# New Isogenies of Elliptic Curves Over Number Fields

Tyler Genao

###### Abstract.

We analyze the fields of definition of cyclic isogenies on elliptic curves to
prove the following uniformity result: for any number field $F_{0}$ which
satisfies an isogeny condition, there exists a constant
$B:=B(F_{0})\in\mathbb{Z}^{+}$ such that for any finite extension $L/F_{0}$
whose degree $[L:F_{0}]$ is coprime to $B$, one has for all elliptic curves
$E_{/F_{0}}$ with $j$-invariant $\neq 0,1728$ that any $L$-rational cyclic
isogeny on $E$ must be $F_{0}$-rational. We also prove unconditional results
for the mod-$\ell$ Galois representations of non-CM elliptic curves with an
$F_{0}$-rational $\ell$-isogeny when $\ell$ is uniformly large.

###### 2010 Mathematics Subject Classification:

11G05

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##  1\. Introduction

Given a number field $F$, it is of great interest to understand isogenies of
elliptic curves which are rational over $F$, in analogy to understanding
torsion points. For example, a classic theorem of Mazur shows that the only
possible prime degrees $\ell$ of $\mathbb{Q}$-rational isogenies are the
primes $\ell\in\\{2,3,5,7,11,13,17,19,37,43,67,163\\}$ [Maz78]. Following
Mazur’s work, Kenku has shown that all possible degrees $n$ of
$\mathbb{Q}$-rational isogenies with cyclic kernel are the integers
$n\in[1,19]\cup\\{21,25,27,37,43,67,163\\}$ [Ken82]. There are also uniform
bounds on the degrees of rational isogenies over number fields without
rational complex multiplication if one assumes the generalized Riemann
hypothesis (GRH), see [LV14] and [Ban23] (as well as Theorem 9).

The study of rational isogenies on elliptic curves is intimately related to
the study of torsion subgroups: for an elliptic curve $E_{/F}$, any finite
subgroup $C\subseteq E(\overline{\mathbb{Q}})$ induces an isogeny from $E$ to
the quotient curve $E/C$, whose kernel is $C$. This isogeny is $F$-rational if
and only if $C$ is stable under the action of the absolute Galois group
$G_{F}:=\textrm{Gal}(\overline{\mathbb{Q}}/F)$, and in such a case one has
that $E/C$ is unique up to $F$-rational isomorphism.

For a number field $F$ and an elliptic curve $E_{/F}$ defined over $F$, if its
mod-$n$ Galois representation $\rho_{E,n}\colon
G_{F}\rightarrow\textrm{GL}_{2}(\mathbb{Z}/n\mathbb{Z})$ surjects onto
$\textrm{GL}_{2}(\mathbb{Z}/n\mathbb{Z})$, then the field of definition of a
isogeny whose kernel $C\subseteq E(\overline{\mathbb{Q}})$ is cyclic of order
$n$ is “as large as possible.” Denoting this field by $F(C)$, it has degree
$[F(C):F]>n$ when $n>1$, whence it follows that $C$ is not $F$-rational. When
$E$ has no complex multiplication (abbreviated as non-CM), by Serre’s open
image theorem the image $\rho_{E,\ell}(G_{F})$ is equal to
$\textrm{GL}_{2}(\mathbb{Z}/\ell\mathbb{Z})$ for all primes
$\ell\in\mathbb{Z}^{+}$ “sufficiently large” with respect to both $E$ and $F$
[Ser72].

Is there a similar result for the field of definition of $n$-isogenies for
elliptic curves over $F$ that is uniform in $F$, i.e., independent of $E$?
When $F=\mathbb{Q}$, this is connected to Serre’s uniformity question, which
asks for an absolute constant $\ell_{0}\in\mathbb{Z}^{+}$ such that when a
prime $\ell\in\mathbb{Z}^{+}$ satisfies $\ell>\ell_{0}$, one has for any non-
CM elliptic curve $E_{/\mathbb{Q}}$ that
$\rho_{E,\ell}(G_{\mathbb{Q}})=\textrm{GL}_{2}(\mathbb{Z}/\ell\mathbb{Z})$.
What is currently known is that for any prime $\ell>37$ and for any non-CM
elliptic curve $E_{/\mathbb{Q}}$, the mod-$\ell$ Galois representation either
surjects onto $\textrm{GL}_{2}(\mathbb{Z}/\ell\mathbb{Z})$ or is the
normalizer of the nonsplit Cartan subgroup of
$\textrm{GL}_{2}(\mathbb{Z}/\ell\mathbb{Z})$ up to conjugacy [FL]. It follows
that for $\ell>37$, any order $\ell$ subgroup $C\subseteq
E(\overline{\mathbb{Q}})$ on any non-CM elliptic curve $E_{/\mathbb{Q}}$
satisfies $[\mathbb{Q}(C):\mathbb{Q}]=\ell+1$. One consequence of this is that
if $L/\mathbb{Q}$ is any odd degree extension, then for any non-CM elliptic
curve $E_{/\mathbb{Q}}$, an $L$-rational $\ell$-isogeny on $E$ must satisfy
$\ell\leq 37$. Surprisingly, this result extends to cyclic isogenies on
elliptic curves whose $j$-invariants have arbitrarily large degree over
$\mathbb{Q}$: if $[L:\mathbb{Q}]$ is odd, then for any elliptic curve $E_{/L}$
that is $\overline{\mathbb{Q}}$-isogenous to a $\mathbb{Q}$-rational elliptic
curve, all $L$-rational $\ell$-isogenies on $E$ must also satisfy $\ell\leq
37$, see [CN21, Theorem 1.1]. This type of result can be used to study torsion
from the family of $\mathbb{Q}$-curves, see also [BN].

Towards better understanding the fields of definition of cyclic isogenies, let
us introduce some terminology. Given an elliptic curve $E_{/F}$ and a cyclic
$n$-isogeny with kernel $C\subseteq E(\overline{\mathbb{Q}})$, we say that $C$
(and the isogeny) is new if $C$ is not $F$-rational. For an extension $L/F$,
we say that $C$ is new for $L/F$ if $C$ is $L$-rational but not $F$-rational;
equivalently, $C$ is new for $L/F$ if $F(C)\subseteq L$ and $[F(C):F]>1$.

The main theorem of this paper is a uniformity result on fields of definition
of new cyclic isogenies for elliptic curves over a fixed number field $F_{0}$.
It is phrased as a generalization of [Gen24, Theorem 1], which concerns new
torsion points over fixed $F_{0}$.

######  Theorem 1.

Assume that $F_{0}$ is a number field which satisfies the LV-hypotheses. Then
there exists a constant $B:=B(F_{0})\in\mathbb{Z}^{+}$ such that for any
finite extension $L/F_{0}$ whose degree $[L:F_{0}]$ is coprime to $B$, there
are no new isogenies for $L/F_{0}$ on any elliptic curve over $F_{0}$ with
$j$-invariant $\neq 0,1728$.

This theorem applies to any number field $F_{0}$ which satisfies a particular
isogeny condition, called the LV-hypotheses:

######  LV-hypotheses for $F_{0}$.

We assume that the generalized Riemann hypothesis (GRH) is true, and that
$F_{0}$ has no rationally defined complex multiplication (RCM), i.e., there do
not exist CM elliptic curves over $F_{0}$ whose endomorphism rings are
contained in $F_{0}$.

A consequence of work of Larson and Vaintrob is that when a number field
$F_{0}$ satisfies the LV-hypotheses, there exists a finite set
$S_{F_{0}}\subseteq\mathbb{Z}^{+}$ of primes such that the following holds: if
there exists an $F_{0}$-rational isogeny of prime degree $\ell$ on an elliptic
curve, then $\ell\in S_{F_{0}}$ [LV14, Theorem 1]; see Theorem 9 for the full
result. Phrased differently, under the LV-hypotheses there exists a uniform
bound on prime degree $F_{0}$-rational isogenies. Thus, Theorem 1 says that if
an extension $L/F_{0}$ has a degree which satisfies certain coprimality
conditions, then prime degrees of $L$-rational isogenies are bounded in terms
of $F_{0}$.

As will be made precise in the following section, the degree of the field of
definition of an $\ell$-isogeny on $E$ is determined by the size of its orbit
under the action of $G_{F_{0}}$, which acts on the set of order $\ell$
subgroups of $E$. When $F_{0}$ satisfies the LV-hypotheses and $\ell$ is
sufficiently large with respect to $F_{0}$, the group $E[\ell]$ of
$\ell$-torsion points on $E$ is an irreducible
$(\mathbb{Z}/\ell\mathbb{Z})$-module, which restricts the possible images
$\rho_{E,\ell}(G_{F_{0}})$ of the mod-$\ell$ Galois representation of $E$ over
$F_{0}$. This implies that the image $\rho_{E,\ell}(G_{F_{0}})$ is “relatively
uniformly large,” as described in Theorem 2 below. This theorem can be viewed
as partial progress towards Serre’s uniformity question for number fields
larger than $\mathbb{Q}$ which satisfy the LV-hypotheses.

######  Theorem 2.

Suppose that $F_{0}$ is a number field which satisfies the LV-hypotheses. Then
there exists a constant $c:=c(F_{0})\in\mathbb{Z}^{+}$ such that for all
primes $\ell>c$, one has the following: for an elliptic curve $E_{/F_{0}}$, if
the image $G:=\rho_{E,\ell}(G_{F_{0}})$ doesn’t equal
$\emph{GL}_{2}(\mathbb{Z}/\ell\mathbb{Z})$, then it is contained in the
normalizer of a Cartan group in $\emph{GL}_{2}(\mathbb{Z}/\ell\mathbb{Z})$ up
to conjugacy.

