%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Notation and auxiliary results}\label{sec:prelim}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

We will be using the conventions and notation from \cite[Section 2]{N69}, which we will essentially
repeat here for the convenience of the reader. We also collect the auxiliary results of \cite{N69} that
we use. They are stated there for number fields of degree $6$, $7$, $8$ and $9$, but most of them are
stated for an arbitrary degree $d$, and their proofs do not use the degree otherwise.

\subsection{Modular curves}\label{subsec:modular}
Throughout, $m\mid n$ are positive integers with $mn>4$. We
denote by $X:=X_1(m,n)$ the modular curve parametrizing triples $(E,P,Q)$ consisting of a
generalized elliptic curve $E$ and points $P,Q$ of orders $m$ and $n$ generating a subgroup of
$E[n]$ isomorphic to $\tg{m}{n}$, such that the Weil pairing of $P$ and $(n/m)Q$ is $\zeta_m$. It is a smooth
projective geometrically irreducible curve over $\Z[\zeta_m,\tfrac1n]$, and it is a fine moduli
space when $n>4$ (see \cite[\S 2]{DN} and \cite[\S 2]{DS}). Every curve $X_1(m,n)$ that we use has
$n\geq 5$, so we use the fine moduli interpretation throughout. We write $Y_1(m,n)$ for the open
subscheme obtained by removing the cusps, and $J:=J_1(m,n)$ for the Jacobian of $X$. We
abbreviate $X_1(n):=X_1(1,n)$ and $J_1(n):=J_1(1,n)$. For $m\leq 2$ the curve $X$ is defined
over $\Q$. For $d\geq1$ we write $X^{(d)}$ for the $d$-th symmetric power of $X$, so that
$X^{(d)}(\Q(\zeta_m))$ is the set of effective $\Q(\zeta_m)$-rational divisors of degree $d$ on $X$. For a curve
$X$ over a field $F$ with a set of cusps, such as $X_1(m,n)$ over $\Q(\zeta_m)$ or a quotient of it, we
write $\ClCusp_F X$ for the subgroup of $J_X(F)$ generated by the classes of the $F$-rational
divisors of degree $0$ supported on the cusps. For a curve $X$ we write
$\gon_k X$ for its gonality over a field $k$, and $J_X$ or $J(X)$ for its Jacobian. For a divisor $D$ on a curve $X$ over a field $k$
we write $L(D):=\{f\in k(X)^\times:\ddiv(f)+D\geq0\}\cup\{0\}$ for its Riemann-Roch space and $\ell(D)=\dim L(D)$.
Similarly, we write $\Omega(D)$ for the space consisting of $0$ and the non-zero differentials
$\omega$ on $X$ with $\ddiv(\omega)\geq D$. Then $\Omega(0)=H^0(X,\Omega^1)$ is the space of
holomorphic differentials on $X$, and if $D=\sum_P m_PP$ is effective, then $\Omega(D)$ is its
subspace of the differentials that vanish to order at least $m_P$ at every $P$. The
Riemann-Roch theorem states that $\ell(D)=\deg D+1-g+\dim\Omega(D)$, where $g$ is the genus
of $X$.

The cusps of $X_1(m,n)$ correspond to N\'eron $k$-gons with $m\mid k\mid n$ together with a level
structure (see \cite[\S 2]{DN}). All cusps are defined over $\Q(\zeta_n)$, and
$\Gal(\Q(\zeta_n)/\Q(\zeta_m))$ acts on them through its action on $\mathbb G_m$. The cuspidal
subscheme is finite \'etale over $\Z[\zeta_m,\tfrac1n]$. We denote by $c_0$ the cusp given by
the N\'eron $n$-gon together with the level structure $(a,b)\mapsto (\zeta_m^a,b)$ (see
\cite[Lemma 2.8]{DN}), which is $\Q(\zeta_m)$-rational, and by $c_\ast$ an arbitrary $\Q(\zeta_m)$-rational
cusp used as a base point. For a cusp $c$
we write $\sigma_q(c)$ for its image under the automorphism $\zeta_n\mapsto\zeta_n^q$ of $\Q(\zeta_n)$.
When we list the degrees of closed cusps or of Galois orbits of cusps, $a^r$ stands for $r$ of them of
degree $a$.

