\end{array}\right)\,$

and

$\,T_{2}^{-1}=\left(\begin{array}[]{cccc}0&0&\frac{1}{4}\,\alpha^{-7/4}&0\\\\[0.6458pt]
0&0&0&1\\\\[0.6458pt] -\frac{48}{5}\,\alpha^{7/4}&0&0&0\\\\[0.6458pt]
0&-\frac{4.48}{5}\,\alpha^{7/2}&0&0\end{array}\right)\,$

takes the system (3.17) into the system

$\displaystyle\tau^{-6}\,V^{\prime}_{3}(\tau)=(T^{-1}_{2}\,P(t)\,T_{2})\,V_{3}(\tau)=P_{1}(\tau)\,V_{3}(\tau)\,,$
(3.18)

where

$\,P_{1}(\tau)=\left(\begin{array}[]{cccc}2\,\tau^{-7}&1&0&0\\\\[0.6458pt]
-\frac{4.32}{5}\,\alpha^{3}\,\tau^{-2}&3\,\tau^{-7}&1&0\\\\[0.6458pt]
0&0&0&1\\\\[0.6458pt]
-\frac{16.48.48}{25}\,\alpha^{6}\,\tau^{-4}&0&\frac{64.7}{5}\,\alpha^{3}\,\tau^{-2}&\tau^{-7}\end{array}\right)\,.\,$

A shearing transformation

$\,V_{3}(\tau)=\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&\tau^{-1}&0&0\\\
0&0&\tau^{-2}&0\\\ 0&0&0&\tau^{-3}\end{array}\right)\,V_{4}(\tau)\,$

with $g=1$ produces the system

$\displaystyle\tau^{-5}\,V^{\prime}_{4}(\tau)=P_{2}(\tau)\,V_{4}(\tau)\,,$
(3.19)

where

$\,P_{2}(\tau)=\left(\begin{array}[]{cccc}2\,\tau^{-6}&1&0&0\\\\[0.6458pt]
-\frac{4.32}{5}\,\alpha^{3}&4\,\tau^{-6}&1&0\\\\[0.6458pt]
0&0&2\,\tau^{-6}&1\\\\[0.6458pt]
-\frac{15.48.48}{25}\,\alpha^{6}&0&\frac{64.7}{5}\,\alpha^{3}&4\,\tau^{-6}\end{array}\right)\,.\,$

Choosing $\alpha^{3}=\frac{5}{64}$ we find that the leading matrix

$\,P_{2}(\infty)=\left(\begin{array}[]{rccc}0&1&0&0\\\ -2&0&1&0\\\ 0&0&0&1\\\
-9&0&7&0\end{array}\right)\,$

has four distinct eigenvalues

$\displaystyle\lambda_{1}=-\frac{i\,\sqrt{6\,\sqrt{5}-10}}{2},\quad\lambda_{2}=\frac{i\,\sqrt{6\,\sqrt{5}-10}}{2},$
(3.20)
$\displaystyle\lambda_{3}=-\frac{\sqrt{6\,\sqrt{5}+10}}{2},\quad\lambda_{4}=\frac{\sqrt{6\,\sqrt{5}+10}}{2}\,.$

Now the transformation $V_{4}(\tau)=T_{3}\,V(\tau)$ with

$\,T_{3}=\left(\begin{array}[]{cccc}\frac{\sqrt{5}\,(\sqrt{5}+3)}{20}&\frac{\sqrt{5}\,(\sqrt{5}+3)}{20}&\frac{\sqrt{5}\,(\sqrt{5}-3)}{20}&\frac{\sqrt{5}\,(\sqrt{5}-3)}{20}\\\\[1.72218pt]
-\frac{i}{\sqrt{6\sqrt{5}-10}}&\frac{i}{\sqrt{6\sqrt{5}-10}}&\frac{1}{\sqrt{6\sqrt{5}+10}}&-\frac{1}{\sqrt{6\sqrt{5}+10}}\\\\[1.72218pt]
\frac{3\sqrt{5}}{10}&\frac{3\sqrt{5}}{10}&-\frac{3\sqrt{5}}{10}&-\frac{3\sqrt{5}}{10}\\\\[1.72218pt]
-\frac{3\,i\,\sqrt{5}\,\sqrt{6\sqrt{5}-10}}{20}&\frac{3\,i\,\sqrt{5}\,\sqrt{6\sqrt{5}-10}}{20}&\frac{3\,\sqrt{5}\,\sqrt{6\sqrt{5}+10}}{20}&-\frac{3\,\sqrt{5}\,\sqrt{6\sqrt{5}+10}}{20}\end{array}\right),\,$

and

$\,T_{3}^{-1}=\left(\begin{array}[]{cccc}1&\frac{2\,i}{\sqrt{6\sqrt{5}-10}}&\frac{\sqrt{5}-3}{6}&-\frac{i\,\sqrt{5}\,\sqrt{6\sqrt{5}-10}}{30}\\\\[1.72218pt]
1&-\frac{2\,i}{\sqrt{6\sqrt{5}-10}}&\frac{\sqrt{5}-3}{6}&\frac{i\,\sqrt{5}\,\sqrt{6\sqrt{5}-10}}{30}\\\\[1.72218pt]
1&-\frac{2}{\sqrt{6\sqrt{5}+10}}&-\frac{\sqrt{5}+3}{6}&\frac{\sqrt{5}\,\sqrt{6\sqrt{5}+10}}{30}\\\\[1.72218pt]
1&\frac{2}{\sqrt{6\sqrt{5}+10}}&-\frac{\sqrt{5}+3}{6}&-\frac{\sqrt{5}\,\sqrt{6\sqrt{5}+10}}{30}\end{array}\right)\,.\,$

changes the differential equation (3.19) into the equation

$\displaystyle\tau^{-5}\,V^{\prime}(\tau)=\left(\begin{array}[]{cccc}\lambda_{1}+3\,\tau^{-6}&-\tau^{-6}&0&0\\\\[0.6458pt]
-\tau^{-6}&\lambda_{2}+3\,\tau^{-6}&0&0\\\\[0.6458pt]
0&0&\lambda_{3}+3\,\tau^{-3}&-\tau^{-6}\\\\[0.6458pt]
0&0&-\tau^{-6}&\lambda_{4}+3\,\tau^{-6}\end{array}\right)\,V(\tau)\,,$ (3.25)

with $\lambda_{j},j=1,2,3,4$ defined by (3.20). The system (3.25) is the
wanted system (3.14).

This ends the proof. ∎

######  Remark 3.2.

Following the Wasow’s book we have to put $\alpha=4^{-4/7}$. Here we choose
$\alpha^{3}=5/64$ in order to simplify the eigenvalues of the matrix
$P_{2}(\infty)$. Our choice is allowed since it does not change the aim of the
shearing transformations, which is to obtain a system of differential
equations with a leading matrix different from $A(\infty)$.

The resulting system (3.25) has two singular points over
${\mathbb{C}}{\mathbb{P}}^{1}$ : $\tau=0$ and $\tau=\infty$, while the
$(\textrm{NVE})$ have only one at $t=\infty$. In the next proposition we show
that $\tau=0$ is an apparent singularity for the system (3.25). This
phenomenon seems to be similar to that one observed at the reduction of second
order linear system of ordinary differential equations to a second order
scalar equation. Recall that in this case if the element $a_{12}(t)$ of the
matrix $A(t)=\\{a_{ij}(t)\\}_{i,j=1}^{2},\,a_{ij}\in{\mathbb{C}}(t)$ for the
system $\dot{x}=A(t)\,x$ has a zero of second order at $t=a$ then $t=a$ is an
apparent singularity for the resulting second order scalar equation.

######  Proposition 3.3.

The point $\tau=0$ is an apparent singularity for the system (3.25).

###### Proof.

The change of the independent variable

$\,\tau^{6}=\eta\,$

takes the differential equation (3.25) into the equation

$\displaystyle\frac{dV(\eta)}{d\eta}=\left(\begin{array}[]{cccc}\frac{\lambda_{1}}{6}+\frac{1}{2\eta}&-\frac{1}{6\eta}&0&0\\\\[0.86108pt]
-\frac{1}{6\eta}&\frac{\lambda_{2}}{6}+\frac{1}{2\eta}&0&0\\\\[0.86108pt]
0&0&\frac{\lambda_{3}}{6}+\frac{1}{2\eta}&-\frac{1}{6\eta}\\\\[0.86108pt]
0&0&-\frac{1}{6\eta}&\frac{\lambda_{4}}{6}+\frac{1}{2\eta}\end{array}\right)\,V(\eta)\,.$
(3.30)

The system (3.30) is reduced to two independent second-order linear systems in
the form

$\displaystyle\frac{du}{d\eta}=\left(\begin{array}[]{cc}\frac{\lambda_{i}}{6}+\frac{1}{2\eta}&-\frac{1}{6\eta}\\\\[1.07639pt]
-\frac{1}{6\eta}&\frac{\lambda_{j}}{6}+\frac{1}{2\eta}\end{array}\right)\,u$
(3.33)

for $(i,j)=(1,2)$ or $(i,j)=(3,4)$. Let $u=(u_{1},u_{2})^{T}$. Since
$\lambda_{1}+\lambda_{2}=\lambda_{3}+\lambda_{4}=0$ the system (3.33) is
equivalent to the second-order scalar equation