Let $N_{s}(\ell)$ and $N_{ns}(\ell)$ denote the normalizer of the split and
nonsplit Cartan group mod-$\ell$, respectively. Then up to a choice of basis
for $E[\ell]$, one of the following cases holds:

  1. 1.

If $G\subseteq N_{s}(\ell)$ then $C_{s}(\ell)^{e}\subseteq G$ for some
$e\in\\{1,2,3,4,6\\}$, and $G$ contains the subgroup of scalar matrices.
Furthermore, the index of $G$ in $N_{s}(\ell)$ satisfies

$[N_{s}(\ell):G]\mid\gcd(\ell-1,e).$

  2. 2.

If $G\subseteq N_{ns}(\ell)$ then $G=N_{ns}(\ell)$ or $G(\ell)$. If
$\ell\equiv 1\pmod{3}$ then $G=N_{ns}(\ell)$.

(All of these subgroups are defined in $\S\ref{Sect_SubgpsOfGL2}$.)

The conclusion of Theorem 2 is an improvement of [Gen24, Theorem 2] in the
case where $G$ is contained in the normalizer $N_{ns}(\ell)$ of the nonsplit
Cartan group mod-$\ell$, up to conjugacy. More specifically, when $\ell$ is
large, if up to conjugacy $G$ is also contained in the nonsplit Cartan group
$C_{ns}(\ell)$, then it follows that $G$ is diagonalizable over the quadratic
extension $\mathbb{F}_{\ell^{2}}$ of $\mathbb{Z}/\ell\mathbb{Z}$. Thus,
$E[\ell]\otimes_{\mathbb{F}_{\ell}}\overline{\mathbb{F}_{\ell}}$ has a degree
1 associated character with values in $\mathbb{F}_{\ell^{2}}$. By Theorem 9
this implies that $F_{0}$ has RCM, or that GRH fails for
$F_{0}[\sqrt{-\ell}]$. Therefore, assuming both GRH and that $F_{0}$ has no
RCM, we may conclude that $G$ cannot be contained in $C_{ns}(\ell)$ up to
conjugacy when $\ell$ is large; thus, Theorem 2 follows at once from [Gen24,
Theorem 2].

Theorem 1 is in part a consequence of the following result, which can be
viewed as a generalization of [CN21, Proposition 3.3] from $\mathbb{Q}$ to
$F_{0}$.

######  Proposition 3.

Given a number field $F_{0}$ which satisfies the LV-hypotheses, there exists a
constant $c:=c(F_{0})\in\mathbb{Z}^{+}$ such that for all primes $\ell>c$ and
for all elliptic curves $E_{/F_{0}}$, the field of definition for an
$\ell$-isogeny of $E$ has even degree over $F_{0}$.

######  Remark 1.

Both Theorem 2 and Proposition 3 use the same constant $c(F_{0})$: this is the
least positive integer such that for all primes $\ell\geq c(F_{0})$, one has
$\ell\geq\max\\{74,15[F_{0}:\mathbb{Q}]+2\\}$, $\ell$ is unramified in $F_{0}$
and $\ell\not\in S_{F_{0}}$ from Theorem 9.

In the final section of this paper, we will prove unconditional results for
the fields of definition of cyclic isogenies of large prime degree which come
from non-CM elliptic curves over a fixed $F_{0}$. This case is excluded when
one assumes the LV-hypotheses.

######  Theorem 4.

Let $F_{0}$ be a number field. Fix a prime
$\ell\geq\max\\{c(F_{0}),72[F_{0}:\mathbb{Q}]\\}$ such that $\ell$ is
unramified in $F_{0}$ and in each imaginary quadratic order contained in
$F_{0}$. Then for an elliptic curve $E_{/F_{0}}$ with exactly one
$F_{0}$-rational $\ell$-isogeny, $E$ must be non-CM, and one has for all order
$\ell$ subgroups $C\subseteq E(\overline{\mathbb{Q}})$ that either
$[F_{0}(C):F_{0}]=1$ or $[F_{0}(C):F_{0}]=\ell$.

Towards proving Theorem 4, we also prove another result which describes the
“relative index” of the mod-$\ell$ Galois representation of an elliptic curve
with exactly one rational $\ell$-isogeny. This result is complementary to
Theorem 2, and its analysis is aided by the description of the isogeny
character in Theorem 9.

######  Theorem 5.

Let $F_{0}$ be a number field. Fix a prime
$\ell\geq\max\\{c(F_{0}),72[F_{0}:\mathbb{Q}]\\}$ such that $\ell$ is
unramified in $F_{0}$ and in each imaginary quadratic order contained in
$F_{0}$. Then for an elliptic curve $E_{/F_{0}}$ with exactly one
$F_{0}$-rational $\ell$-isogeny, $E$ must be non-CM, and up to conjugacy the
image $G:=\rho_{E,\ell}(G_{F_{0}})$ is contained in $B(\ell)$, the Borel
subgroup of $\emph{GL}_{2}(\mathbb{Z}/\ell\mathbb{Z})$. Furthermore, if
$G\subseteq B(\ell)$ then one has

$[B(\ell):G]\mid 864[F_{0}:\mathbb{Q}].$

##  2\. Galois Action on Torsion and Isogenies

In this section, we will set up some notation and terminology for the rest of
this paper, as well as describe and prove some basic results.

###  2.1. The action of Galois on $n$-torsion points of elliptic curves

For an integer $n>1$, we let
$\textrm{GL}_{2}(n):=\textrm{GL}_{2}(\mathbb{Z}/n\mathbb{Z})$ be the group of
$2\times 2$ invertible matrices over $\mathbb{Z}/n\mathbb{Z}$. If we let
$e_{1}:=\begin{bmatrix}1\\\ 0\end{bmatrix}$ and $e_{2}:=\begin{bmatrix}0\\\
1\end{bmatrix}$ be the unit column vectors over $\mathbb{Z}/n\mathbb{Z}$, then
$\textrm{GL}_{2}(n)$ acts by left multiplication on the
$\mathbb{Z}/n\mathbb{Z}$-module $V$ generated by $e_{1}$ and $e_{2}$, which is
free of rank two.

Once and for all, fix an algebraic closure $\overline{\mathbb{Q}}$ of
$\mathbb{Q}$. Given an elliptic curve $E_{/F}$ and an integer
$n\in\mathbb{Z}^{+}$, the $n$-torsion subgroup
$E[n]:=E(\overline{\mathbb{Q}})[n]$ is a free rank two
$\mathbb{Z}/n\mathbb{Z}$-module [Sil09, Corollary 6.4]; a choice of basis
$\\{P,Q\\}$ for $E[n]$ gives a $\mathbb{Z}/n\mathbb{Z}$-module isomorphism
$E[n]\cong V$. In fact, the absolute Galois group
$G_{F}:=\textrm{Gal}(\overline{\mathbb{Q}}/F)$ acts
$\mathbb{Z}/n\mathbb{Z}$-linearly on $E[n]$. With respect to a basis
$\\{P,Q\\}$ of $E[n]$, this action is explicitly described by the mod-$n$
Galois representation

$\rho_{E,n,P,Q}\colon G_{F}\rightarrow\textrm{GL}_{2}(n).$

This map is as follows: if $\sigma\in G_{F}$ acts on $E[n]$ via
$P^{\sigma}=aP+cQ$ and $Q^{\sigma}=bP+dQ$, then one has
$\rho_{E,n,P,Q}(\sigma)=\begin{bmatrix}a&b\\\ c&d\end{bmatrix}$. By the Weil
pairing, the image of the determinant of $\rho_{E,n,P,Q}(G_{F})$ is equal to
the image of the mod-$n$ cyclotomic character on $G_{F}$.

We will usually write $G:=\rho_{E,n}(G_{F})$ instead of
$\rho_{E,n,P,Q}(G_{F})$ for the image of the mod-$n$ Galois representation
over $F$; this notation is well-defined up to conjugacy, but we will often
work with an implicitly fixed basis. One always has
$G\cong\textrm{Gal}(F(E[n])/F)$, where $F(E[n])$ is the $n$-division field of
$E$ over $F$.

###  2.2. Cyclic isogenies of elliptic curves

Given an algebraic extension $F/\mathbb{Q}$ and two elliptic curves $E_{/F}$
and ${E_{2}}_{/F}$, an $F$-isogeny from $E$ to $E_{2}$ is a nontrivial
$F$-rational morphism of curves $\phi\colon E\rightarrow E_{2}$ which fixes
the point at infinity: $\phi(O_{E})=O_{E_{2}}$. Such an isogeny induces a
group homomorphism $\phi\colon E(F)\rightarrow E_{2}(F)$ [Sil09, Theorem 4.8].
“Being $F$-isogenous” defines an equivalence relation on elliptic curves over
$F$, and the equivalence class of an elliptic curve is called its $F$-isogeny
class. When $F=\overline{\mathbb{Q}}$, we also call this the geometric isogeny
class of $E$.