For a congruence subgroup $\Gamma\subseteq\SL_2(\Z)$ we write $\overline\Gamma$ for its image
in $\PSL_2(\Z)$. The curve $X_1(m,n)_\C$ is the modular curve of the group
$\Gamma_1(n)\cap\Gamma(m)$, and for $n>2$ we have
$[\PSL_2(\Z):\overline{\Gamma_1(n)\cap\Gamma(m)}]=\tfrac m2\,n^2\prod_{\ell\mid n}(1-\ell^{-2})$,
the product being over the primes dividing $n$. We write $i(m,n)$ for this index, and $i(n):=i(1,n)$.

For $a\in(\torz n)^\times$ we write $\dm a$ for the diamond automorphism
$(E,P,Q)\mapsto (E,a^{-1}P,aQ)$ of $X_1(m,n)$, which preserves the Weil pairing. For $m\leq 2$
it is the usual diamond operator, which is $(E,P)\mapsto(E,aP)$ on $X_1(n)$ and
$(E,P,Q)\mapsto(E,P,aQ)$ on $X_1(2,n)$. It depends only on $a$ modulo
$\pm1$, and we call $\Delta(n):=(\torz n)^\times/\{\pm1\}$ the group of diamond operators. We write $\Z[\Delta(n)]$ for its group ring, which acts on divisors, on divisor classes and on $J$.
For $a_1,\dots,a_r\in(\torz n)^\times$ we write $X/\dm{a_1,\dots,a_r}$ for the quotient of $X$ by
the subgroup of $\Delta(n)$ generated by $\dm{a_1},\dots,\dm{a_r}$.
For a prime $q\nmid n$ with $q\equiv 1\pmod m$ we write $T_q$ for the Hecke correspondence
$T_q(E,P,Q)=\sum_{G}(E/G,P\bmod G,Q\bmod G)$, the sum being over the $q+1$ subgroups
$G\subseteq E[q]$ of order $q$, and we set
\[
A_q:=T_q-q\dm q-1 .
\]
The congruence $q\equiv1\pmod m$ ensures that $T_q$ preserves the Weil pairing, and that $q$
splits completely in $\Q(\zeta_m)$. The operator $A_q$ commutes with the diamond operators. For a prime $q\mid n$ we
write $U_q$ for the Hecke operator at $q$.

An automorphism of $X_1(m,n)$ of the form $(E,P,Q)\mapsto(E,P',Q')$, where $P',Q'$ are integral linear
combinations of $P$ and $Q$, will be called a \textit{level automorphism}. The diamond operators are level
automorphisms. Some of the level automorphisms are defined over $\Q(\zeta_m)$, namely those that preserve
the Weil pairing.

We will use reduction modulo primes of good reduction also for quotients of $X_1(m,n)$, by the following
result of Liu and Lorenzini.

\begin{lemma}[{\cite[Corollary 4.10]{LL}}]\label{lem:goodred}
Let $f:X\to Y$ be a finite morphism of curves over a number field $F$, and let $\mathfrak p$ be a prime of
$F$. Assume that $X$ has good reduction at $\mathfrak p$ and that $Y$ has positive genus. Then $Y$ has good
reduction at $\mathfrak p$, and $f$ extends to a finite morphism of the smooth models over the localization
of the ring of integers of $F$ at $\mathfrak p$.
\end{lemma}

In particular, every quotient of $X_1(m,n)$ of positive genus by a group of automorphisms defined over
$\Q(\zeta_m)$ has good reduction at the primes above every prime $p\nmid n$, and we use these smooth models
for reduction modulo the primes above $p$.

\subsection{Trace divisors}\label{subsec:trace}
Let $K$ be a number field of degree $10$ and $E/K$ an elliptic curve with
$\tg{m}{n}\subseteq E(K)$. By the Weil pairing $\Q(\zeta_m)\subseteq K$, so $\varphi(m)\mid 10$, and we
put $d_m:=10/\varphi(m)=[K:\Q(\zeta_m)]$. Hence $m\in\{1,2,3,4,6,11,22\}$, and $d_m=10$ for $m\leq2$, $d_m=5$
for $m\in\{3,4,6\}$ and $d_m=1$ for $m\in\{11,22\}$. Choosing generators $P,Q$ with the right Weil pairing we
obtain a point $x\in Y_1(m,n)(K)$. We write $x^{(d_m)}$ for the sum of the
$\Q(\zeta_m)$-conjugates of $x$, each counted $[K:\Q(\zeta_m)(x)]$ times. It is an effective
$\Q(\zeta_m)$-rational divisor of degree $d_m$ on $X$ whose support contains no cusp, and we call it the
\textit{trace divisor} of $x$, usually denoted by $D$. All the arguments below are applied to trace divisors,
so they also cover the case in which the residue field $\Q(\zeta_m)(x)$ is smaller than $K$.