$\displaystyle\frac{d^{2}u_{1}}{d\eta^{2}}-\left[-\frac{\lambda_{i}\,\lambda_{j}}{36}+\frac{\lambda_{i}-\lambda_{j}}{12\,\eta}-\frac{2}{9\,\eta^{2}}\right]\,u_{1}=0\,.$
(3.34)

The point $\eta=0$ is a regular singularity for the equation (3.34). The
characteristic equation for $\eta=0$ is given by

$\,\rho\,(\rho-1)+\frac{2}{9}=0\,.\,$

The corresponding characteristic exponents are $\rho_{1}=\frac{2}{3}$ and
$\rho_{2}=\frac{1}{3}$. Since $\rho_{1}-\rho_{2}\notin{\mathbb{Z}}$ the local
theory of regular singularities ensures that the equation (3.34) has two
independent local solutions $u_{1}(\eta)$ and $\widetilde{u}_{1}(\eta)$ near
the origin in the form

$\,u_{1}(\eta)=\eta^{2/3}\,f_{1}(\eta)\quad\textrm{and}\quad\widetilde{u}_{1}(\eta)=\eta^{1/3}\,f_{2}(\eta)\,.\,$

The functions $f_{1}(\eta)$ and $f_{2}(\eta)$ are holomorphic functions near
the origin. Then the system (3.30) admits a local fundamental matrix solution
$\Psi(\eta)$ near the origin $\eta=0$ in the form

$\,\Psi(\eta)=U(\eta)\,\eta^{J}\,,\,$

where $U(\eta)$ is a holomorphic matrix-function near the origin, and

$\,J={\mathrm{diag}}\left(\frac{2}{3},\frac{1}{3},\frac{2}{3},\frac{1}{3}\right)\,.\,$

Therefore the system (3.25) admits a local fundamental matrix solution near
the origin in the form

$\,\Psi(\tau)=U(\tau^{6})\,\tau^{J_{1}}\,,\,$

where $J_{1}=6\,J={\mathrm{diag}}(4,2,4,2)$. Thus the origin $\tau=0$ is an
apparent singularity for the system (3.25).

This ends the proof. ∎

######  Proposition 3.4.

The connected component $(G_{1})^{0}$ of the unit element of the differential
Galois group $G_{1}$ of the system (3.25) is a direct product of two groups
$SL_{2}({\mathbb{C}})$, i.e. it has the form
$(G_{1})^{0}=SL_{2}({\mathbb{C}})\times SL_{2}({\mathbb{C}})$.

###### Proof.

We will show that the differential Galois group of each systems (3.33)
coincides with $\textrm{SL}_{2}({\mathbb{C}})$. The change of the variable

$\displaystyle\eta=\frac{3}{\sqrt{-\lambda_{i}\,\lambda_{j}}}\,\zeta$ (3.35)

takes the equation (3.34) into the equation

$\displaystyle\frac{d^{2}u_{1}}{d\zeta^{2}}-\left[\frac{1}{4}-\frac{\lambda_{j}-\lambda_{i}}{4\,\sqrt{-\lambda_{i}\,\lambda_{j}}}\,\frac{1}{\zeta}-\frac{2}{9\,\zeta^{2}}\right]\,u_{1}=0.$
(3.36)

The equation (3.36) is the Whittaker equation [21]

$\,\frac{d^{2}u_{1}}{d\zeta^{2}}-\left[\frac{1}{4}-\frac{\kappa}{\zeta}+\frac{4\,\mu^{2}-1}{4\zeta^{2}}\right]\,u_{1}=0\,$

with

$\,\kappa=\frac{\lambda_{j}-\lambda_{i}}{4\,\sqrt{-\lambda_{i}\,\lambda_{j}}}=\frac{1}{2},\qquad\mu=\frac{1}{6}\,.\,$

In [4] Martinet and Ramis show that the Whittaker equation has two Stokes
matrices at $\zeta=\infty$

$\,St_{\pi}=\left(\begin{array}[]{cc}1&0\\\ \mu_{1}&1\end{array}\right),\quad
St_{0}=\left(\begin{array}[]{cc}1&\mu_{2}\\\ 0&1\end{array}\right)\,,\,$

which correspond to the singular directions $\theta_{1}=\pi$ and
$\theta_{2}=0$. In Proposition 3.3.1 in [4] they prove that

  1. (1)

$\mu_{1}=0$ if and only if $\kappa-\mu\in\frac{1}{2}+{\mathbb{N}}$ or
$\kappa+\mu\in\frac{1}{2}+{\mathbb{N}}$

  2. (2)

$\mu_{2}=0$ if and only if $-\kappa-\mu\in\frac{1}{2}+{\mathbb{N}}$ or
$-\kappa+\mu\in\frac{1}{2}+{\mathbb{N}}$.

Since $\kappa\pm\mu=\frac{1}{2}\pm\frac{1}{6}\notin\frac{1}{2}+{\mathbb{N}}$
and $-\kappa\pm\mu=-\frac{1}{2}\pm\frac{1}{6}\notin\frac{1}{2}+{\mathbb{N}}$
we have that the equation (3.36) has two non-trivial Stokes matrices. Since
the exponential torus ${\mathcal{T}}$ are given by [4]

$\,{\mathcal{T}}=\left\\{\left(\begin{array}[]{cc}a&0\\\
0&a^{-1}\end{array}\right)\,|\,a\in{\mathbb{C}}^{*}\right\\}\,$

we conclude that the local differential Galois group of the equation (3.36) at
$\zeta=\infty$ and therefore its global Galois group is the group
$SL_{2}({\mathbb{C}})$, [4] . The substitution (3.35) changes the Galois group
of the equation (3.34) but remains its connected component of the unit
element. Thus the connected component of the unit element of the differential
Galois group of the system (3.33) is the group $SL_{2}({\mathbb{C}})$.
Therefore connected component of the unit element of the differential Galois
group of the systems (3.30) is a direct product of two groups
$SL_{2}({\mathbb{C}})$, i.e. it has the form $SL_{2}({\mathbb{C}})\times
SL_{2}({\mathbb{C}})$. From Proposition 3.3 it follows that the differential
Galois group of the system (3.25) is its local Galois group at $\tau=\infty$.
The transformation $\tau^{6}=\eta$ changes the the differential Galois group
$G_{1}$ of the system (3.25) but its connected component $(G_{1})^{0}$ of the
unit element remains the same . Thus $(G_{1})^{0}=SL_{2}({\mathbb{C}})\times
SL_{2}({\mathbb{C}})$.

This ends the proof. ∎

######  Remark 3.5.

In fact both of the systems (3.33) are not equivalent to the Whittaker
equation (3.36) in the same variable $\zeta$. The transformation

$\,\eta=\frac{3}{\sqrt{-\lambda_{1}\,\lambda_{2}}}\,\zeta\,$

takes only the system (3.33) for $(i,j)=(1,2)$ into the equation (3.36). At
the same time this transformation takes the second system for $(i,j)=(3,4)$
into the equation

$\displaystyle\frac{d^{2}u_{1}}{d\zeta^{2}}-\left[\frac{7+3\,\sqrt{5}}{8}+\frac{i\,(3+\sqrt{5})}{4}\,\frac{1}{\zeta}-\frac{2}{9\,\zeta^{2}}\right]\,u_{1}=0.$
(3.37)

However, the equation (3.37) is not solvable and its differential Galois group
is $SL_{2}({\mathbb{C}})$ since the change

$\,\zeta=\sqrt{\frac{7-3\,\sqrt{5}}{2}}\,\omega\,$

takes it into the non-solvable Whittaker equation

$\,\frac{d^{2}u_{1}}{d\omega^{2}}-\left[\frac{1}{4}+\frac{i}{2\omega}-\frac{2}{9\,\omega^{2}}\right]\,u_{1}=0.\,$

Thanks to Proposition 3.4 we establish the main result of this section.

######  Theorem 3.6.

Assume that $\alpha_{0}=2/5,\alpha_{1}=1/5,\alpha_{2}=1/10$. Then the Sasano
system of type $A_{2}^{(4)}$ (1) is not integrable by meromorphic first
integrals which are rational functions in $t$.

###### Proof.

The transformation (3.13) defined in the proof of Theorem 3.1 changes the
differential Galois group $G$ of the $(\textrm{NVE})$ but remains the
corresponding connected component of the unit element $G^{0}$ [15]. From
Proposition 3.4 it follows that $G^{0}=SL_{2}({\mathbb{C}})\times
SL_{2}({\mathbb{C}})$ which is not Abelian group. Thus from the Morales-Ramis
theory it follows that when $\alpha_{0}=2/5,\alpha_{1}=1/5,\alpha_{2}=1/10$
the Sasano system (1) is not integrable by meromorphic first integral which
are rational functions in $t$.

This ends the proof. ∎

##  4\. Bäcklund transformations and generalization

The Sasano systems (1) admits an affine Weyl group of symmetry of type
$A^{(2)}_{4}$ which acts on it as a group of Bäcklund transformations. In this
section using the Bäcklund transformations of the system (1) we extend the
non-integrable result of Theorem 3.6 to the entire orbit of the parameters
$\alpha_{0},\alpha_{1}$ and $\alpha_{2}$.