Given an isogeny $\phi\colon E\rightarrow E_{2}$, the kernel $C:=\ker\phi$ is
a finite subgroup of
$E[\textrm{tors}]:=E(\overline{\mathbb{Q}})[\textrm{tors}]$; if $C$ is cyclic
of order $n$, we say that $\phi$ is a cyclic $n$-isogeny. Since $\phi$ is
$F$-rational, it follows that $C$ is stable under the action of $G_{F}$: that
is, for all $\sigma\in G_{F}$ one has $C^{\sigma}=C$. Conversely, any cyclic
order $n$ subgroup $C\subseteq E[\textrm{tors}]$ which is stable under the
action of $G_{F}$ induces an $F$-rational isogeny on $E$ whose kernel is $C$
[Sil09, Exercise 3.13.e]. For this reason, we sometimes refer to cyclic order
$n$ subgroups of $E[n]$ as cyclic $n$-isogenies; when $n=\ell$ is prime, an
$\ell$-isogeny is automatically cyclic.

The action of $G_{F}$ on $E[n]$ extends to an action on its set of cyclic
order $n$ subgroups. In general, if we let $\textrm{Cyc}(V)$ denote the set of
nontrivial cyclic $\mathbb{Z}/n\mathbb{Z}$-submodules of the free rank two
$\mathbb{Z}/n\mathbb{Z}$-module $V:=\langle e_{1},e_{2}\rangle$, then the
action of $\textrm{GL}_{2}(n)$ on $V$ extends to an action on
$\textrm{Cyc}(V)$: for $C=\langle v\rangle\in\textrm{Cyc}(V)$ and
$m\in\textrm{GL}_{2}(n)$, we set $m\cdot C:=\langle m\cdot v\rangle$. We will
denote the set of cyclic order $n$ subgroups in $E$ by
$\textrm{Cyc}(E,n):=\textrm{Cyc}(E[n])$. Matters simplify in the case where
$n=\ell$ is prime: one has $\\#\textrm{Cyc}(V)=\ell+1$, since the nontrivial
order $\ell$ subgroups of $V$ are $\langle e_{1}\rangle$ and $C_{k}:=\langle
ke_{1}+e_{2}\rangle$, where $0\leq k<\ell$.

###  2.3. Orbits and fields of definition

Given an elliptic curve $E_{/F}$ and a torsion point $R\in E[\textrm{tors}]$,
the field of definition of $R$ over $F$, denoted by $F(R)$, is the extension
of $F$ of least degree over which $R$ is rational. Equivalently, it is the
fixed field of the stabilizer of $R$ under the action of $G_{F}$ on $E[n]$.

Say $R\in E(\overline{\mathbb{Q}})$ has order $n$, and let
$G:=\rho_{E,n}(G_{F})$. We will write $\mathcal{O}_{G}(R)$ to denote the orbit
of $R$ under $G$ (also called the $G$-orbit of $R$). A key observation
regarding orbits and fields of definition is the following. Since the
stabilizer of $R$ under $G$ is $\textrm{Gal}(F(E[n])/F(R))$, the orbit-
stabilizer theorem implies that

$\\#\mathcal{O}_{G}(R)=[F(R):F].$

This size is invariant under a change of basis for $E[n]$.

A similar description applies to the field of definition of a cyclic isogeny.
Given an elliptic curve $E_{/F}$ and a cyclic $n$-isogeny
$C\in\textrm{Cyc}(E,n)$, the field of definition of $C$ over $F$, denoted by
$F(C)$, is the least extension of $F$ over which $C$ is stable under the
action of Galois. By Galois theory, $F(C)$ is also the fixed field of the
stabilizer of $C$ under the action of $G_{F}$ on $\textrm{Cyc}(E,n)$. The
orbit-stabilizer theorem then implies that

$\\#\mathcal{O}_{G}(C)=[F(C):F],$

which is invariant under a change of basis for $E[n]$.

We will first prove Theorem 1, assuming that Proposition 3 holds. Our proof of
Proposition 3 will analyze the degree $[F(C):F]$ by studying the size of the
orbit $\mathcal{O}_{G}(C)$. We will show that if $F=F_{0}$ satisfies the LV-
hypotheses, then if $n=\ell$ is prime and sufficiently large (specifically
$\ell\geq c(F_{0})$, see Remark 1), we must have that $\\#\mathcal{O}_{G}(C)$
is even.

###  2.4. Subgroups of $\textrm{GL}_{2}(\ell)$

In this subsection, we will define the groups which appear in Theorem 2.

For $\ell\geq 5$, let us define the split Cartan group mod-$\ell$ as the
subgroup of diagonal matrices,

$C_{s}(\ell):=\left\\{\begin{bmatrix}a&0\\\
0&d\end{bmatrix}\in\textrm{GL}_{2}(\ell)\right\\}.$

Its normalizer is denoted as $N_{s}(\ell)$; explicitly, we have

$\displaystyle N_{s}(\ell)$
$\displaystyle=C_{s}(\ell)\cup\begin{bmatrix}0&1\\\
1&0\end{bmatrix}C_{s}(\ell)$ $\displaystyle=\left\\{\begin{bmatrix}a&0\\\
0&d\end{bmatrix},\begin{bmatrix}0&d\\\
a&0\end{bmatrix}\in\textrm{GL}_{2}(\ell)\right\\}.$

We will also define the non-split Cartan group mod-$\ell$, denoted
$C_{ns}(\ell)$, as follows. For a prime $\ell\in\mathbb{Z}^{+}$, we will write
$\mathbb{F}_{\ell}:=\mathbb{Z}/\ell\mathbb{Z}$. Fix the least positive integer
$\epsilon$ which generates $\mathbb{F}_{\ell}^{\times}$: then we have

$C_{ns}(\ell)=\left\\{\begin{bmatrix}a&b\epsilon\\\
b&a\end{bmatrix}\in\textrm{GL}_{2}(\ell)\right\\}.$

This is the regular representation of
$\mathbb{F}_{\ell}[\sqrt{\epsilon}]^{\times}$ acting on itself via
multiplication with respect to the basis $\\{1,\sqrt{\epsilon}\\}$. Its
normalizer is

$\displaystyle N_{ns}(\ell$
$\displaystyle)=C_{ns}(\ell)\cup\begin{bmatrix}1&0\\\
0&-1\end{bmatrix}C_{ns}(\ell)$
$\displaystyle=\left\\{\begin{bmatrix}a&b\epsilon\\\
b&a\end{bmatrix},\begin{bmatrix}a&b\epsilon\\\
-b&-a\end{bmatrix}\in\textrm{GL}_{2}(\ell)\right\\}.$

Let us also define the Borel subgroup mod-$\ell$ as the subgroup of upper
triangular matrices,

$B(\ell):=\left\\{\begin{bmatrix}a&b\\\
0&d\end{bmatrix}\in\textrm{GL}_{2}(\ell)\right\\}.$

Also appearing in Theorem 2 is the mod-$\ell$ subgroup

$G(\ell):=\left\langle\begin{bmatrix}1&0\\\
0&-1\end{bmatrix},C_{ns}(\ell)^{3}\right\rangle$

which has index $3$ in $N_{ns}(\ell)$. Finally, we will use $Z(\ell)$ to
denote the subgroup of scalar matrices in $\textrm{GL}_{2}(\ell)$.

###  2.5. Some elementary bounds on fields of definition

In this subsection, we will prove some basic but useful facts that will help
us describe the degrees of fields of definition of torsion points and cyclic
isogenies, which will be used in the proof of Proposition 3.

######  Lemma 6.

Let $G$ be a finite group, and let $H,K$ and $M$ be subgroups of $G$ such that
$K$ is a normal subgroup of $H$. Then we have the index divisibility

$[M\cap H:M\cap K]\mid[H:K].$

###### Proof.

Since $M\cap K=(M\cap H)\cap K$, we see at once that

$\frac{|M\cap H||K|}{|M\cap K|}=|(M\cap H)\cdot K|.$

Since $H$ is in the normalizer of $K$, so is $M\cap H$, whence it follows that
$(M\cap H)\cdot K$ is a subgroup of $H$. We deduce by Lagrange’s Theorem that

$\frac{|M\cap H||K|}{|M\cap K|}\mid|H|.$

Dividing both sides by $|K|$ proves our result. ∎

Using Lemma 6, we will prove Lemma 7 which describes the degree of the
extension $F(P)/F(C)$, where $P$ is a torsion point on an elliptic curve
$E_{/F}$ and $C$ is the subgroup generated by $P$.

######  Lemma 7.

Fix an integer $n\in\mathbb{Z}^{+}$ and an elliptic curve $E_{/F}$. Fix a
point $P\in E[n]$ of exact order $n$, and let $C:=\langle P\rangle$ be the
subgroup generated by $P$. Then one has

$[F(P):F(C)]\mid\varphi(n),$

where $\varphi\colon\mathbb{Z}^{+}\rightarrow\mathbb{Z}^{+}$ is Euler’s
totient function.

Before proving this, we will define some auxiliary subgroups of
$\textrm{GL}_{2}(n)$. Let us set

$B_{1}(n):=\left\\{\begin{bmatrix}1&b\\\
0&d\end{bmatrix}\in\textrm{GL}_{2}(n)\right\\}.$

We will also let $B(n)$ denote the subgroup of upper triangular matrices in
$\textrm{GL}_{2}(n)$. We see that $B_{1}(n)$ is a normal subgroup of $B(n)$,
as it is the kernel of the map $\begin{bmatrix}a&b\\\ 0&d\end{bmatrix}\mapsto
a$. One also checks that $\\#B(n)=n\varphi(n)^{2}$ and
$\\#B_{1}(n)=n\varphi(n)$.