If $\mathfrak p$ is a prime of $\Q(\zeta_m)$ above a prime $p\nmid n$, the reduction of $D$ modulo $\mathfrak p$ is
$\sum_{\wp\mid\mathfrak p}e(\wp/\mathfrak p)[k_\wp:\kappa(x_\wp)]\,x_\wp$, where $\wp$ runs over the primes of $K$ above
$\mathfrak p$, $k_\wp$ is the residue field of $\wp$, and $x_\wp$ is the closed point of $X_{k_{\mathfrak p}}$ given by the
reduction of $x$ modulo $\wp$, with residue field $\kappa(x_\wp)$. In particular, a non-cuspidal closed point which occurs
in the reduction with multiplicity $1$ is $x_\wp$ for a prime $\wp$ which is unramified over $\mathfrak p$ and satisfies
$k_\wp=\kappa(x_\wp)$.

Two pairs $(K,E)$ and $(K',E')$, consisting of a number field and an elliptic curve over it, are
\textit{isomorphic} if there is a field isomorphism $\sigma:K\to K'$ such that $E^\sigma\simeq E'$ over $K'$.

\subsection{Hecke operators and reduction}\label{subsec:hecke}

\begin{proposition}[{\cite[Proposition 2.2]{N69}}]\label{prop:tools}
Let $X$ be $X_1(m,n)$ or its quotient by a subgroup of $\Delta(n)$, and let $q\nmid 2n$ be a prime with
$q\equiv1\pmod m$.
\begin{enumerate}
\item[a)] \textup{(Eichler-Shimura)} $T_{q,\F_q}=\Frob_q+\dm q_{\F_q,*}\Ver_q$ on $J_{X,\F_q}$, where
      $\F_q$ is the residue field of a prime of $\Q(\zeta_m)$ above $q$.
\item[b)] If $L\supseteq \Q(\zeta_m)$ is a number field in which $q$ splits completely, then
      $A_q\bigl(J_X(L)_\tors\bigr)=0$.
\item[c)] $A_q(C)=0$ as a divisor for every $\Q(\zeta_m)$-rational cuspidal divisor $C$.
\item[d)] If $J_X(\Q(\zeta_m))$ is finite, then $[A_q(D)]=0$ for every $\Q(\zeta_m)$-rational divisor $D$ on $X$. In
      particular $A_q(D)$ is principal for every $D\in X^{(d)}(\Q(\zeta_m))$ and every $d$.
\end{enumerate}
\end{proposition}

For $m\leq2$ these are \cite[(2.2), Proposition 2.6, Lemma 2.8 and Lemma 5.4]{DN}. We also need the action
of $T_q$ on all the cusps, not only on the rational ones.

\begin{lemma}[{\cite[Lemma 2.3]{N69}}]\label{lem:cuspHecke}
Let $X$ be $X_1(m,n)$ or its quotient by a subgroup of $\Delta(n)$, let $q\nmid n$ be a prime with
$q\equiv1\pmod m$, and let $\sigma_q\in\Gal(\Q(\zeta_n)/\Q(\zeta_m))$ be the automorphism with
$\sigma_q(\zeta_n)=\zeta_n^q$. For every cusp $c$ of $X$ we have
\[
T_qc=\sigma_q(c)+q\dm q\sigma_q^{-1}(c)
\]
as divisors. Hence $A_qC=(\sigma_qC-C)+q\dm q(\sigma_q^{-1}C-C)$ for every cuspidal divisor $C$, and $A_qC=0$ if
$\sigma_q(C)=C$. The same holds for the reductions modulo a prime $p\nmid qn$, where $\sigma_q$ acts on the
closed cusps through the reduction of the cusps, and in particular $A_qC=0$ for every cuspidal divisor $C$ on
$X_{\F_{p^f}}$ if $q\equiv p^{fk}\pmod n$ for some $k$.
\end{lemma}