The Bäcklund transformations of the Sasano system (1) are defined by

$\displaystyle s_{0}\,:\,(x,y,z,w,t;\alpha_{0},\alpha_{1},\alpha_{2})$
$\displaystyle\rightarrow$
$\displaystyle\left(x,y,z+\frac{\alpha_{0}}{w},w,t;-\alpha_{0},\alpha_{1}+\alpha_{0},\alpha_{2}\right),$
(4.38) $\displaystyle s_{1}\,:\,(x,y,z,w,t;\alpha_{0},\alpha_{1},\alpha_{2})$
$\displaystyle\rightarrow$
$\displaystyle\left(x,y-\frac{\alpha_{1}}{x+z^{2}},z,w-\frac{2\alpha_{1}\,z}{x+z^{2}},t;\alpha_{0}+2\alpha_{1},-\alpha_{1},\alpha_{2}+\alpha_{1}\right),$
$\displaystyle s_{2}\,:\,(x,y,z,w,t;\alpha_{0},\alpha_{1},\alpha_{2})$
$\displaystyle\rightarrow$
$\displaystyle\left(x+\frac{2\alpha_{2}\,y}{f_{2}}-\frac{\alpha_{2}^{2}}{f^{2}_{2}},y-\frac{\alpha_{2}}{f_{2}},z+\frac{\alpha_{2}}{f_{2}},w,t;\alpha_{0},\alpha_{1}+2\alpha_{2},-\alpha_{2}\right)\,,$

where $f_{2}:=x+y^{2}+w+t$.

######  Remark 4.1.

When one of the invariant divisors $f_{0}=w,f_{1}=x+z^{2}$ or
$f_{2}=x+y^{2}+w+t$ vanishes the parameters become $\alpha_{0}=0,\alpha_{1}=0$
or $\alpha_{2}=0$, respectively. So, when one of the divisors $f_{i},i=0,1,2$
vanishes the corresponding transformation $s_{i},i=0,1,2$ acts as an identity
transformation.

The transformations (4.38) satisfy the relations

$\,s_{i}^{2}=1,\quad(s_{0}\,s_{1})^{4}=(s_{1}\,s_{0})^{4}=1,\quad(s_{0}\,s_{2})^{2}=(s_{2}\,s_{0})^{2}=1,\quad(s_{1}\,s_{2})^{4}=(s_{2}\,s_{1})^{4}=1.\,$

Hence the transformations $s_{i},i=0,1,2$ generate the affine Weyl group
$W(A^{(2)}_{4})$.

In [5] Matsuda proves that acting on the solution (3.6) by Bäcklund
transformations (4.38) we obtain a rational solution of the Sasano system (1)
with parameters satisfying one of the following condition:

$\displaystyle 5\alpha_{2}-1/2\equiv 0\,\textrm{ mod}\,5,$ $\displaystyle
5\alpha_{1}-1\equiv 0,2\,\textrm{mod}\,5,$ $\displaystyle
5\alpha_{2}-1/2\equiv 1\,\textrm{ mod}\,5,$ $\displaystyle 5\alpha_{1}-1\equiv
2,3\,\textrm{mod}\,5,$ (4.39) $\displaystyle 5\alpha_{2}-1/2\equiv 3\,\textrm{
mod}\,5,$ $\displaystyle 5\alpha_{1}-1\equiv 0,1\,\textrm{mod}\,5,$
$\displaystyle 5\alpha_{2}-1/2\equiv 4\,\textrm{ mod}\,5,$ $\displaystyle
5\alpha_{1}-1\equiv 1,3\,\textrm{mod}\,5\,.$

Then we have the main result of this paper

######  Theorem 4.2.

Assume that the parameters $\alpha_{i},i=0,1,2$ satisfy one of the conditions
(4). Then the Sasano system of type $A^{(2)}_{4}$ (1) is not integrable by
meromorphic first integrals which are rational functions in $t$.

###### Proof.

The proof follows from the fact that the Bäcklund transformations (4.38) are
canonical transformations which are rational in all of the canonical variables
[9]. ∎

Since all of the rational solutions of the Sasano system (1) can be obtained
from the solution (3.6) by Bäcklund transformations [5] Theorem 4.2 states
that the Sasano system (1) is not integrable in the Liouville-Arnold sense by
meromorphic first integrals which are rational functions in $t$.

Acknowledgments. The author was partially supported by Grant KP-06-N 62/5 of
the Bulgarian Found ”Scientific research”.

## References

  * [1] Arnol’d, V. I., Mathematical methods in classical mechanics, 2nd ed., New York: Springer, 1989
  * [2] Fuji, K. and Sizuki, T., Higher order Painlevé system of type $D^{(1)}_{2n+2}$ arising from integrable hierarchy, Int. Math. Res. Not., 2008, vol. 2008, pp. 1-21.
  * [3] Horozov E. and Stoyanova T., Non-integrability of some Painlevé VI-equations and dilogarithms, Regul. Chaotic Dyn. 12(6), 2007, pp. 622-629.
  * [4] Martinet, J. and Ramis, J.-P., Théorie de Galois différentielle et resommation, In Computer Algebra and Differential Equations, E. Tournier. Ed., Academic Press, London, 1988, pp. 117-214.
  * [5] Matsuda, K., Rational solutions of the Sasano system of type $A^{(2)}_{4}$, J. Phys. A : Math. Theor., 44 (40) , 2011, 20 pp.
  * [6] Morales-Ruiz, J., Differential Galois theory and non-integrability of Hamiltonian systems, Basle : Birkhäuser, 1995.
  * [7] Morales-Ruiz J. and Ramis, J.-P., Galois obstruction to integrability of Hamiltonian systems.I, II, Methods and Appl. Anal., 2001, vol. 8, no. 1, pp. 33-95, 97-111.
  * [8] Sasano, Y., Higher order Painlevé equations of type $D^{(1)}_{l}$, RIMS Kôkyûroku, 2006, vol. 1473, pp. 143-163.
  * [9] Sasano, Y., Coupled Painlevé systems in dimension four with affine Weyl group symmetry of types $A^{(2)}_{4}$ and $A^{(1)}_{1}$, arXiv : 0809.2399.
  * [10] Sasano, Y., Symmetry in the Painlevé systems and their extensions to four-dimensional systems, Func. Ekvacioj, Ser. Int. 51 (3), 2008, pp. 351-369.
  * [11] Sasano, Y., Coupled Painlevé VI systems in dimension four with affine Weyl group symmetry of type $D^{(1)}_{6}$. II. , RIMS Kôkyûroku Bassatsu B5, 2008, pp. 137-152.
  * [12] Sasano, Y., Coupled Painlevé II systems in dimension four and the symmetry of type $A^{(1)}_{4}$ , Tohoku Math. J. (2), 58 (4), 2006, pp. 529-548.
  * [13] Sasano, Y., Coupled Painlevé IV systems in dimension four, Kumamoto J. Math. 20, 2007, pp. 13-31.
  * [14] Sasano, Y., Coupled Painlevé systems in dimension 4, Funkc. Ekvacioj, Se. Int. 49 (1), 2006, pp. 133-161.
  * [15] Singer, M., Introduction to the Galois theory of liner differential equations, Algebraic theory of differential equations, London Math. Soc. Lecture Note Ser., no. 357, Cambridge Univ. Press, Cambridge, 2009, pp. 1-82.
  * [16] Stoyanova, Ts., Non-integrability of the Painlevé VI equations in the Liouville sense, Nonlinearity 22 (9), 2009, pp. 2201-2230.
  * [17] Stoyanova, Ts., Non-integrability of the fourth Painlevé equation in the Liouville-Arnols sense, Nonlinearity 27 (5), 2014, pp. 1029-1044.
  * [18] Stoyanova, Ts., Nonintegrability of the Painlevé IV equation in the Liouville-Arnols sense and Stokes phenomena, Stud. Appl. Math. 151 (4), 2023, pp. 1380-1405.
  * [19] Stoyanova, Ts., Non-integrability of the Sasano systemof type $D^{(1)}_{5}$ and Stokes phenomena, arXiv : 2306.07062
  * [20] Wasow, W., Asymptotic expansions for ordinary differential equations, New York:Dover, 1965.
  * [21] Whittaker, E. and Watson, E., A course of modern analysis, Cambridge Univ. Press, 1989.


# Lower bounds for the number of number fields with Galois group
$\operatorname{GL}_{2}(\mathbb{F}_{\ell})$

Anwesh Ray  Centre de recherches mathématiques, Université de Montréal,
Pavillon André-Aisenstadt, 2920 Chemin de la tour, Montréal (Québec) H3T 1J4,
Canada<EMAIL_ADDRESS>
###### Abstract.

Let $\ell\geq 5$ be a prime number and $\mathbb{F}_{\ell}$ denote the finite
field with $\ell$ elements. We show that the number of Galois extensions of
the rationals with Galois group isomorphic to
$\operatorname{GL}_{2}(\mathbb{F}_{\ell})$ and absolute discriminant bounded
above by $X$ is asymptotically at least
$\frac{X^{\frac{\ell}{12(\ell-1)\\#\operatorname{GL}_{2}(\mathbb{F}_{\ell})}}}{\log
X}$. We also obtain a similar result for the number of surjective
homomorphisms $\rho\mathrel{\mathop{\mathchar
58\relax}}\operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow\operatorname{GL}_{2}(\mathbb{F}_{\ell})$
ordered by the prime to $\ell$ part of the Artin conductor of $\rho$.