######  Remark 2.

Given an elliptic curve $E_{/F}$ and an integer $n\in\mathbb{Z}^{+}$, we find
that $E$ has an $F$-rational order $n$ point $P\in E(\overline{\mathbb{Q}})$
if and only if $\rho_{E,n,P,Q}(G_{F})\subseteq B_{1}(n)$ for some $Q\in E[n]$.
Similarly, $E$ has an $F$-rational cyclic order $n$ subgroup $C=\langle
P\rangle\subseteq E(\overline{\mathbb{Q}})$ if and only if
$\rho_{E,n,P,Q}(G_{F})\subseteq B(n)$ for some $Q\in E[n]$.

Let us now prove Lemma 7.

###### Proof.

Following the notation in Lemma 6, let us set $G:=\textrm{GL}_{2}(n)$,
$H:=B(n)$, $K:=B_{1}(n)$ and $M:=\rho_{E,n,P,Q}(G_{F})$, where $Q\in E[n]$ is
chosen such that $\\{P,Q\\}$ is a basis for $E[n]$. Then Lemma 6 implies that

(1) $[\rho_{E,n}(G_{F})\cap B(n):\rho_{E,n}(G_{F})\cap
B_{1}(n)]\mid[B(n):B_{1}(n)]=\varphi(n).$

By our choice of basis, we see that the stabilizer of $C$ under
$\rho_{E,n}(G_{F})$ is $\rho_{E,n}(G_{F})\cap B(n)$; by Galois theory, this
implies that $\textrm{Gal}(F(E[n])/F(C))\cong\rho_{E,n}(G_{F})\cap B(n)$.
Similarly, the stabilizer of $P$ under $\rho_{E,n}(G_{F})$ is
$\rho_{E,n}(G_{F})\cap B_{1}(n)$, whence we have
$\textrm{Gal}(F(E[n])/F(P))\cong\rho_{E,n}(G_{F})\cap B_{1}(n)$. Since

$[\textrm{Gal}(F(E[n])/F(C)):\textrm{Gal}(F(E[n])/F(P))]=[F(P):F(C)],$

Equation (1) becomes

$[F(P):F(C)]\mid\varphi(n).$

This completes our proof. ∎

###  2.6. Orbits under a subgroup

In this subsection, we will describe how orbits under the action of a finite
group compare to orbits under its subgroups. This is particularly important
for Theorems 4 and 5 (and the part of the proof of Theorem 2 from [Gen24]).
Given a finite group $G$ which acts on a set $X$, one has for any subgroup
$H\subseteq G$ that

(2) $\\#\mathcal{O}_{G}(x)\mid[G:H]\cdot\\#\mathcal{O}_{H}(x).$

As noted in $\S\ref{Subsect_FieldsOfDefn}$, for an elliptic curve $E_{/F}$ and
an integer $n\in\mathbb{Z}^{+}$, for any cyclic subgroup
$C\in\textrm{Cyc}(E,n)$ one has
$\\#\mathcal{O}_{\rho_{E,n}(G_{F})}(C)=[F(C):F]$. Thus, for any subgroup
$N\subseteq\textrm{GL}_{2}(n)$ with $\rho_{E,n}(G_{F})\subseteq N$, we have by
Equation (2) that

$\\#\mathcal{O}_{N}(C)\mid[N:\rho_{E,n}(G_{F})]\cdot[F(C):F].$

In particular, if a prime $p\mid\\#\mathcal{O}_{N}(C)$ does not divide
$[N:\rho_{E,n}(G_{F})]$, then $p\mid[F(C):F]$. In Theorem 5 we will take
$n=\ell$ to be a uniformly large prime, and the subgroup $N$ to be either
$B(\ell)$ or a Cartan subgroup. In both cases, $\\#\mathcal{O}_{N}(C)$ is easy
to compute. With a finer analysis, one can also produce bounds on the index
$[N:\rho_{E,\ell}(G_{F})]$ (see Theorems 2 and 5), and thus deduce information
about the degrees $[F(C):F]$. For example, in Theorem 4 we use this to show
that $[F(C):F]\in\\{1,\ell\\}$ when $\ell$ is uniformly large, $G\subseteq
B(\ell)$ and $G$ is not diagonalizable.

##  3\. The Proofs of Theorem 1 and Proposition 3

###  3.1. Proof of Theorem 1

First, we will prove Theorem 1 using Proposition 3.

###### Proof of Theorem 1.

Given the constant $c:=c(F_{0})$ from Proposition 3, let
$B:=B(F_{0})\in\mathbb{Z}^{+}$ be the product of the primes
$p\in\mathbb{Z}^{+}$ with $p\leq c$.

Let $E_{/F_{0}}$ be an elliptic curve. Fix an extension $L/F_{0}$ such that
$[L:F_{0}]$ is coprime to $B$. Let $C\subseteq E(\overline{\mathbb{Q}})$ be a
finite cyclic subgroup that is $L$-rational; we will show that $C$ must be
$F_{0}$-rational. For each divisor $d\mid|C|$, we let $C_{d}:=\frac{|C|}{d}C$
denote the unique subgroup of order $d$; this is also $L$-rational.

Assume that $j(E)\neq 0,1728$. Then by [Cla, Proposition 3.5], we have a
“factorization” of the field of definition $F_{0}(C)$ from the $\ell$-primary
parts of $|C|$, via

$F_{0}(C)=\prod_{\ell^{n}\parallel|C|}F_{0}(C_{\ell^{n}}).$

In particular, if each Sylow $\ell$-subgroup $C_{\ell^{n}}$ of $C$ is
$F_{0}$-rational, then so is $C$. Thus, without loss of generality we will
assume that $|C|=\ell^{n}$ is a prime power.

We will proceed by induction on $n\geq 1$. If $n=1$, then $C$ is an
$L$-rational $\ell$-isogeny, and the degree $[F_{0}(C):F_{0}]\leq\ell+1$. If
$\ell\leq c$, then each prime factor $p\in\mathbb{Z}^{+}$ of
$[F_{0}(C):F_{0}]$ is at most $c$, which implies that all primes which divide
$[F_{0}(C):F_{0}]$ also divide $B$. However, since $F_{0}(C)\subseteq L$ and
$[L:F_{0}]$ is coprime to $B$, so is $[F_{0}(C):F_{0}]$, which forces
$[F_{0}(C):F_{0}]=1$. If $\ell>c$, then by Proposition 3 we have that
$2\mid[F_{0}(C):F_{0}]$ and thus $2\mid[L:F_{0}]$, which is impossible since
$B$ is even and $[L:F_{0}]$ is coprime to $B$. We conclude that $C$ is
$F_{0}$-rational when $n=1$. (We have also just shown that the prime degrees
of $L$-rational isogenies are uniformly bounded by $c(F_{0})$.)

Assume next that the result is true for cyclic $\ell^{k}$-isogenies, where
$1\leq k<n$. This inductive hypothesis implies that $C_{\ell^{n-1}}=\ell C$ is
$F_{0}$-rational, so that $[F_{0}(C):F_{0}]=[F_{0}(C):F_{0}(\ell C)]$. Thus,
to show that $C$ is $F_{0}$-rational, it suffices to show that if
$[F_{0}(C):F_{0}(\ell C)]\neq 1$, then $[F_{0}(C):F_{0}(\ell C)]$ is divisible
by a prime $\ell\leq c$, an impossibility. Let us fix a generator $P$ of $C$.
Then we have the following diagram of field extensions:

${F_{0}(P)}$${F_{0}(\ell P)}$${F_{0}(C)}$${F_{0}(\ell C)}$

It follows that

$[F_{0}(C):F_{0}(\ell C)]=\frac{[F_{0}(P):F_{0}(\ell P)]\cdot[F_{0}(\ell
P):F_{0}(\ell C)]}{[F_{0}(P):F_{0}(C)]}.$

By Lemma 7 we have $[F_{0}(\ell P):F_{0}(\ell
C)]\mid\varphi(\ell^{n-1})=\ell^{n-2}(\ell-1)$, and by [GJN20, Proposition
4.6] we know that $[F_{0}(P):F_{0}(\ell P)]\mid\ell^{2}(\ell-1)$, whence it
follows that

$[F_{0}(C):F_{0}(\ell C)]\mid\ell^{n}(\ell-1)^{2}.$

Suppose for contradiction that $F_{0}(C)\neq F_{0}$. Thus, there exists a
common prime divisor $p$ of $\ell^{n}(\ell-1)^{2}$ and $[L:F_{0}]$, i.e.,
there exists prime $p\mid\ell(\ell-1)$ with $p\mid[L:F_{0}]$. If $\ell\leq c$
then $p\leq c$, which implies $p\mid B$ – this is impossible since $[L:F_{0}]$
is coprime to $B$. We then must have $\ell>c$, which implies by Proposition 3
that $[F_{0}(\ell^{n-1}C):F_{0}]$ is even; however, this is impossible since
$\ell^{n-1}C$ is $F_{0}$-rational by the inductive hypothesis. We thus
conclude that $C$ is $F_{0}$-rational. This completes the proof by induction.
∎

###  3.2. Proof of Proposition 3

As seen above, Theorem 1 will now follow from Proposition 3. Before proving
this proposition, we will need a small lemma.

######  Lemma 8.

Fix an integer $n\in\mathbb{Z}^{+}$, and let $V$ be the
$\mathbb{Z}/n\mathbb{Z}$-span of unit column vectors $e_{1},e_{2}$ over
$\mathbb{Z}/n\mathbb{Z}$. Then for all integers $1\leq k,m<n$, one has that
$\langle ke_{1}+e_{2}\rangle=\langle me_{1}+e_{2}\rangle$ if and only if
$k=m$.