Some of our computations use models of $X_1(n)$ and its quotients given by $q$-expansions, which
parametrize the dual objects $(E/\dm P,\ \text{the induced }\mu_n\text{-structure})$. The isomorphism
between the two models conjugates $\dm a$ to $\dm{a}^{-1}$, and on the dual model the
Eichler-Shimura relation reads $T_q=\dm q\Frob_q+q\Frob_q^{-1}$, with annihilator $T_q-\dm q-q$, and the formula of
\Cref{lem:cuspHecke} reads $T_qc=\dm q\sigma_q(c)+q\sigma_q^{-1}(c)$ (see \cite[Section 2.3]{N69}).

We will also use the following results.
\begin{itemize}
\item (\textit{Manin-Drinfeld \cite{manin,drinfeld}}) the class of a degree $0$ cuspidal divisor
      on $X$ is torsion in $J(\Qbar)$;
\item (\textit{Katz \cite[Appendix]{katz}}) if $p>2$ is a prime and $A$ is an abelian variety over a
      number field $L$ with good reduction at a prime $\wp$ of $L$ above $p$ which is unramified
      in $L$, then reduction modulo $\wp$ is injective on $A(L)_\tors$;
\item (\textit{Parent \cite[Proposition 2.3]{parent}}) if $A$ is an abelian variety over a
      number field $L$ with good reduction at a prime $\lambda$ above $2$ which is unramified
      in $L$, then the kernel of reduction modulo $\lambda$ on $A(L)_\tors$ is trivial or of
      exponent $2$ (see \cite[Section 2.3]{N69});
\item (\textit{Abramovich \cite[proof of Theorem 0.1]{abramovich}, with the bound
      $\lambda_1\geq 975/4096$ of Kim-Sarnak \cite[Appendix 2]{kim}}) for a congruence subgroup
      $\Gamma\subseteq\PSL_2(\Z)$,
      \[
        \gon_\C X_\Gamma\geq\tfrac{325}{2^{15}}\,[\PSL_2(\Z):\Gamma];
      \]
\item (\textit{Kato \cite[Corollary 14.3]{kato}}) if $A$ is a quotient of $J_1(n)$ over $\Q$ with
      $L(A,1)\neq0$, then $A(\Q)$ is finite.
\end{itemize}

We use Kato's theorem in the form of the \textit{winding quotient}, as in \cite[Section 2.3]{N69}: we compute, for
each Galois orbit of even Dirichlet characters $\chi$ modulo $n$, the winding submodule of the space of
modular symbols with character $\chi$, and if it is the whole space, then all the factors with
character $\chi$ have finitely many rational points.

\begin{lemma}[{\cite[Lemma 2.4]{N69}}]\label{lem:additive}
Let $K$ be a number field, let $E/K$ be an elliptic curve with $\tg{m}{n}\subseteq E(K)$, and
let $\wp$ be a prime of $K$ above a rational prime $p\nmid n$. Then $E$ does not have additive
reduction at $\wp$.
\end{lemma}

The following proposition is \cite[Proposition 5.2]{DN}, stated also for quotients of $X_1(m,n)$ and for
one prime of $\Q(\zeta_m)$ above $p$ at a time, as in \cite[Proposition 2.5]{N69}.

\begin{proposition}[{\cite[Proposition 2.5]{N69}}]\label{prop:main}
Let $F=\Q(\zeta_m)$, let $p\nmid n$ be a prime, let $X$ be $X_1(m,n)$ or its quotient of positive genus by a
subgroup of $\Delta(n)$, and let $d\geq1$. Assume that $J_X(F)$ is finite and
that one of the following holds:
\begin{enumerate}
\item[a)] $p>2$ and $\gon_FX>d$;
\item[b)] $p=2$ and $\gon_FX>2d$.
\end{enumerate}
Then no effective $F$-rational divisor of degree $d$ on $X$ whose support contains no cusp reduces to a
cuspidal divisor modulo a prime of $F$ above $p$. In particular, if $X=X_1(m,n)$, then every elliptic curve $E$
over a degree $d$ extension $K$ of $F$ with $\tg{m}{n}\subseteq E(K)$ has, for every prime $\mathfrak p$ of $F$
above $p$, good reduction at at least one prime of $K$ above $\mathfrak p$.
\end{proposition}