##  1\. Introduction

Let $G$ be a transitive subgroup of $S_{n}$. Given a number field $K$ with
$[K\mathrel{\mathop{\mathchar 58\relax}}\mathbb{Q}]=n$, let $\tilde{K}$ denote
its Galois closure over $\mathbb{Q}$. We view
$\operatorname{Gal}(\tilde{K}/\mathbb{Q})$ as a transitive permutation
subgroup of $S_{n}$, where the permutation action is defined by its action on
the $n$ embeddings of $K$ into $\mathbb{C}$. For any real number $X>0$, let
$N(G;X)$ be the number of number fields $K$ of degree $n$, with absolute
discriminant at most $X$, such that $\operatorname{Gal}(\tilde{K}/\mathbb{Q})$
is isomorphic to $G$ as a permutation subgroup of $S_{n}$. In [Mal02], Malle
made a precise conjecture about the asymptotic growth of $N(G;X)$ as a
function of $X$. The conjecture posits that

$N(G;X)\sim X^{\frac{1}{a(G)}}(\log X)^{b(G)-1},$

where $a(G)$ and $b(G)$ are numbers which depend on the permutation
representation of $G$. Malle’s conjecture has been proven for the abelian
groups [Mäk85], $S_{n}$ for $n\leq 5$ [DH71, Bha05, BW08, Bha10, BF10], the
dihedral group $D_{4}$ [CDO02], and various groups of the form $A\times
S_{n}$, where $A$ is a finite abelian group and $n\leq 5$, cf. [Wan21]. This
list is not exhaustive. Motivated by such developments, various asymptotic
lower bounds for the groups $S_{n}$ and $A_{n}$ have been proven, cf. [EV06,
PTBW20].

###  1.1. Main result

A large class of finite groups of interest are the groups of the form
$G(\mathbb{F}_{q})$, where $G$ is an algebraic group over the finite field
$\mathbb{F}_{q}$. These are groups that arise naturally in the study of Galois
representations associated with abelian varieties, motives and automorphic
forms. Let $\ell\geq 5$ be an odd prime number and let $\mathbb{F}_{\ell}$ be
the finite field with $\ell$ elements. One of the simplest examples to
consider are the groups $G=\operatorname{GL}_{2}(\mathbb{F}_{\ell})$. We view
$\operatorname{GL}_{2}(\mathbb{F}_{\ell})$ as a subgroup of
$S_{\\#\operatorname{GL}_{2}(\mathbb{F}_{\ell})}$, taken with respect to the
regular representation, and prove asymptotic lower bounds for the number of
Galois number fields $L/\mathbb{Q}$ with Galois group
$\operatorname{Gal}(L/\mathbb{Q})$ isomorphic to
$\operatorname{GL}_{2}(\mathbb{F}_{\ell})$, and bounded absolute discriminant.

Given a number field $L$, set $\Delta_{L}$ to denote the discriminant of
$L/\mathbb{Q}$. For a finite group $G$, set $\mathcal{M}(G)$ to denote the set
of all pairs $(L,\rho)$, where $L/\mathbb{Q}$ is a Galois extension with
Galois group $\operatorname{Gal}(L/\mathbb{Q})$ isomorphic to $G$, and
$\rho\mathrel{\mathop{\mathchar
58\relax}}\operatorname{Gal}(L/\mathbb{Q})\xrightarrow{\sim}G$ is an
isomorphism. Taking $G=\operatorname{GL}_{2}(\mathbb{F}_{\ell})$, set
$N_{\rho}$ to denote the prime to $\ell$ part of the Artin conductor of
$\rho\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{\mathbb{Q}}\rightarrow\operatorname{GL}_{2}(\mathbb{F}_{\ell})$.
This quantity is also known as the _optimal level_ , with reference to Serre’s
conjecture, cf. [RS99, section 1.3]. Note that if two isomorphisms
$\rho,\rho^{\prime}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{Gal}(L/\mathbb{Q})\xrightarrow{\sim}\operatorname{GL}_{2}(\mathbb{F}_{\ell})$
differ by an inner automorphism of $\operatorname{GL}_{2}(\mathbb{F}_{\ell})$,
then, $N_{\rho}=N_{\rho^{\prime}}$. Taking $X\in\mathbb{R}_{>0}$, one is
interested in obtaining asymptotic lower bounds for the cardinalities of the
following sets

$\begin{split}&\mathcal{M}(\operatorname{GL}_{2}(\mathbb{F}_{\ell});X)\mathrel{\mathop{\mathchar
58\relax}}=\\{(L,\rho)\in\mathcal{M}(\operatorname{GL}_{2}(\mathbb{F}_{\ell}))\mid
N_{\rho}\leq X\\},\\\
&\mathcal{F}(\operatorname{GL}_{2}(\mathbb{F}_{\ell});X)\mathrel{\mathop{\mathchar
58\relax}}=\\{L\mid L/\mathbb{Q}\text{ is Galois with
}\operatorname{Gal}(L/\mathbb{Q})\simeq\operatorname{GL}_{2}(\mathbb{F}_{\ell})\text{,
and }|\Delta_{L}|\leq X\\}.\end{split}$

We note that $\\#\mathcal{F}(\operatorname{GL}_{2}(\mathbb{F}_{\ell});X)$ is
simply $N(\operatorname{GL}_{2}(\mathbb{F}_{\ell});X)$ with respect to the
natural inclusion of $\operatorname{GL}_{2}(\mathbb{F}_{\ell})$ into
$S_{\\#\operatorname{GL}_{2}(\mathbb{F}_{\ell})}$. Given a set $S$, let
$\\#S\in\mathbb{Z}_{\geq 0}\cup\\{\infty\\}$ denote its cardinality. Given
functions $f(X)$ and $g(X)$ taking on non-negative values, we set $f(X)\gg
g(X)$ (resp. $f(X)=O(g(X))$) to mean that there is a constant $C>0$,
independent of $X$, such that $f(X)\geq Cg(X)$ (resp. $f(X)\leq Cg(X)$). The
main result is stated below.

######  Theorem A.

Let $\ell\geq 5$ be a prime number. Then,

$\begin{split}&\\#\mathcal{M}(\operatorname{GL}_{2}(\mathbb{F}_{\ell});X)\gg\frac{X^{\frac{1}{12}}}{\log
X};\\\
&N(\operatorname{GL}_{2}(\mathbb{F}_{\ell});X)=\\#\mathcal{F}(\operatorname{GL}_{2}(\mathbb{F}_{\ell});X)\gg\frac{X^{\frac{\ell}{12(\ell-1)\\#\operatorname{GL}_{2}(\mathbb{F}_{\ell})}}}{\log
X},\end{split}$

where the implied constant depends only on $\ell$.

Lipnowski [Lip21] proposed a heuristic for counting
$\operatorname{GL}_{2}(\mathbb{F}_{\ell})$-extensions of $\mathbb{Q}$, that
satisfy the same properties as the extensions that arise from modular Galois
representations. In _loc. cit._ , a homomorphism
$\rho\mathrel{\mathop{\mathchar
58\relax}}\operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow\operatorname{GL}_{2}(\mathbb{F}_{\ell})$
is said to be _locally modular_ , if it satisfies the same local conditions as
that of a Galois representation associated with a Hecke eigencuspform of
weight $2$. This conjecture is based on Bhargava’s local-global heuristics for
counting $G$-number fields [Bha07], and predicts that if $G(X)$ is the number
of locally modular Galois representations with squarefree conductor
$N\in[X,2X]$, then,

$G(X)\sim X\log X\prod_{v}\left(1-3v^{-2}+2v^{-3}\right).$

The representations constructed in this article arise from elliptic curves,
and thus are locally modular.

For any finite group $G$ with $n$ elements and inclusion $G\hookrightarrow
S_{n}$ with respect to the regular representation, the value of $a(G)$ is
$\left(\frac{p-1}{p}\right)\\#G$, where $p$ is the smallest prime divisor of
$G$ (cf. [Mal02, p. 318]). Therefore, the power of $X$ in the predicted
asymptotic for $N(\operatorname{GL}_{2}(\mathbb{F}_{\ell});X)$ is
$\frac{2}{\\#\operatorname{GL}_{2}(\mathbb{F}_{\ell})}$. In our main result
for $N(\operatorname{GL}_{2}(\mathbb{F}_{\ell});X)$, the power of $X$ obtained
is thus $\frac{1}{24}\left(\frac{\ell}{\ell-1}\right)$ times the conjectured
value. In particular the conjecture predicts that
$\frac{1}{a(\operatorname{GL}_{2}(\mathbb{F}_{\ell}))}\sim\frac{2}{\ell^{4}}$
(as $\ell\rightarrow\infty$), and the power of $X$ in our bound is
$\sim\frac{1}{12\ell^{4}}$. Thus, the correct power of $\ell$ has been
achieved, and we are off by a factor of $24$.