###### Proof.

If $\langle ke_{1}+e_{2}\rangle=\langle me_{1}+e_{2}\rangle$, then
$me_{1}+e_{2}\in\langle ke_{1}+e_{2}\rangle$, and so for some $x\in\mathbb{Z}$
we have

$me_{1}+e_{2}=x(ke_{1}+e_{2})=xke_{1}+xe_{2}.$

By $\mathbb{Z}/n\mathbb{Z}$-linear independence of $e_{1}$ and $e_{2}$, it
follows that $m\equiv xk\pmod{n}$ and $1\equiv x\pmod{n}$; we deduce that
$m\equiv k\pmod{n}$, whence from $1\leq k,m<n$ we conclude that $k=m$. ∎

###### Proof of Proposition 3.

Fix a prime $\ell\in\mathbb{Z}^{+}$ such that
$\ell\geq\max\\{74,15[F_{0}:\mathbb{Q}]+2\\}$, $\ell$ is unramified in $F_{0}$
and $\ell\not\in S_{F_{0}}$. We must show that for an elliptic curve
$E_{/F_{0}}$, for any cyclic subgroup $C\in\textrm{Cyc}(E,\ell)$ one has that
$[F_{0}(C):F_{0}]$ is even. By Theorem 2, our assumptions on $\ell$ imply
that, up to conjugacy, $G:=\rho_{E,\ell}(G_{F_{0}})$ either equals one of
$\textrm{GL}_{2}(\ell)$, $N_{ns}(\ell)$ or $G(\ell)$, or that $G$ is contained
in $N_{s}(\ell)$. The former three groups act transitively on
$E[\ell]\smallsetminus\\{O\\}$, and thus act transitively on Cyc$(E,\ell)$,
which implies that $[F_{0}(C):F_{0}]=\ell+1$ for all $C\in$ Cyc$(E,\ell)$ when
$G\in\\{\textrm{GL}_{2}(\ell),N_{ns}(\ell),G(\ell)\\}$.

Let us assume then that $G\subseteq N_{s}(\ell)$ with respect to a basis
$\\{P,Q\\}$. By Theorem 2, we find that $C_{s}(\ell)^{e}\subseteq G$ for some
$e\in\\{1,2,3,4,6\\}$, and that $G$ contains $Z(\ell)$, the subgroup of
scalars of $\textrm{GL}_{2}(\ell)$. Note that $G\not\subseteq C_{s}(\ell)$
since otherwise $E$ has (two distinct) $F_{0}$-rational $\ell$-isogenies,
which contradicts $\ell\not\in S_{F_{0}}$.

Observe that the subgroups $\langle P\rangle$ and $\langle Q\rangle$ have
quadratic fields of definition over $F_{0}$, since the subgroup $H:=G\cap
C_{s}(\ell)$ has index two in $G$; consequently, both $[F_{0}(\langle
P\rangle):F_{0}]$ and $[F_{0}(\langle Q\rangle):F_{0}]$ are even. Thus, we
will restrict our attention to showing that the remaining order $\ell$
subgroups $C_{k}:=\langle kP+Q\rangle$, where $0<k<\ell$, have even degree
fields of definition over $F_{0}$.

Since $G\not\subseteq C_{s}(\ell)$, there exists an antidiagonal matrix
$\gamma:=\begin{bmatrix}0&b\\\ c&0\end{bmatrix}\in G$. The action of $\gamma$
on $\textrm{Cyc}(E,\ell)$ sends $C_{k}$ to $C_{bc^{-1}k^{-1}}$; thus we have
$C_{bc^{-1}k^{-1}}\in\mathcal{O}_{G}(C_{k})$. One can check that if
$\gamma\cdot C_{k}=\gamma\cdot C_{m}$, then
$C_{bc^{-1}k^{-1}}=C_{bc^{-1}m^{-1}}$, which by Lemma 8 implies that $k=m$.
Thus, if $C_{k}\neq C_{k^{-1}bc^{-1}}$ for all $0<k<\ell$, then all orbits
$\mathcal{O}_{G}(C_{k})$ have even size, as each pair of distinct subgroups
$C_{k}$ and $\gamma\cdot C_{k}$ lie in the same $G$-orbit; this would then
imply that the field of definition of each $C_{k}$ has even degree over
$F_{0}$, which finishes the proof. Suppose then that for some $k$ we have
$C_{k}=C_{k^{-1}bc^{-1}}$; by Lemma 8, this forces $c\equiv
bk^{-2}\pmod{\ell}$, whence we have $\gamma=\begin{bmatrix}0&b\\\
bk^{-2}&0\end{bmatrix}$. Since $Z(\ell)\subseteq G$, we can scale $\gamma$ by
$b^{-1}k$ and deduce that $\begin{bmatrix}0&k\\\ k^{-1}&0\end{bmatrix}\in G$.

To this end, we may assume that $\gamma:=\begin{bmatrix}0&b\\\
b^{-1}&0\end{bmatrix}\in G$ for some $1\leq b<\ell$. We check that $\gamma$
sends any subgroup $C_{k}$ to $C_{b^{2}k^{-1}}$. By Lemma 8, we know that
$C_{k}=C_{b^{2}k^{-1}}$ if and only if $k\equiv\pm b\pmod{\ell}$. Therefore,
to show that all orbits of $\textrm{Cyc}(E,\ell)$ have even size, it is enough
to show that $C_{b}$ and $C_{-b}$ are in the same $G$-orbit (note that
$C_{-b}:=C_{\ell-b}$ is distinct from $C_{b}$).

With some calculation, one can show that $C_{b}$ and $C_{-b}$ are in the same
$G$-orbit if and only if $\begin{bmatrix}1&0\\\ 0&-1\end{bmatrix}\in G$ or
$\begin{bmatrix}0&b\\\ -b^{-1}&0\end{bmatrix}\in G$. Since
$\begin{bmatrix}0&b\\\ -b^{-1}&0\end{bmatrix}=\begin{bmatrix}1&0\\\
0&-1\end{bmatrix}\cdot\gamma$, we deduce that $C_{b}$ and $C_{-b}$ are in the
same $G$-orbit if and only if $\gamma_{0}:=\begin{bmatrix}1&0\\\
0&-1\end{bmatrix}\in G$.

We will show that $\gamma_{0}\in G$. To do this, we will make a few more
reductions. Let us set $H:=G\cap C_{s}(\ell)$. By Theorem 2, we have
$C_{s}(\ell)^{e}\subseteq H$ for some $e\in\\{1,2,3,4,6\\}$. It follows that
$\gamma_{0}\in H$ if $-1$ is an $e$’th power modulo $\ell$. This latter part
is true if and only if $\ell=2$ or $\frac{\ell-1}{\gcd(e,\ell-1)}$ is even.
Thus, $-1$ is an $e$’th power modulo $\ell$ unless $e\in\\{2,4,6\\}$ and
$\ell\equiv 3\pmod{4},$ or $e=4$ and $\ell\equiv 5\pmod{8}$.

We will show that in either of these two cases, we still have $\gamma_{0}\in
H$. Let $\det(H)\subseteq\mathbb{F}_{\ell}^{\times}$ denote the subgroup of
determinants of matrices in $H$, and let $\mu_{4}(\mathbb{F}_{\ell})$ denote
the subgroup of the fourth roots of unity in $\mathbb{F}_{\ell}$. First, we
will show that if $\mu_{4}(\mathbb{F}_{\ell})\subseteq\det(H)$ then
$\gamma_{0}\in H$.

Assume first that $\ell\equiv 3\pmod{4}$. Observe that $-1\in\det(H)$ if and
only if there exists $m=\begin{bmatrix}a&0\\\ 0&-a^{-1}\end{bmatrix}\in H$ for
some $a\in\mathbb{F}_{\ell}^{\times}$. When this happens, it follows that
$am=\begin{bmatrix}a^{2}&0\\\ 0&-1\end{bmatrix}\in H$, noting that $H$
contains $Z(\ell)$. Since $\ell\equiv 3\pmod{4}$, we find that $a^{2}$ has odd
order. Writing this order as $|a^{2}|$, it follows that
$(am)^{|a^{2}|}=\gamma_{0}\in H$. Next, assume that $e=4$ and $\ell\equiv
5\pmod{8}$; let us write $i\in\mathbb{F}_{\ell}$ for a primitive fourth root
of unity. Then we have $i\in\det(H)$ iff there exists $m=\begin{bmatrix}a&0\\\
0&ia^{-1}\end{bmatrix}\in H$, and in such a case we have
$am=\begin{bmatrix}a^{2}&0\\\ 0&i\end{bmatrix}\in H$. Thus
$(am)^{2}=\begin{bmatrix}a^{4}&0\\\ 0&-1\end{bmatrix}\in H$. Since
$8\nmid\ell-1$ we have $8\nmid|a|$, and thus $|a^{4}|$ is odd, so that
$(am)^{2|a^{4}|}=\gamma_{0}\in H$. We conclude that in any case, if
$\mu_{4}(\mathbb{F}_{\ell})\subseteq\det(H)$ then $\gamma_{0}\in G$.

We will show that $\det(H)=\mathbb{F}_{\ell}^{\times}$. Fix a prime
$\mathcal{L}\mid\ell$ in $F_{0}$ and let $K:=(F_{0})_{\mathcal{L}}$ denote the
$\mathcal{L}$-adic completion of $F_{0}$ at $\mathcal{L}$. Let us use $I_{K}$
to denote the inertia group of $K$ (throughout this, we are working with a
fixed algebraic closure $\overline{\mathbb{Q}_{\ell}}$ of
$\mathbb{Q}_{\ell}$). Since $\det\rho_{E,\ell}(I_{K})=\chi_{\ell}(I_{K})$ and
$\ell$ is unramified in $F_{0}$, it follows that $\det\rho_{E,\ell}(I_{K})$
surjects onto $\mathbb{F}_{\ell}^{\times}$.