The proof of \Cref{prop:main} shows a little more, which we will use when the gonality bound is not
available. If $J_X(F)$ is finite, $p$ is odd and $D$ reduces to an effective cuspidal divisor $\overline C$ modulo
a prime above $p$, then $D\sim C$ over a number field unramified above $p$, where $C$ is the lift of $\overline C$,
so $\ell(\overline C)\geq\ell(C)\geq2$ by semicontinuity. For $p=2$ we use the following lemma, which is
\cite[Lemma 5.3]{N69}, whose proof works for any prime
$\mathfrak p$ of $F$ above $2$.

\begin{lemma}[{\cite[Lemma 5.3]{N69}}]\label{lem:doubled}
Let $m\mid n$ with $n$ odd and $mn>4$, let $F=\Q(\zeta_m)$, let $\mathfrak p$ be a prime of $F$ above $2$ with
residue field $k$, and let $d\geq1$. Let $X=X_1(m,n)$, and assume that $J_X(F)$ is finite and that
\[
\ell(2\overline C)=1
\]
for every effective cuspidal divisor $\overline C$ of degree $d$ on $X_k$. Then no effective $F$-rational
divisor of degree $d$ on $X$ whose support contains no cusp reduces to a cuspidal divisor modulo $\mathfrak p$.
\end{lemma}

\begin{lemma}[{\cite[Lemma 2.6]{N69}, \cite[Lemma 5.3]{DN}}]\label{lem:badred}
Let $p\nmid n$ be a prime and let $K\supseteq \Q(\zeta_m)$ be a number field. Suppose that for every
prime $\wp$ of $K$ above $p$ there is no elliptic curve $E'$ over the residue field $\F_\wp$
with $\tg{m}{n}\subseteq E'(\F_\wp)$. Then no elliptic curve $E/K$ with
$\tg{m}{n}\subseteq E(K)$ has good reduction at a prime of $K$ above $p$.
\end{lemma}

Whenever we invoke \Cref{lem:badred}, the hypothesis follows from the Hasse bound, from
Waterhouse's description \cite[Theorem 4.1]{waterhouse} of the possible traces of Frobenius,
or from an explicit enumeration of all elliptic curves over the relevant finite fields.

\subsection{Divisors and level automorphisms}\label{subsec:divisors}

\begin{lemma}[{\cite[Lemma 2.7]{N69}}]\label{lem:closed}
Let $X$ be a curve over a field $F$, and let $D$ and $A$ be effective $F$-rational divisors of
degree $d$ on $X$ with $A\sim D$.
\begin{enumerate}
\item[a)] If $\gon_F X>d$, then $A=D$.
\item[b)] If $D$ is a closed point of degree $d$ and $A\neq D$, then there is a function on $X$ over $F$ of degree
      exactly $d$.
\end{enumerate}
\end{lemma}

\begin{lemma}[{\cite[Lemma 2.8]{N69}}]\label{lem:orbits}
Let $G$ be a finite group of level automorphisms of $X_1(m,n)$, and let $D$ be a divisor on $X_1(m,n)$ whose
support consists of non-cuspidal points without complex multiplication. Then no element $g\neq1$ of $G$ fixes
a geometric point in the support of $D$. If moreover $gD=D$ for all $g\in G$, then the geometric points in the
support of $D$ form a union of $G$-orbits of length $\#G$, and $D=\sum_O m_O\sum_{x\in O}x$, where $O$ runs
over these orbits. In particular $\#G\mid\deg D$.
\end{lemma}

\begin{lemma}[{\cite[Lemma 2.9]{N69}}]\label{lem:stab}
Let $F=\Q(\zeta_m)$, let $p\nmid n$ be a prime, and let $X$ be $X_1(m,n)$ or its quotient of positive genus by a
subgroup of $\Delta(n)$. Let $g$ be a level automorphism of $X$ defined
over $F$, which maps cusps to cusps and extends to the smooth model of $X$ over the primes above $p$, such
that $(g-1)J_X(F)$ is finite. Let $D$ be an effective $F$-rational divisor of degree $d$ on $X$, and let $C$
be an effective cuspidal divisor of degree $e$ defined over a number field $L\supseteq F$ which is unramified
above $p$, such that $g$ fixes the reduction of $D-C$ modulo a prime above $p$. Assume that one of the following
holds:
\begin{enumerate}
\item[a)] $p$ is odd, $C$ is $F$-rational and $\gon_F X>d+e$,
\item[b)] $p$ is odd and $\gon_\C X>d+e$,
\item[c)] $p=2$ and $\gon_\C X>2(d+e)$,
\item[d)] $p=2$, $C$ is $F$-rational, $g$ has odd order and acts trivially on $J_X(F)[2]$, and
      $\gon_F X>d+e$.
\end{enumerate}
Then $gD+C=D+gC$. In particular, if the support of $D$ contains no cusp, then $gD=D$ and $gC=C$.
\end{lemma}