###  1.2. Method of proof

Let us briefly describe the methodology behind the proof of Theorem A. Given
an elliptic curve $E_{/\mathbb{Q}}$ and a prime number $\ell$, one considers
the associated Galois representation

$\bar{\rho}_{E,\ell}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow\operatorname{GL}_{2}(\mathbb{F}_{\ell})$

on the $\ell$-torsion subgroup of $E(\bar{\mathbb{Q}})$. The $\ell$ division
field $L_{\ell}(E)$ is the field extension of $\mathbb{Q}$, which is fixed by
the kernel of $\bar{\rho}_{E,\ell}$. The number field $L_{\ell}(E)$ is Galois
over $\mathbb{Q}$, and the Galois group
$\operatorname{Gal}(L_{\ell}(E)/\mathbb{Q})$ is isomorphic to
$\operatorname{GL}_{2}(\mathbb{F}_{\ell})$ precisely when
$\bar{\rho}_{E,\ell}$ is surjective. We consider the family of all elliptic
curves over $\mathbb{Q}$, ordered according to _height_ , and relate it to the
family of $\operatorname{GL}_{2}(\mathbb{F}_{\ell})$-extensions of
$\mathbb{Q}$, ordered by discriminant (or Artin conductor). It follows from a
theorem of Duke [Duk97] that for almost all elliptic curves $E_{/\mathbb{Q}}$,
the residual representation $\bar{\rho}_{E,\ell}$ is surjective. Given a
Galois extension $L/\mathbb{Q}$ with Galois group isomorphic to
$\operatorname{GL}_{2}(\mathbb{F}_{\ell})$, we obtain an asymptotic upper
bound for the number of elliptic curves $E_{/\mathbb{Q}}$ with bounded height,
for which $L_{\ell}(E)=L$, cf. Proposition 3.3. This argument involves a
variant of Gallagher’s large sieve. We then prove Theorem A via a counting
argument.

###  1.3. Outlook

Galois representations arise from motives and automorphic forms. The Galois
representations that arise from elliptic curves are the most natural examples
to consider. We expect that this methodology has the potential to lead to
interesting generalizations.

###  1.4. Related work

The author was notified when the earlier version of the manuscript was made
public, Vittoria Crisante is studying related questions for her thesis, and
some similar results have been obtained independently and simultaneously by
her.

### Acknowledgment

The author’s research is supported by the CRM Simons postdoctoral fellowship.
He would like to thank Chris Wuthrich for his comments in response to a
question he posted on MathOverflow. He would like to thank Robert Lemke-Oliver
for pointing out a calculation error in the previous version.

##  2\. Elliptic curves and division fields

###  2.1. Parametrizing $\operatorname{GL}_{2}(\mathbb{F}_{\ell})$-extensions
of $\mathbb{Q}$

Let $\ell$ be an odd prime number. In this section, we recall some well known
results about the division field associated with the $\ell$-torsion of an
elliptic curve $E_{/\mathbb{Q}}$. Throughout, $\bar{\mathbb{Q}}$ will be a
fixed algebraic closure of $\mathbb{Q}$, the absolute Galois group of
$\mathbb{Q}$ is denoted
$\operatorname{G}_{\mathbb{Q}}\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$. The field with
$\ell$ elements is denoted by $\mathbb{F}_{\ell}$. Given a finite group $G$
and a homomorphism $\rho\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{\mathbb{Q}}\rightarrow G$, the field
$\mathbb{Q}(\rho)$ denotes $\bar{\mathbb{Q}}^{\operatorname{ker}\rho}$, and is
referred to as the field _cut out by $\rho$_. We note that
$\mathbb{Q}(\rho)/\mathbb{Q}$ is a Galois extension, and that the Galois group
$G_{\rho}\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{Gal}(\mathbb{Q}(\rho)/\mathbb{Q})$ injects into $G$
via $\rho$, and we may identify $G_{\rho}$ with the image of $\rho$. In
particular, if $\rho$ is surjective, then, $\mathbb{Q}(\rho)$ is a Galois
extension of $\mathbb{Q}$ with Galois group $G_{\rho}$ isomorphic to $G$. We
shall refer to such an extension as a $G$-extension of $\mathbb{Q}$.

Let $E_{/\mathbb{Q}}$ be an elliptic curve and $\ell$ be a prime number.
Denote by $E[\ell]$ the $\ell$-torsion subgroup of $E(\bar{\mathbb{Q}})$, and
note that $E[\ell]$ is a $2$-dimensional $\mathbb{F}_{\ell}$-vector space.
Choosing an $\mathbb{F}_{\ell}$-basis for $E[\ell]$, we identify the
automorphism group of $E[\ell]$ with
$\operatorname{GL}_{2}(\mathbb{F}_{\ell})$. Set
$\bar{\rho}_{E,\ell}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{\mathbb{Q}}\rightarrow\operatorname{GL}_{2}(\mathbb{F}_{\ell})$
to be the Galois representation on $E[\ell]$ and denote by $L_{\ell}(E)$ the
subfield of $\bar{\mathbb{Q}}$ fixed by the kernel of $\bar{\rho}_{E,\ell}$.
In other words, $L_{\ell}(E)$ is the field cut out by $\bar{\rho}_{E,\ell}$.
Set $G_{\ell}(E)\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{Gal}(L_{\ell}(E)/\mathbb{Q})$; note that if
$\bar{\rho}_{E,\ell}$ is surjective, then, $G_{\ell}(E)$ is isomorphic to
$\operatorname{GL}_{2}(\mathbb{F}_{\ell})$.

###  2.2. Elliptic curves ordered by height

An elliptic curve $E_{/\mathbb{Q}}$ admits a unique Weierstrass equation
$y^{2}=x^{3}+Ax+B$, where $(A,B)\in\mathbb{Z}^{2}$ is such that $p^{4}\nmid A$
and $p^{6}\nmid B$ for all primes $p$ (cf. [Duk97]). The Weierstrass equation
$y^{2}=x^{3}+Ax+B$ is minimal at every prime, and the naive height of $E$ is
defined as follows

$H(E)\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{max}\\{|A|^{3},B^{2}\\}.$

Let $\mathcal{C}$ be the set of isomorphism classes of elliptic curves
$E_{/\mathbb{Q}}$ and let $\mathcal{C}(X)$ be the set of elliptic curves with
$H(E)\leq X^{6}$. We identify $\mathcal{C}(X)$ with the set of tuples
$(A,B)\in\mathbb{Z}^{2}$ such that $p^{4}\nmid A$ and $p^{6}\nmid B$ for all
primes $p$, and $|A|\leq X^{2}$ and $|B|\leq X^{3}$. It is easy to see that

(2.1) $\\#\mathcal{C}(X)=\frac{4}{\zeta(10)}X^{5}+O(X^{3}),$

cf. [Bru92, Lemma 4.3]. Let $\mathcal{S}$ be a set of isomorphism classes of
elliptic curves $E_{/\mathbb{Q}}$. For $X>0$, set
$\mathcal{S}(X)\mathrel{\mathop{\mathchar
58\relax}}=\mathcal{S}\cap\mathcal{C}(X)$, and let $\delta(\mathcal{S})$ to
denote the limit

$\delta(\mathcal{S})=\lim_{X\rightarrow\infty}\left(\frac{\\#\mathcal{S}(X)}{\\#\mathcal{C}(X)}\right)=\frac{\zeta(10)}{4}\lim_{X\rightarrow\infty}\left(\frac{\\#\mathcal{S}(X)}{X^{5}}\right),$

provided it exists. Denote by $\Delta(E)\mathrel{\mathop{\mathchar
58\relax}}=-16(4A^{3}+27B^{2})$, and note that if $H(E)\leq X^{6}$, then,
$|\Delta(E)|\leq 496X^{6}$. Let $N_{E}$ denote the conductor of $E$. It
follows from Ogg’s formula that $N_{E}$ divides $\Delta_{E}$, cf. [Sil94,
p.389]. The elliptic curve $E$ is modular, and the associated modular form $f$
has level $N_{E}$. Set $N_{E}^{\prime}\mathrel{\mathop{\mathchar
58\relax}}=N_{\bar{\rho}_{E,\ell}}$; results of Carayol (cf. [RS99]) imply
that $N_{E}^{\prime}$ divides $N_{E}$, and hence, $N_{E}^{\prime}\leq
496X^{6}$.

######  Definition 2.1.

We shall denote by $\mathcal{S}_{\ell}(X)$ the subset of $\mathcal{C}(X)$ for
which $E$

  1. (1)

has semistable reduction at all primes $p$, except possibly at $p=2,3$,

  2. (2)

and $\bar{\rho}_{E,\ell}$ is surjective.

For $E\in\mathcal{S}_{\ell}(X)$, let $\gamma_{\ell}(E)$ be the pair
$(L_{\ell}(E),\bar{\rho}_{E,\ell})$.

######  Proposition 2.2.

Set $Y_{1}(X)\mathrel{\mathop{\mathchar
58\relax}}=\left(\frac{X}{496}\right)^{\frac{1}{6}}$. For
$E\in\mathcal{S}_{\ell}\left(Y_{1}(X)\right)$, the pair $\gamma_{\ell}(E)$
belongs to $\mathcal{M}(\operatorname{GL}_{2}(\mathbb{F}_{\ell});X)$.

###### Proof.

By the discussion above, $N_{E}^{\prime}\leq|\Delta_{E}|\leq
496Y_{1}(X)^{6}=X$, and thus, the result follows. ∎

Set $\Delta_{E}^{\prime}\mathrel{\mathop{\mathchar
58\relax}}=\Delta_{L_{\ell}(E)}$; we study the relationship between
$\Delta(E)$ and $\Delta_{E}^{\prime}$. Write $|\Delta_{E}^{\prime}|$ to be a
product of prime powers $\prod_{p}p^{m_{\ell}(p)}$. At each prime number $p$,
let $v_{p}$ be the valuation normalized by setting $v_{p}(p)=1$.