We will argue that the same holds once we replace $F_{0}$ by the fixed field
of $H$, which we denote by $M_{H}$. Fix a prime $\mathcal{Q}\mid\mathcal{L}$
in $M_{H}$, and let $L:=(M_{H})_{\mathcal{Q}}$ denote the completion of
$M_{H}$ at $\mathcal{Q}$. Then the inertia group $I_{L}$ is contained in $H$.
In particular, it follows that $\ell$ is unramified in $M_{H}$, see also
[Ser72, Lemme 2]. Consequently, $L/K$ is unramified, and so $I_{K}\subseteq
I_{L}$. Thus, from $\det\rho_{E,\ell}(I_{K})=\mathbb{F}_{\ell}^{\times}$ we
deduce that $\det\rho_{E,\ell}(I_{L})=\mathbb{F}_{\ell}^{\times}$, whence we
conclude that $\det(H)=\mathbb{F}_{\ell}^{\times}$.

From the work above, we conclude that $\gamma_{0}\in G$. In particular,
$C_{b}$ and $C_{-b}$ are distinct and in the same $G$-orbit, and so all
$G$-orbits of $\textrm{Cyc}(E,\ell)$ have even size, which is equivalent to
the degree $[F_{0}(C):F_{0}]$ being even for all $C\in\textrm{Cyc}(E,\ell)$.
This concludes our proof. ∎

##  4\. Fields of Definition of Cyclic Isogenies in the Borel Case

In this section we will prove Theorems 4 and 5, which allow us to describe the
mod-$\ell$ Galois representation of a non-CM elliptic curve over a number
field with a rational $\ell$-isogeny when $\ell$ is sufficiently large. These
results are unconditional, in that they are not predicated on the LV-
hypotheses for $F_{0}$.

Recall that if an $\ell$-isogeny $C=\langle P\rangle\subseteq
E(\overline{\mathbb{Q}})$ is $G_{F_{0}}$-stable, then the action of
$G_{F_{0}}$ on $C$ induces a homomorphism $r\colon
G_{F_{0}}\rightarrow\mathbb{F}_{\ell}^{\times}$ called the isogeny character
of $C$. This character is described as follows: for $\sigma\in G_{F_{0}}$, one
has $P^{\sigma}=r(\sigma)P$.

The proof of Theorem 4 will follow from Theorem 5. The proof of this latter
theorem analyzes the isogeny character of an elliptic curve over $F_{0}$,
using the following uniformity result from Larson and Vaintrob.

######  Theorem 9.

[LV14, Theorem 1] Let $F_{0}$ be a number field. Then there exists a finite
set $S_{F_{0}}$ of prime numbers such that the following holds. Let
$\ell\in\mathbb{Z}^{+}$ be a prime not in $S_{F_{0}}$, and let $E_{/F_{0}}$ be
an elliptic curve. Suppose that
$E[\ell]\otimes_{\mathbb{F}_{\ell}}\overline{\mathbb{F}_{\ell}}$ is reducible,
with a degree $1$ associated character $r\colon
G_{F_{0}}\rightarrow\overline{\mathbb{F}_{\ell}}^{\times}$. Then one of the
following two cases holds.

  1. (1)

There exists a CM elliptic curve $E^{\prime}_{/F_{0}}$ with rational CM over
$F_{0}$, and associated characters $\psi,\psi^{\prime}\colon
G_{F_{0}}\rightarrow\overline{\mathbb{F}_{\ell}}^{\times}$ such that

$\rho_{E^{\prime},\ell}(G_{F_{0}})\otimes_{\mathbb{F}_{\ell}}\overline{\mathbb{F}_{\ell}}=\emph{im}\begin{bmatrix}\psi&0\\\
0&\psi^{\prime}\end{bmatrix}$

up to conjugation, and

$r^{12}=\psi^{12}.$

  2. (2)

GRH fails for $F_{0}[\sqrt{-\ell}]$, and one has

$r^{12}=\chi_{\ell}^{6}$

where $\chi_{\ell}\colon G_{F_{0}}\rightarrow\mathbb{F}_{\ell}^{\times}$ is
the mod-$\ell$ cyclotomic character. Moreover, in this case one has that
$E[\ell]$ is reducible over $\mathbb{F}_{\ell}$, and $\ell\equiv 3\pmod{4}$.

######  Remark 3.

Under the LV-hypotheses, an effectively computable bound on the product of the
primes from $S_{F_{0}}$ is given in [LV14, Theorem 7.9]. For certain quadratic
fields, a significantly better bound on the primes in $S_{F_{0}}$ has been
given by Banwait [Ban23, Corollary 1.9].

######  Remark 4.

For $\ell\not\in S_{F_{0}}$, if an elliptic curve $E_{/F_{0}}$ has an
$F_{0}$-rational $\ell$-isogeny, then $E[\ell]$ is reducible and the
conclusions hold for the isogeny character $r\colon
G_{F_{0}}\rightarrow\mathbb{F}_{\ell}^{\times}$. In case (1) of Theorem 9, if
we assume that $\ell$ is unramified in the endomorphism ring of $E^{\prime}$,
we find that the CM mod-$\ell$ Galois representation
$G^{\prime}:=\rho_{E^{\prime},\ell}(G_{F_{0}})$ is contained in the split or
nonsplit Cartan subgroup of $\textrm{GL}_{2}(\ell)$, up to conjugacy. In the
split case, $E^{\prime}[\ell]$ is a reducible $G_{F_{0}}$-module, and the
associated character $\psi$ takes values in $\mathbb{F}_{\ell}^{\times}$; in
particular, $\psi$ is the isogeny character of an $F_{0}$-rational
$\ell$-isogeny on $E^{\prime}$. However, in the nonsplit case,
$E^{\prime}[\ell]$ is irreducible over $\mathbb{F}_{\ell}$ but absolutely
reducible: we have that
$G^{\prime}\otimes_{\mathbb{F}_{\ell}}\mathbb{F}_{\ell}[\sqrt{\epsilon}]$ is
diagonalizable, since the Eigenvalues of matrices in $C_{ns}(\ell)$ come in
pairs $a\pm b\sqrt{\epsilon}$; thus, $\psi$ takes values in
$\mathbb{F}_{\ell}[\sqrt{\epsilon}]^{\times}$.

Let us first prove Theorem 5, which provides a relative uniform bound on the
index of the mod-$\ell$ Galois representation of an elliptic curve
$E_{/F_{0}}$ with exactly one $F_{0}$-rational $\ell$-isogeny when $\ell$ is
sufficiently large; since $\rho_{E,\ell}(G_{F_{0}})$ is non-diagonalizable,
such an elliptic curve must be non-CM. A similar analysis of isogeny
characters can be found in [Gen22].

###### Proof of Theorem 5.

Let $\langle P\rangle$ be an $F_{0}$-rational $\ell$-isogeny on $E$. Then we
can choose $Q\in E[\ell]$ such that $\\{P,Q\\}$ is a basis for $E[\ell]$, and
with respect to this basis we have

$G:=\rho_{E,\ell,P,Q}(G_{F_{0}})=\textrm{im}\thinspace\begin{bmatrix}r&*\\\
0&\chi_{\ell}r^{-1}\end{bmatrix},$

where $\chi_{\ell}\colon G_{F_{0}}\rightarrow\mathbb{F}_{\ell}^{\times}$ is
the mod-$\ell$ cyclotomic character, and $r$ is the isogeny character of
$\langle P\rangle$. Let $G^{\textrm{ss}}$ be the semisimplification of $G$.
Explicitly, we have

$G^{\textrm{ss}}=\textrm{im}\thinspace\begin{bmatrix}r&0\\\
0&\chi_{\ell}r^{-1}\end{bmatrix}.$

Since $E$ has exactly one $F_{0}$-rational $\ell$-isogeny, we know that $G$ is
not diagonalizable, thus by [Gen22, Lemma 4] we have $G^{\textrm{ss}}\subseteq
G$. It also follows by the proof of [Gen22, Lemma 4] that
$\gamma:=\begin{bmatrix}1&1\\\ 0&1\end{bmatrix}\in G$, and thus
$G=\langle\gamma,G^{\textrm{ss}}\rangle$. Since we also have
$B(\ell)=\langle\gamma,C_{s}(\ell)\rangle$, we deduce that

(3) $[B(\ell):G]=[C_{s}(\ell):G^{\textrm{ss}}].$

The rest of this proof will be showing that $[C_{s}(\ell):G^{\textrm{ss}}]\mid
864[F_{0}:\mathbb{Q}]$.

First, let us assume that GRH is true. Then by Theorem 9, the existence of the
$F_{0}$-rational $\ell$-isogeny $\langle P\rangle$ implies the following:
there exists a CM elliptic curve $E^{\prime}_{/F_{0}}$ whose CM field $K$ is
contained in $F_{0}$, such that $E^{\prime}$ has an associated character
$\psi\colon G_{F_{0}}\rightarrow\overline{\mathbb{F}_{\ell}}^{\times}$ which
satisfies

(4) $r^{12}=\psi^{12}.$

By our initial hypotheses, $\ell$ is unramified in the ring
$\mathcal{O}:=\textrm{End}(E^{\prime})$, regarded as an order of the CM field
$K$. By [BC20, Corollary 1.5], up to conjugation we have the bound

(5) $[C_{\ell}(\mathcal{O}):G^{\prime}]\mid 6[F_{0}:\mathbb{Q}]$

where $G^{\prime}:=\rho_{E^{\prime},\ell}(G_{F_{0}})$ and
$C_{\ell}(\mathcal{O})$ is either $C_{s}(\ell)$ or $C_{ns}(\ell)$ depending on
whether $\ell$ is split or inert in $\mathcal{O}$, respectively.