Every level automorphism of $X$ defined over $F$ satisfies the assumption on $g$ if $J_X(F)$ is finite. For
$m=1$ and $g=\dm a$ the assumption holds if all the factors of $J_X$ on which $\dm a$ acts non-trivially have
finitely many rational points, in particular if every character $\chi$ for which the winding submodule is not
the whole space satisfies $\chi(a)=1$.

\begin{lemma}[{\cite[Lemma 2.10]{N69}}]\label{lem:quadchar}
Let $C$ be a smooth projective curve over $\F_q$ with $q$ odd, and let $f$ be a function on $C$ with $\ddiv(f)=2A$ for a divisor
$A$. For a divisor $Z=\sum n_PP$ whose support is disjoint from that of $\ddiv(f)$ put
$\chi_f(Z)=\prod_P N_{\F_q(P)/\F_q}(f(P))^{n_P}$ modulo $\F_q^{\times2}$. Then $\chi_f$ induces a homomorphism
$\chi_f:\Pic^0(C)\to\F_q^\times/\F_q^{\times2}$, which is trivial on every subgroup of odd order.
\end{lemma}

\begin{lemma}[{\cite[Proposition 2.2]{DSt}, \cite[Lemma 2.13]{N69}}]\label{lem:tree}
Let $X$ be $X_1(n)$ or its quotient by a subgroup of $\Delta(n)$, and let $q\nmid n$ be a prime. Let
$L$ be a divisor on $X_\C$ whose support consists of non-cuspidal points at which the elliptic curve has no
complex multiplication. If $A_qL=0$, then $L=0$.
\end{lemma}

\begin{lemma}[{\cite[Lemma 8.1]{N69}}]\label{lem:mult}
Let $X$ be a smooth projective geometrically integral curve over a field $k$, and let $B=T-U$
with $T,U$ effective divisors of the same degree. Let $\mathcal L,\mathcal M$ be line bundles on $X$,
and choose $0\neq F\in H^0(X,\mathcal L(-T))$ and $H\in H^0(X,\mathcal M(-T))$. If the linear map
\[
H^0(X,\mathcal L)\longrightarrow H^0(U,\mathcal L|_U)\oplus \frac{H^0(X,\mathcal L\otimes\mathcal M)}{F\cdot H^0(X,\mathcal M)},\qquad G\longmapsto\bigl(G|_U,\ GH \bmod F\cdot H^0(X,\mathcal M)\bigr)
\]
is injective, then $\ell(B)=0$. The same holds if several sections $H$ are used.
\end{lemma}

We will use the following consequence of the results of Clark, Corn, Rice and Stankewicz on torsion of elliptic curves
with complex multiplication.

\begin{lemma}\label{lem:CM}
Let $K$ be a number field whose degree divides $10$, and let $E/K$ be an elliptic curve with complex multiplication.
Then $E(K)_\tors$ is isomorphic to $\torz m$ for $m\in\{1,2,3,4,6,7,10,11,22,31,50\}$, to $\tg2m$ for
$m\in\{2,4,6,22\}$, or to $\tg33$.
\end{lemma}

\begin{proof}
For $[K:\Q]=10$ this is \cite[Section 4.10]{CCRS14}. If $[K:\Q]$ is a proper divisor of $10$, then by \Cref{lem:base}
there is a number field $K'\supseteq K$ of degree $10$ with $E(K')_\tors=E(K)_\tors$.
\end{proof}

All these groups lie in $\mathcal G_{10}$ (\Cref{sec:groups}), and none of them contains a group of $\mathcal M_{10}$ or one
of the groups $\tg{2}{26}$, $\tg{3}{15}$, $\tg{3}{18}$ and $\tg{6}{12}$. Hence, in the trace divisor of a point giving such a
subgroup over a number field of degree $10$, every point corresponds to an elliptic curve without complex multiplication,
as its residue field has degree dividing $10$, and \Cref{lem:orbits} applies.