######  Proposition 2.3.

Let $E_{/\mathbb{Q}}$ be an elliptic curve and $\ell$ be a prime number. The
following assertions hold.

  1. (1)

For a prime number $p\neq\ell$ at which $E$ has semistable reduction, let
$m_{\ell}(p)\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{ord}_{p}(\Delta_{E}^{\prime})$ defined above. Then,
we have that
$m_{\ell}(p)\leq\left(\frac{\ell-1}{\ell}\right)\\#\operatorname{GL}_{2}(\mathbb{F}_{\ell})$.

  2. (2)

Suppose on the other hand that $E$ has additive reduction at $p$, and
$p\in\\{2,3\\}$, then, $m_{\ell}(p)\leq(3^{2}-1)(3^{2}-3)\times 68=3264$.

  3. (3)

We have that $m_{\ell}(\ell)\leq C_{\ell}$, where $C_{\ell}>0$ is a constant
which depends only on $\ell$.

###### Proof.

The [CK02, Theorem 1, part (a)] implies that $m_{\ell}(p)=dD/e$, where
$d=[\mathbb{Q}(E[\ell])\mathrel{\mathop{\mathchar
58\relax}}\mathbb{Q}]\leq\\#\operatorname{GL}_{2}(\mathbb{F}_{\ell})$ and $e$
is the index of ramification in $\mathbb{Q}_{p}(E[\ell])/\mathbb{Q}_{p}$. The
number

$D=\begin{cases}0&\text{ if }\ell\mid v_{p}(j(E));\\\ (\ell-1)&\text{ if
}\ell\nmid v_{p}(j(E)).\end{cases}$

When $\ell\nmid v_{p}(j(E))$, it follows from the theory of the Tate curve
that $e=\ell$. This proves part (1).

For part (2), we note that by results in [CK02], $D\leq 68$, and
$d\leq\\#\operatorname{GL}_{2}(\mathbb{F}_{3})=(3^{2}-3)(3^{2}-3)$, from which
the result follows.

When $p=\ell$, the result follows from the recipe in [Kra99]. ∎

######  Proposition 2.4.

Set $Y_{2}(X)\mathrel{\mathop{\mathchar
58\relax}}=\left(\frac{X}{6^{3264}\ell^{C_{\ell}}}\right)^{\frac{\ell}{6(\ell-1)\\#\operatorname{GL}_{2}(\mathbb{F}_{\ell})}}$.
For $E\in\mathcal{S}_{\ell}\left(Y_{2}(X)\right)$, the pair $L_{\ell}(E)$
belongs to $\mathcal{F}(\operatorname{GL}_{2}(\mathbb{F}_{\ell});X)$.

###### Proof.

From Proposition 2.3, we arrive at the following estimates

$\begin{split}|\Delta_{E}^{\prime}|=&2^{m_{\ell}(2)}3^{m_{\ell}(3)}\ell^{m_{\ell}(\ell)}\left(\prod_{p\notin\\{2,3,\ell\\}}p^{m_{\ell}(p)}\right)\\\
\leq&6^{3264}\ell^{C_{\ell}}\left(\prod_{p\mid\Delta_{E}}p\right)^{\left(\frac{\ell-1}{\ell}\right)\\#\operatorname{GL}_{2}(\mathbb{F}_{\ell})}\\\
\leq&6^{3264}\ell^{C_{\ell}}|\Delta_{E}|^{\left(\frac{\ell-1}{\ell}\right)\\#\operatorname{GL}_{2}(\mathbb{F}_{\ell})}\\\
\leq&6^{3264}\ell^{C_{\ell}}Y_{2}(X)^{6\left(\frac{\ell-1}{\ell}\right)\\#\operatorname{GL}_{2}(\mathbb{F}_{\ell})}\leq
X.\end{split}$

This proves the result. ∎

##  3\. Density results

In this section, we prove density results. We give a proof of Theorem A at the
end of the section.

###  3.1. A lower bound for the lower density of $\mathcal{S}_{\ell}$

Let $E_{/\mathbb{Q}}$ be an elliptic curve. A prime number $\ell$ is said to
be _exceptional_ if $\bar{\rho}_{E,\ell}$ is not surjective.

######  Theorem 3.1 (Duke [Duk97]).

The set of elliptic curves $E_{/\mathbb{Q}}$ with no exceptional primes is
$1$.

Recall from the previous section that $\mathcal{S}_{\ell}(X)$ consists of
elliptic curves $E_{/\mathbb{Q}}$ with $H(E)\leq X^{6}$, such that

  1. (1)

has semistable reduction at all primes $p$, except possibly at $p=2,3$,

  2. (2)

and $\bar{\rho}_{E,\ell}$ is surjective.

We thus write $\mathcal{S}_{\ell}(X)$ as the intersection of sets
$\mathcal{D}_{\ell}(X)$ and $\mathcal{E}_{\ell}(X)$, where

$\begin{split}&\mathcal{D}_{\ell}(X)\mathrel{\mathop{\mathchar
58\relax}}=\\{E\in\mathcal{C}(X)\mid E\text{ has semistable reduction at all
primes }p\neq 2,3\\};\\\ &\mathcal{E}_{\ell}(X)\mathrel{\mathop{\mathchar
58\relax}}=\\{E\in\mathcal{C}(X)\mid\bar{\rho}_{E,\ell}\text{ is
surjective}\\}.\\\ \end{split}$

It follows from Theorem 3.1 that $\mathcal{E}_{\ell}(X)$ has density $1$.

######  Proposition 3.2.

Let $C\in\left(0,\frac{4}{\zeta(2)}\right)$, then, for all large enough values
of $X$,

$\\#\mathcal{S}_{\ell}(X)>CX^{5}.$

In particular, there is a constant $C_{1}>0$, such that for all $X>0$,

$\\#\mathcal{S}_{\ell}(X)>C_{1}X^{5}.$

###### Proof.

We make a few reductions, and prove the result as a consequence of theorems of
Duke [Duk97] and Sadek [Sad17]. It suffices to show that

$\liminf_{X\rightarrow\infty}\frac{\\#\mathcal{S}_{\ell}(X)}{X^{5}}\geq\frac{4}{\zeta(2)}.$

Recall from (2.1) that $\\#\mathcal{C}(X)\sim\frac{4}{\zeta(10)}X^{5}$, and
therefore, the statement requires that

$\liminf_{X\rightarrow\infty}\frac{\\#\mathcal{S}_{\ell}(X)}{\\#\mathcal{C}(X)}\geq\frac{\zeta(10)}{\zeta(2)}.$

Since $\mathcal{E}_{\ell}$ has density $1$ and
$\mathcal{S}_{\ell}=\mathcal{D}_{\ell}\cap\mathcal{E}_{\ell}$, the above
inequality is equivalent to showing that

$\liminf_{X\rightarrow\infty}\frac{\\#\mathcal{D}_{\ell}(X)}{\\#\mathcal{C}(X)}\geq\frac{\zeta(10)}{\zeta(2)}.$

Clearly, this is equivalent to proving that

$\liminf_{X\rightarrow\infty}\frac{\\#\mathcal{D}_{\ell}(X^{1/6})}{\\#\mathcal{C}(X^{1/6})}\geq\frac{\zeta(10)}{\zeta(2)};$

which is the statement of [Sad17, Theorem 9.3] 111The work [Sad17] is
superseded by [CS20]. However, the conventions in [CS20] are different from
those used in this article. The author is willing to reproduce the arguments
here for greater clarity (if the referee insists).. ∎

###  3.2. Counting elliptic curves with prescribed division field

Throughout the rest of this section, we fix a Galois extension $L/\mathbb{Q}$
such that $\operatorname{Gal}(L/\mathbb{Q})$ is isomorphic to
$\operatorname{GL}_{2}(\mathbb{F}_{\ell})$. Let $\mathcal{D}_{L}$ be the set
of elliptic curves $E\in\mathcal{C}$ such that $L_{\ell}(E)=L$. The following
is the main result of this section.

######  Proposition 3.3.

Let $L/\mathbb{Q}$ be as above. Then, there exists a positive constant
$C_{2}$, which only depends on $\ell$, and is independent of the extension
$L/\mathbb{Q}$, such that for all $X>0$,

$\\#\mathcal{D}_{L}(X)\leq C_{2}X^{9/2}\log X.$

We provide the proof at the end of this section. Let $X>0$ and let us assume
without loss of generality that there exists an elliptic curve
$A\in\mathcal{D}_{L}$. Note that since $L_{\ell}(A)=L$, it follows that
$\bar{\rho}_{A,\ell}$ is surjective. We make a choice of
$\mathbb{F}_{\ell}$-basis $\mathcal{B}_{A}$ for $A[\ell]$. Suppose that
$E\in\mathcal{D}_{L}(X)$ is another elliptic curve, and $\mathcal{B}_{E}$ be a
choice of $\mathbb{F}_{\ell}$-basis for $E[\ell]$. Then, the residual
representations give rise to isomorphisms, taken with respect to
$\mathcal{B}_{A}$ and $\mathcal{B}_{E}$ respectively

$\begin{split}&\bar{\rho}_{E,\ell}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{Gal}(L/\mathbb{Q})\xrightarrow{\sim}\operatorname{GL}_{2}(\mathbb{F}_{\ell}),\\\
&\bar{\rho}_{A,\ell}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{Gal}(L/\mathbb{Q})\xrightarrow{\sim}\operatorname{GL}_{2}(\mathbb{F}_{\ell}).\end{split}$

Therefore, there exists
$\eta\in\operatorname{Aut}\left(\operatorname{GL}_{2}(\mathbb{F}_{\ell})\right)$,
such that $\bar{\rho}_{A,\ell}=\eta\circ\bar{\rho}_{E,\ell}$. We set
$\operatorname{Inn}\left(\operatorname{GL}_{2}(\mathbb{F}_{\ell})\right)$ to
denote the inner automorphisms of $\operatorname{GL}_{2}(\mathbb{F}_{\ell})$,
and set

$\operatorname{Out}\left(\operatorname{GL}_{2}(\mathbb{F}_{\ell})\right)\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{Aut}\left(\operatorname{GL}_{2}(\mathbb{F}_{\ell})\right)/\operatorname{Inn}\left(\operatorname{GL}_{2}(\mathbb{F}_{\ell})\right).$

The class

$\bar{\eta}_{E}\mathrel{\mathop{\mathchar
58\relax}}=\eta\operatorname{Inn}\left(\operatorname{GL}_{2}(\mathbb{F}_{\ell})\right)\in\operatorname{Out}\left(\operatorname{GL}_{2}(\mathbb{F}_{\ell})\right)$

associated to $E\in\mathcal{D}_{L}(X)$ depends only on $E$ (and is independent
of the choice of basis $\mathcal{B}_{E}$).