We will show that $\ell$ must be split in $\mathcal{O}$. Assume to the
contrary that $\ell$ is inert in $\mathcal{O}$; then as noted in Remark 4, the
elements of $G^{\prime}$ have Eigenvalues over
$\mathbb{F}_{\ell^{2}}:=F_{\ell}[\sqrt{\epsilon}]$ which are Galois conjugates
of one another. Writing

$G^{\prime}\otimes_{\mathbb{F}_{\ell}}\mathbb{F}_{\ell^{2}}=\textrm{im}\thinspace\begin{bmatrix}\psi&0\\\
0&\psi^{\prime}\end{bmatrix},$

it follows that

(6) $\\#G^{\prime}=\\#\psi(G_{F_{0}}).$

Now, for any nonzero torsion point $R^{\prime}\in E^{\prime}[\ell]$, since
$G^{\prime}\subseteq C_{ns}(\ell)$ we have by Equations (2) and (5) the orbit
size divisibility

$\\#\mathcal{O}_{C_{ns}(\ell)}(R^{\prime})\mid
6[F_{0}:\mathbb{Q}]\cdot[F_{0}(R^{\prime}):F_{0}],$

noting that $\\#\mathcal{O}_{G^{\prime}}(R)=[F_{0}(R^{\prime}):F_{0}]$. Since
the action of $C_{ns}(\ell)$ on $E^{\prime}[\ell]\smallsetminus\\{O\\}$ is
transitive, this implies that

$\ell^{2}-1\mid 6[F_{0}:\mathbb{Q}]\cdot[F_{0}(R^{\prime}):F_{0}].$

However, from $r^{12}=\psi^{12}$ we find that

$\gcd(\\#\psi(G_{F_{0}}),12)\cdot\\#r(G_{F_{0}})=\gcd(\\#r(G_{F_{0}}),12)\cdot\\#\psi(G_{F_{0}}),$

and thus

$\\#\psi(G_{F_{0}})\mid 12\cdot\\#r(G_{F_{0}})\mid 12(\ell-1).$

Since $[F_{0}(R^{\prime}):F_{0}]\mid\\#G^{\prime}$, and because
$\\#G^{\prime}=\\#\psi(G_{F_{0}})$ by Equation (6), we deduce that

$\ell^{2}-1\mid 6[F_{0}:\mathbb{Q}]\cdot 12(\ell-1),$

and so

$\ell+1\mid 72[F_{0}:\mathbb{Q}].$

However, this is impossible since $\ell\geq 72[F_{0}:\mathbb{Q}]$. We deduce
that $\ell$ splits in $\mathcal{O}$.

Since $\ell$ splits in $\mathcal{O}$, we can choose a basis for
$E^{\prime}[\ell]$ such that $G^{\prime}$ has the form

$G^{\prime}=\textrm{im}\thinspace\begin{bmatrix}\psi&0\\\
0&\chi_{\ell}\psi^{-1}\end{bmatrix}.$

Thus, from Equation (4) it follows that
$(G^{\textrm{ss}})^{12}=(G^{\prime})^{12}$. With this, we have the
containments $C_{s}(\ell)\supseteq
G^{\textrm{ss}}\supseteq(G^{\textrm{ss}})^{12}=(G^{\prime})^{12}$, whence it
follows that

$[C_{s}(\ell):G^{\textrm{ss}}]\mid[C_{s}(\ell):(G^{\prime})^{12}].$

Since $[G^{\prime}:(G^{\prime})^{12}]\mid 144$, this shows that

$[C_{s}(\ell):G^{\textrm{ss}}]\mid 144[C_{s}(\ell):G^{\prime}].$

By Equation (5), we thus conclude that

$[C_{s}(\ell):G^{\textrm{ss}}]\mid 864[F_{0}:\mathbb{Q}].$

Since $[B(\ell):G]=[C_{s}(\ell):G^{\textrm{ss}}]$ by Equation (3), this
concludes our proof assuming that GRH is true.

Suppose next that GRH fails (for $F_{0}[\sqrt{-\ell}]$). Then by Theorem 9 we
have $r^{12}=\chi_{\ell}^{6}$, i.e., $r^{6}=(\chi_{\ell}r^{-1})^{6}$. It
follows that

(7) $(G^{\textrm{ss}})^{6}=r^{6}(G_{F_{0}})I.$

Since $\ell$ is unramified in $F_{0}$, we find that $\chi_{\ell}\colon
G_{F_{0}}\rightarrow\mathbb{F}_{\ell}^{\times}$ is surjective; thus, from
$r^{12}=\chi_{\ell}^{6}$ it follows that

$\\#r(G_{F_{0}})\cdot\gcd(\ell-1,6)=\gcd(\\#r(G_{F_{0}}),12)\cdot(\ell-1).$

This implies that the image $r(G_{F_{0}})$ of our isogeny character is
“relatively uniformly large” in $\mathbb{F}_{\ell}^{\times}$:

$[\mathbb{F}_{\ell}^{\times}:r(G_{F_{0}})]=\frac{\gcd(\ell-1,6)}{\gcd(\\#r(G_{F_{0}}),12)},$

so that $[\mathbb{F}_{\ell}^{\times}:r(G_{F_{0}})]\mid 6$. It follows by
Equation (7) that $[C_{s}(\ell):(G^{\textrm{ss}})^{6}]\mid 36$. Therefore, we
have

$[C_{s}(\ell):G^{\textrm{ss}}]=\frac{[C_{s}(\ell):(G^{\textrm{ss}})^{6}]}{[G^{\textrm{ss}}:(G^{\textrm{ss}})^{6}]}\mid
36.$

Since $36\mid 864[F_{0}:\mathbb{Q}]$, this concludes our proof. ∎

We are now in a position to prove Theorem 4.

###### Proof of Theorem 4.

With our assumptions, we can choose a basis $\\{P,Q\\}$ of $E[\ell]$ such that
$G:=\rho_{E,\ell,P,Q}(G_{F_{0}})\subseteq B(\ell)$. Since $E$ has exactly one
$F_{0}$-rational $\ell$-isogeny, we know that $G$ is not diagonalizable. It
follows by the proof of Theorem 5 that
$[B(\ell):G]=[C_{s}(\ell):G^{\textrm{ss}}]$, where $G^{\textrm{ss}}$ denotes
the semisimplification of $G$, see also Equation (3). Thus, Equation (2)
implies that for each subgroup $C\in\textrm{Cyc}(E,\ell)$, one has the orbit
size divisibility

$\\#\mathcal{O}_{B(\ell)}(C)\mid[C_{s}(\ell):G^{\textrm{ss}}]\cdot[F_{0}(C):F_{0}].$

By Theorem 5, this implies that

$\\#\mathcal{O}_{B(\ell)}(C)\mid 864[F_{0}:\mathbb{Q}]\cdot[F_{0}(C):F_{0}].$

The orbits of $\textrm{Cyc}(E,\ell)$ under $B(\ell)$ are easy to compute. We
check that $\mathcal{O}_{B(\ell)}(\langle P\rangle)=\\{\langle P\rangle\\}$
and $\mathcal{O}_{B(\ell)}(\langle P+Q\rangle)=\\{\langle kP+Q\rangle:0\leq
k<\ell\\}$, with the latter orbit having size $\ell$. Since
$\ell>\max\\{2,3,[F_{0}:\mathbb{Q}]\\}$ (from $\ell\geq c(F_{0})$), we have
$\ell\nmid 864[F_{0}:\mathbb{Q}]$; thus, for $C\neq\langle P\rangle$ we have
$\ell\mid[F_{0}(C):F_{0}]$, which by $[F_{0}(C):F_{0}]\leq\ell+1$ forces
$[F_{0}(C):F_{0}]=\ell$. ∎