Fixing $(A,\mathcal{B}_{A})$ for the entire family $\mathcal{D}_{L}$, we
decompose $\mathcal{D}_{L}$ into a disjoint union

$\mathcal{D}_{L}=\bigsqcup_{\zeta\in\operatorname{Out}\left(\operatorname{GL}_{2}(\mathbb{F}_{\ell})\right)}\mathcal{D}_{L,\zeta},$

where

$\mathcal{D}_{L,\zeta}\mathrel{\mathop{\mathchar
58\relax}}=\\{E\in\mathcal{D}_{L}\mid\eta_{E}=\zeta\\}.$

Let $E_{1},E_{2}\in\mathcal{D}_{L,\zeta}$, and let $\mathcal{B}_{E_{i}}$ be a
choice of $\mathbb{F}_{\ell}$-basis of $E_{i}[\ell]$. We find that

$\bar{\rho}_{A,\ell}=\eta_{i}\circ\bar{\rho}_{E_{i},\ell},$

where
$\eta_{1},\eta_{2}\in\operatorname{Aut}\left(\operatorname{GL}_{2}(\mathbb{F}_{\ell})\right)$.
Since $\eta_{E_{1}}=\eta_{E_{2}}$, there is an inner automorphism $\tau$ of
$\operatorname{GL}_{2}(\mathbb{F}_{\ell})$ such that
$\eta_{1}=\eta_{2}\circ\tau$. This implies that
$\bar{\rho}_{E_{2},\ell}=\tau\circ\bar{\rho}_{E_{1},\ell}$, i.e.,

$\bar{\rho}_{E_{1},\ell}\simeq\bar{\rho}_{E_{2},\ell}.$

######  Definition 3.4.

Let $E_{/\mathbb{Q}}$ be an elliptic curve, and let $\mathcal{D}_{E}$ to be
the set of elliptic curves $E^{\prime}_{/\mathbb{Q}}$ for which
$\bar{\rho}_{E^{\prime},\ell}\simeq\bar{\rho}_{E,\ell}$.

######  Proposition 3.5.

With respect to notation above, there is a set of elliptic curves
$E_{1},\dots,E_{t}$ with
$t\leq\\#\operatorname{Out}\left(\operatorname{GL}_{2}(\mathbb{F}_{\ell})\right)$,
such that $\mathcal{D}_{L}$ is contained in the union
$\bigcup_{i=1}^{t}\mathcal{D}_{E_{i}}$.

###### Proof.

For each
$\zeta\in\operatorname{Out}\left(\operatorname{GL}_{2}(\mathbb{F}_{\ell})\right)$,
such that $\mathcal{D}_{L,\zeta}$ is nonempty, pick an elliptic curve
$E_{\zeta}\in\mathcal{D}_{L,\zeta}$. Then, from the discussion above, we find
that $\mathcal{D}_{L,\zeta}$ is contained in $\mathcal{D}_{E_{\zeta}}$, and
the result follows. ∎

Suppose that $E_{1}$ and $E_{2}$ are elliptic curves over $\mathbb{Q}$ such
that $\bar{\rho}_{E_{1},\ell}\simeq\bar{\rho}_{E_{2},\ell}$. Let $N_{i}$
denote the conductor of $E_{i}$. For any prime number $p\nmid N_{1}N_{2}\ell$,
note that $\bar{\rho}_{E_{i},\ell}$ is unramified at $p$. Let $\sigma_{p}$ be
a choice of Frobenius at $p$, and set $t_{p}(E_{i})\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{trace}\left(\bar{\rho}_{E_{i},\ell}(\sigma_{p})\right)$.
We find that $t_{p}(E_{1})=t_{p}(E_{2})$ for all prime numbers $p\nmid
N_{1}N_{2}\ell$.

Let $(t_{1},t_{2},d)\in\mathbb{F}_{\ell}^{3}$ be such that $d\neq 0$. Let
$(E_{1},E_{2})$ be a pair of elliptic curves defined over $\mathbb{Q}$, and
let $\pi_{E_{1},E_{2}}(X,t_{1},t_{2},d,\ell)$ to be the number of prime
numbers $p\leq X$ such that

  1. (1)

$p\nmid N_{1}N_{2}$,

  2. (2)

$p\equiv d\pmod{\ell}$,

  3. (3)

$t_{p}(E_{i})=t_{i}$ for $i=1,2$.

Let $\pi(X,d,\ell)$ be the number of prime numbers $p\leq X$ such that
$p\equiv d\pmod{\ell}$. Note that if
$\bar{\rho}_{E_{1},\ell}\simeq\bar{\rho}_{E_{2},\ell}$, then, the relation
$t_{p}(E_{1})=t_{p}(E_{2})$ implies that

$\pi_{E_{1},E_{2}}(X,t_{1},t_{2},d,\ell)=0,$

unless $t_{1}=t_{2}$.

######  Proposition 3.6.

Let $(t_{1},t_{2},d)\in\mathbb{F}_{\ell}^{3}$ be such that $d\neq 0$, and
$(E_{1},E_{2})$ be any pair of elliptic curves defined over $\mathbb{Q}$.
Then, there are constants $\delta,C>0$, such that for all $X>0$,

$\frac{1}{\\#\mathcal{C}(X)^{2}}\sum_{(E_{1},E_{2})\in\mathcal{C}(X)^{2}}\left(\pi_{E_{1},E_{2}}(X,t_{1},t_{2},d,\ell)-\delta\pi(X,d,\ell)\right)^{2}\leq
CX.$

###### Proof.

This result follows from a variant of Gallagher’s large sieve, and is similar
argument to that of [Duk97, Theorem 2]. For a proof of the result, we refer to
[RW23, Proposition 3.2]. ∎

######  Proposition 3.7.

Let $E_{/\mathbb{Q}}$ be an elliptic curve. Then, there exists a constant
$C_{3}>0$, which are independent of $E$, such that for all $X>0$,

$\\#\mathcal{D}_{E}(X)\leq C_{3}X^{9/2}\log X.$

###### Proof.

Let $E_{1},E_{2}$ be elliptic curves in $\mathcal{D}_{E}$; notice that
$\bar{\rho}_{E_{1},\ell}\simeq\bar{\rho}_{E_{2},\ell}$. Taking
$(t_{1},t_{2},d)\in\mathbb{F}_{\ell}^{3}$ such that $d\neq 0$ and $t_{1}\neq
t_{2}$, we find that

$\pi_{E_{1},E_{2}}(X,t_{1},t_{2},d,\ell)=0.$

Proposition 3.6 asserts that

$\frac{1}{\\#\mathcal{C}(X)^{2}}\sum_{(E_{1},E_{2})\in\mathcal{C}(X)^{2}}\left(\pi_{E_{1},E_{2}}(X,t_{1},t_{2},d,\ell)-\delta\pi(X,d,\ell)\right)^{2}\leq
CX.$

Restricting the sum to $\mathcal{D}_{E}(X)^{2}$, we find that

$\frac{1}{\\#\mathcal{C}(X)^{2}}\sum_{(E_{1},E_{2})\in\mathcal{D}_{E}(X)^{2}}\left(\pi_{E_{1},E_{2}}(X,t_{1},t_{2},d,\ell)-\delta\pi(X,d,\ell)\right)^{2}\leq
CX.$

Since

$\pi_{E_{1},E_{2}}(X,t_{1},t_{2},d,\ell)=0,$

this simplifies to give us

$\left(\frac{\\#\mathcal{D}_{E}(X)}{\\#\mathcal{C}(X)}\right)^{2}\leq\frac{C}{\delta^{2}}\frac{X}{\pi(X,d,\ell)^{2}},$

and therefore,

$\left(\frac{\\#\mathcal{D}_{E}(X)}{\\#\mathcal{C}(X)}\right)\leq\frac{\sqrt{C}}{\delta}\frac{X^{1/2}}{\pi(X,d,\ell)}.$

According to Dirichlet’s theorem,
$\pi(X,d,\ell)\sim\frac{1}{\varphi(\ell)}\frac{X}{\log X}$, and therefore,