## References

  * [Ban23] B. Banwait, _Explicit isogenies of prime degree over quadratic fields_ , Int. Math. Res. Not. IMRN 14 (2023), 11829–11876.
  * [BC20] A. Bourdon and P.L. Clark, _Torsion points and isogenies on CM elliptic curves_ , J. Lond. Math. Soc. (2) 102 (2020), 580–622.
  * [BN] A. Bourdon and F. Najman, _Sporadic points of odd degree on $X_{1}(N)$ coming from $\mathbb{Q}$-curves,_ preprint, https://arxiv.org/abs/2107.10909.
  * [Cla] P.L. Clark, _CM Elliptic Curves: Volcanoes, Reality and Applications_ , preprint, https://arxiv.org/abs/2212.13316.
  * [CN21] J. Cremona and F. Najman, _$\mathbb{Q}$ -curves over odd degree number fields,_ Res. Number Theory 7 (2021), Paper No. 62, 30.
  * [FL] L. Furio and D. Lombardo, _Serre’s uniformity question and proper subgroups of $C_{ns}^{+}(p)$_, preprint, https://arxiv.org/abs/2305.17780.
  * [Gen22] T. Genao, _Typically bounding torsion on elliptic curves with rational $j$-invariant,_ J. Number Theory 238 (2022), 823–841.
  * [Gen24] T. Genao, _Growth of torsion groups of elliptic curves upon base change from number fields_ , Ramanujan J. 63 (2024), 2, 409–429.
  * [GJN20] E. González-Jiménez and F. Najman, _Growth of torsion groups of elliptic curves upon base change_ , Math. Comp. 89 (2020), 1457–1485.
  * [Ken82] M.A. Kenku, _On the number of ${\bf Q}$-isomorphism classes of elliptic curves in each ${\bf Q}$-isogeny class_, J. Number Theory 15 (1982), no 2., 199–202.
  * [LV14] E. Larson and D. Vaintrob, _Determinants of subquotients of Galois representations associated with abelian varieties, with an appendix by Brian Conrad,_ J. Inst. Math. Jussieu 13 (2014), 517–559.
  * [Maz78] B. Mazur, _Rational isogenies of prime degree (with an appendix by D. Goldfeld)_ , Invent. Math. 44 (1978), no. 2, 129–162.
  * [Mom95] F. Momose, _Isogenies of prime degree over number fields_ , Compositio Math. 97 (1995), no. 3, 329–348.
  * [Ser72] J.-P. Serre, _Propriétés galoisiennes des points d’ordre fini des courbes elliptiques_ , Invent. Math. 15 (1972), 259–331.
  * [Sil09] J. Silverman, _The arithmetic of elliptic curves_ , 2nd Ed., Graduate Texts in Mathematics, vol. 106, Springer, Dordrecht (2009).


Presentation was changed in Section 3 to emphasize the focus on the conjugate
model. Detailed derivation were pushed to the appendices and more focus was
put on big picture ideas. Arbitrary perturbations parts were made explicit.
Table 1 was added and more focus was put on training algorithms.

  3. 3.

We added experiments using leave-one-out estimation to predict generalization
on unseen test data during traininig. We also added results to study the
evolution of sensitivities during training using MPE with iBLR.


* [25] K. Jensen and A. O’Bannon, “Constraint on defect and boundary renormalization group flows,” Phys. Rev. Lett. 116 no. 9, (2016) 091601, arXiv:1509.02160.
  * [26] Y. Wang, “Surface defect, anomalies and b-extremization,” JHEP 11 (2021) 122, arXiv:2012.06574.
  * [27] L. Bianchi, M. Meineri, R. C. Myers, and M. Smolkin, “Rényi entropy and conformal defects,” JHEP 07 (2016) 076, arXiv:1511.06713.
  * [28] N. Drukker, M. Probst, and M. Trépanier, “Defect CFT techniques in the 6d $\mathcal{N}=(2,0)$ theory,” JHEP 03 (2021) 261, arXiv:2009.10732.
  * [29] K. Jensen, A. O’Bannon, B. Robinson, and R. Rodgers, “From the Weyl anomaly to entropy of two-dimensional boundaries and defects,” Phys. Rev. Lett. 122 no. 24, (2019) 241602, arXiv:1812.08745.
  * [30] A. Lewkowycz and E. Perlmutter, “Universality in the geometric dependence of Renyi entropy,” JHEP 01 (2015) 080, arXiv:1407.8171.
  * [31] N. Drukker and M. Trépanier, “Ironing out the crease,” JHEP 08 (2022) 193, arXiv:2204.12627.
  * [32] A. Raviv-Moshe and S. Zhong, “Phases of surface defects in scalar field theories,” arXiv:2305.11370.
  * [33] S. Giombi and B. Liu, “Notes on a surface defect in the $O(N)$ model,” arXiv:2305.11402.
  * [34] J. Henriksson, “The critical $O(N)$ CFT: Methods and conformal data,” Phys. Rept. 1002 (2023) 1–72, arXiv:2201.09520.
  * [35] H. Kleinert and V. Schulte-Frohlinde, Critical properties of $\phi^{4}$ theories.  World Scientific, 2001.
  * [36] S. S. Gubser, A. Nellore, S. S. Pufu, and F. D. Rocha, “Thermodynamics and bulk viscosity of approximate black hole duals to finite temperature quantum chromodynamics,” Phys. Rev. Lett. 101 (2008) 131601, arXiv:0804.1950.
  * [37] V. Gorbenko, S. Rychkov, and B. Zan, “Walking, weak first-order transitions, and complex CFTs,” JHEP 10 (2018) 108, arXiv:1807.11512.
  * [38] N. Drukker, Z. Kong, and G. Sakkas, “Broken global symmetries and defect conformal manifolds,” Phys. Rev. Lett. 129 no. 20, (2022) 201603, arXiv:2203.17157.
  * [39] C. P. Herzog and V. Schaub, “The tilting space of boundary conformal field theories,” arXiv:2301.10789.
  * [40] M. Moshe and J. Zinn-Justin, “Quantum field theory in the large $N$ limit: a review,” Phys. Rept. 385 (2003) 69–228, arXiv:hep-th/0306133.
  * [41] L. Fei, S. Giombi, and I. R. Klebanov, “Critical $O(N)$ models in $6-\epsilon$ dimensions,” Phys. Rev. D 90 no. 2, (2014) 025018, arXiv:1404.1094.
  * [42] A. Petkou, “Conserved currents, consistency relations and operator product expansions in the conformally invariant $O(N)$ vector model,” Annals Phys. 249 (1996) 180–221, arXiv:hep-th/9410093.
  * [43] A. C. Petkou, “$C(T)$ and $C(J)$ up to next-to-leading order in $1/N$ in the conformally invariant $O(N)$ vector model for $2<d<4$,” Phys. Lett. B 359 (1995) 101–107, arXiv:hep-th/9506116.
  * [44] M. Goykhman and M. Smolkin, “Vector model in various dimensions,” Phys. Rev. D 102 no. 2, (2020) 025003, arXiv:1911.08298.
  * [45] K. Lang and W. Rühl, “The critical $O(N)$ $\sigma$-model at dimensions $2<d<4$: a list of quasi-primary fields,” Nuclear Physics B 402 no. 3, (1993) 573–603. https://www.sciencedirect.com/science/article/pii/055032139390119A.
  * [46] Y. Deng, H. W. J. Blöte, and M. P. Nightingale, “Surface and bulk transitions in three-dimensional $\mathrm{O}(n)$ models,” Phys. Rev. E 72 (Jul, 2005) 016128.
  * [47] P. Liendo, L. Rastelli, and B. C. van Rees, “The bootstrap program for boundary $CFT_{d}$,” JHEP 07 (2013) 113, arXiv:1210.4258.
  * [48] L. Fei, S. Giombi, I. R. Klebanov, and G. Tarnopolsky, “Three loop analysis of the critical $O(N)$ models in $6-\epsilon$ dimensions,” Phys. Rev. D 91 no. 4, (2015) 045011, arXiv:1411.1099.
  * [49] J. A. Gracey, “Four loop renormalization of $\phi^{3}$ theory in six dimensions,” Phys. Rev. D 92 no. 2, (2015) 025012, arXiv:1506.03357.
  * [50] M. Kompaniets and A. Pikelner, “Critical exponents from five-loop scalar theory renormalization near six-dimensions,” Phys. Lett. B 817 (2021) 136331, arXiv:2101.10018.
  * [51] M. Borinsky, J. A. Gracey, M. V. Kompaniets, and O. Schnetz, “Five-loop renormalization of $\phi^{3}$ theory with applications to the Lee-Yang edge singularity and percolation theory,” Phys. Rev. D 103 no. 11, (2021) 116024, arXiv:2103.16224.
  * [52] D. Rodriguez-Gomez, “A scaling limit for line and surface defects,” JHEP 06 (2022) 071, arXiv:2202.03471.
  * [53] I. C. n. Bolla, D. Rodriguez-Gomez, and J. G. Russo, “RG flows and stability in defect field theories,” arXiv:2303.01935.
  * [54] A. Söderberg Rousu, “Fusion of conformal defects in interacting theories,” arXiv:2304.10239.
  * [55] M. Henningson and K. Skenderis, “Weyl anomaly for Wilson surfaces,” JHEP 06 (1999) 012, arXiv:hep-th/9905163.
  * [56] A. Gustavsson, “Conformal anomaly of Wilson surface observables: a field theoretical computation,” JHEP 07 (2004) 074, arXiv:hep-th/0404150.
  * [57] S. Gukov and E. Witten, “Gauge theory, ramification, and the geometric Langlands program,” hep-th/0612073.
  * [58] A. Pelissetto and E. Vicari, “Critical phenomena and renormalization group theory,” Phys. Rept. 368 (2002) 549–727, arXiv:cond-mat/0012164.
  * [59] H. Osborn and A. Stergiou, “Seeking fixed points in multiple coupling scalar theories in the $\epsilon$ expansion,” JHEP 05 (2018) 051, arXiv:1707.06165.
  * [60] S. Rychkov and A. Stergiou, “General properties of multiscalar RG flows in $d=4-\epsilon$,” SciPost Phys. 6 no. 1, (2019) 008, arXiv:1810.10541.
  * [61] W. H. Pannell and A. Stergiou, “Line defect RG flows in the $\epsilon$ expansion,” arXiv:2302.14069.
  * [62] I. R. Klebanov and A. M. Polyakov, “AdS dual of the critical $O(N)$ vector model,” Phys. Lett. B 550 (2002) 213–219, arXiv:hep-th/0210114.
  * [63] E. Lauria, P. Liendo, B. C. Van Rees, and X. Zhao, “Line and surface defects for the free scalar field,” JHEP 01 (2021) 060, arXiv:2005.02413.