$\left(\frac{\\#\mathcal{D}_{E}(X)}{\\#\mathcal{C}(X)}\right)=O\left(\frac{\log
X}{\sqrt{X}}\right).$

Since $\\#\mathcal{C}(X)\sim\frac{4}{\zeta(10)}X^{5}$, we deduce that

$\\#\mathcal{D}_{E}(X)=O\left(X^{9/2}\log X\right),$

which proves the result. ∎

###### Proof of Proposition 3.3.

Proposition 3.5 asserts that there is a set of elliptic curves
$E_{1},\dots,E_{t}$ with
$t\leq\\#\operatorname{Out}\left(\operatorname{GL}_{2}(\mathbb{F}_{\ell})\right)$,
such that $\mathcal{D}_{L}$ is contained in the union
$\bigcup_{i=1}^{t}\mathcal{D}_{E_{i}}$. Proposition 3.7 implies that there is
a constant $C_{3}>0$, such that

$\\#\mathcal{D}_{E_{i}}(X)\leq C_{3}X^{9/2}\log X.$

Therefore, setting $C_{2}\mathrel{\mathop{\mathchar
58\relax}}=\\#\operatorname{Out}\left(\operatorname{GL}_{2}(\mathbb{F}_{\ell})\right)C_{3}$,
we find that

$\\#\mathcal{D}_{L}(X)\leq C_{2}X^{9/2}\log X.$

∎

###  3.3. Proof of the main result

In this section, we prove the main result. The main ingredients will be
Propositions 2.2, 2.4, 3.2 and 3.3.

###### Proof of Theorem A.

Setting $Y_{1}(X)\mathrel{\mathop{\mathchar
58\relax}}=\left(\frac{X}{496}\right)^{\frac{1}{6}}$, we recall that
Proposition 2.2 asserts that for any elliptic curve
$E\in\mathcal{S}_{\ell}\left(Y_{1}(X)\right)$, the pair $\gamma_{\ell}(E)$
belongs to $\mathcal{M}(\operatorname{GL}_{2}(\mathbb{F}_{\ell});X)$. This
implies that

$\begin{split}\\#\mathcal{S}_{\ell}(Y_{1}(X))&\leq\sum_{(L,\rho)\in\mathcal{M}(\operatorname{GL}_{2}(\mathbb{F}_{\ell});X)}\\#\\{E\in\mathcal{C}(Y_{1}(X))\mid\gamma_{\ell}(E)=(L,\rho)\\};\\\
&\leq\sum_{(L,\rho)\in\mathcal{M}(\operatorname{GL}_{2}(\mathbb{F}_{\ell});X)}\\#\\{E\in\mathcal{C}(Y_{1}(X))\mid
L_{\ell}(E)=L\\};\\\
&=\sum_{(L,\rho)\in\mathcal{M}(\operatorname{GL}_{2}(\mathbb{F}_{\ell});X)}\\#\mathcal{D}_{L}(Y_{1}(X))\\\
&\leq C_{2}Y_{1}(X)^{9/2}\log
Y_{1}(X)\\#\mathcal{M}(\operatorname{GL}_{2}(\mathbb{F}_{\ell});X);\end{split}$

where in the final inequality, we invoke Proposition 3.3. We reiterate here
that $C_{2}$ depends only on $\ell$, and is independent of the extension
$L/\mathbb{Q}$. It follows from Proposition 3.2 that there is a constant
$C_{1}>0$, such that for all $X>0$, we have that
$\\#\mathcal{S}_{\ell}(X)>C_{1}X^{5}$. Thus, we find that

$\begin{split}\\#\mathcal{M}(\operatorname{GL}_{2}(\mathbb{F}_{\ell});X)&\geq\frac{\\#\mathcal{S}_{\ell}(Y_{1}(X))}{C_{2}Y_{1}(X)^{9/2}\log
Y_{1}(X)};\\\ &\geq\frac{C_{1}(Y_{1}(X))^{5}}{C_{2}Y_{1}(X)^{9/2}\log
Y_{1}(X)}\geq C\frac{X^{\frac{1}{12}}}{\log X},\end{split}$

for some constant $C>0$.

Setting $Y_{2}(X)\mathrel{\mathop{\mathchar
58\relax}}=\left(\frac{X}{6^{3264}\ell^{C_{\ell}}}\right)^{\frac{\ell}{6(\ell-1)\\#\operatorname{GL}_{2}(\mathbb{F}_{\ell})}}$,
we recall that Proposition 2.4 asserts that for any elliptic curve
$E\in\mathcal{S}_{\ell}\left(Y_{2}(X)\right)$, the division field
$L_{\ell}(E)$ belongs to
$\mathcal{F}(\operatorname{GL}_{2}(\mathbb{F}_{\ell});X)$. By the same
argument as above, we find that

$\begin{split}\\#\mathcal{F}(\operatorname{GL}_{2}(\mathbb{F}_{\ell});X)&\geq\frac{\\#\mathcal{S}_{\ell}(Y_{2}(X))}{C_{2}Y_{2}(X)^{9/2}\log
Y_{2}(X)};\\\ &\geq\frac{C_{1}(Y_{2}(X))^{5}}{C_{2}Y_{2}(X)^{9/2}\log
Y_{2}(X)}\geq
C\frac{X^{\frac{\ell}{12(\ell-1)\\#\operatorname{GL}_{2}(\mathbb{F}_{\ell})}}}{\log
X},\end{split}$

for some constant $C>0$. ∎

## References

  * [BF10] Karim Belabas and Étienne Fouvry.  Discriminants cubiques et progressions arithmétiques.  International Journal of Number Theory, 6(07):1491–1529, 2010.
  * [Bha05] Manjul Bhargava.  The density of discriminants of quartic rings and fields.  Annals of Mathematics, pages 1031–1063, 2005.
  * [Bha07] Manjul Bhargava.  Mass formulae for extensions of local fields, and conjectures on the density of number field discriminants.  International Mathematics Research Notices, 2007(9):rnm052–rnm052, 2007.
  * [Bha10] Manjul Bhargava.  The density of discriminants of quintic rings and fields.  Annals of mathematics, pages 1559–1591, 2010.
  * [Bru92] Armand Brumer.  The average rank of elliptic curves I.  Inventiones mathematicae, 109(1):445–472, 1992.
  * [BW08] Manjul Bhargava and Melanie Wood.  The density of discriminants of $S_{3}$-sextic number fields.  Proceedings of the American Mathematical Society, 136(5):1581–1587, 2008.
  * [CDO02] Henri Cohen, Francisco Diaz Y Diaz, and Michel Olivier.  Enumerating quartic dihedral extensions of.  Compositio Mathematica, 133(1):65–93, 2002.
  * [CK02] Élie Cali and Alain Kraus.  Sur la $p$-différente du corps des points de $\ell$-torsion des courbes elliptiques, $\ell\neq p$.  Acta Arithmetica, 104:1–21, 2002.
  * [CS20] JE Cremona and M Sadek.  Local and global densities for weierstrass models of elliptic curves.  arXiv preprint arXiv:2003.08454, 2020.
  * [DH71] Harold Davenport and Hans Arnold Heilbronn.  On the density of discriminants of cubic fields. ii.  Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences, 322(1551):405–420, 1971.
  * [Duk97] William Duke.  Elliptic curves with no exceptional primes.  Comptes rendus de l’Académie des sciences. Série 1, Mathématique, 325(8):813–818, 1997.
  * [EV06] Jordan S Ellenberg and Akshay Venkatesh.  The number of extensions of a number field with fixed degree and bounded discriminant.  Annals of mathematics, pages 723–741, 2006.
  * [Kra99] Alain Kraus.  On the p-différent of the body of the p-torsion points of the elliptical curves.  Bulletin of the Australian Mathematical Society, 60(3):407–428, 1999.
  * [Lip21] Michael Lipnowski.  On bhargava’s heuristics for $\operatorname{GL}_{2}(\mathbb{F}_{p})$-number fields and the number of elliptic curves of bounded conductor.  Experimental Mathematics, 30(2):221–234, 2021.
  * [Mäk85] Sirpa Mäki.  On the density of abelian number fields, volume 54.  Suomalainen tiedeakatemia, 1985.
  * [Mal02] Gunter Malle.  On the distribution of galois groups.  Journal of Number Theory, 92(2):315–329, 2002.
  * [PTBW20] Lillian B Pierce, Caroline L Turnage-Butterbaugh, and Melanie Matchett Wood.  An effective chebotarev density theorem for families of number fields, with an application to $\ell$-torsion in class groups.  Inventiones mathematicae, 219(2):701–778, 2020.
  * [RS99] Kenneth A Ribet and William A Stein.  Lectures on serre’s conjectures.  Arithmetic algebraic geometry (Park City, UT, 1999), 9:143–232, 1999.
  * [RW23] Anwesh Ray and Tom Weston.  Diophantine stability for elliptic curves on average.  arXiv preprint arXiv:2304.09742, 2023.
  * [Sad17] Mohammad Sadek.  Counting elliptic curves with bad reduction over a prescribed set of primes.  arXiv preprint arXiv:1704.02056, 2017.
  * [Sil94] Joseph H Silverman.  Advanced topics in the arithmetic of elliptic curves, volume 151\.  Springer Science & Business Media, 1994.
  * [Wan21] Jiuya Wang.  Malle’s conjecture for.  Compositio Mathematica, 157(1):83–121, 2021.