The following lemma generalizes \cite[Lemma 11.7]{N69}.

\begin{lemma}\label{lem:section}
Let $F=\Q(\zeta_m)$, let $X=X_1(m,n)$ with $J_X(F)$ finite, let $\mathfrak p$ be a prime of $F$ above an odd prime
$p\nmid n$, and let $g$ be a level automorphism of $X$ defined over $F$, of order $r$. Let $D$ be an effective
$F$-rational divisor on $X$ whose support contains no cusp, and suppose that its reduction modulo $\mathfrak p$ is
$D'+C'$, where $D'$ has non-cuspidal support and satisfies $gD'=D'$, and $C'$ is an effective cuspidal divisor. If
$$\ell\Bigl(D'+\sum_{j=0}^{r-1}g^jC'\Bigr)=1$$
on $X_{k_{\mathfrak p}}$, then $gD=D$ and $gC'=C'$. In particular, if $g$ acts on the geometric cusps of $X$ with all
orbits of length $r$, then $r\mid\deg C'$.
\end{lemma}

\begin{proof}
The cusps of $X$ are defined over $\Q(\zeta_n)$, which is unramified above $p$, so $C'$ lifts to an effective
cuspidal divisor $C$ over a number field $L\supseteq F$ which is unramified above $p$. As in \cite[proof of Lemma 2.9]{N69},
the class $[gD-D-gC+C]\in J_X(L)$ is torsion, by the finiteness of $J_X(F)$ and the Manin-Drinfeld theorem. It
reduces to $[gD'-D']=0$ modulo a prime above $\mathfrak p$, so it is $0$ by Katz's result. Put
$\Omega=\sum_{j=0}^{r-1}g^jC$. The divisors $D+\Omega-C$ and $gD+\Omega-gC$ are effective and linearly equivalent, and
both reduce to $D'+\sum_jg^jC'$. By semicontinuity $\ell(D+\Omega-C)\leq1$, so $D+\Omega-C=gD+\Omega-gC$, and hence
$D-gD=C-gC$. The left side has non-cuspidal support and the right side is cuspidal, so $gD=D$ and $gC=C$, and hence $gC'=C'$. If
all the orbits of $g$ on the geometric cusps have length $r$, then the coefficients of the $g$-invariant divisor $C'$ are
constant on these orbits, so $r\mid\deg C'$.
\end{proof}

\subsection{The Hecke sieve and push-forwards}\label{subsec:sieve}

Let $X=X_1(m,n)$ with $J(\Q(\zeta_m))$ finite, let $\mathfrak p$ be a prime of $\Q(\zeta_m)$ above a
prime $p\nmid n$, with residue field $k=\F_{p^f}$, and let $D$ be the trace divisor of a point
$x\in Y_1(m,n)(K)$, where $[K:\Q]=10$. Reducing modulo $\mathfrak p$ we write
\[
D_k=D'+C',
\]
where $D'$ is effective of degree $e$ with non-cuspidal support and $C'$ is effective and cuspidal
of degree $d_m-e$. Note that $e\geq 1$ if and only if the elliptic curve corresponding to $x$ has
good reduction at some prime of $K$ above $\mathfrak p$. As in \cite[\S 6]{N5} and \cite[Section 6.1]{N69}
we call the multiset of degrees of the closed points occurring in $D'$, a partition of $e$, the
\textit{type} of the decomposition, and write it in brackets, as in $[3,3]$.

The Hecke sieve \cite[Proposition 5.5]{DN} says that if $[A_q(D')]\neq 0$ and $A_q(C')=0$ for some
prime $q\nmid 2pn$ with $q\equiv1\pmod m$, then this decomposition cannot occur, since $[A_q(D)]=0$ by
\Cref{prop:tools}~d) and its reduction is $[A_q(D')]+[A_q(C')]$. By \Cref{lem:cuspHecke},
$A_q(C')=0$ if $\sigma_q$ fixes every closed cusp occurring in $C'$, and we then call $q$
\textit{admissible} for $C'$. Every prime $q\equiv p^{fj}\pmod n$, for some $j\geq0$, is admissible for
every $C'$, and every $q$ is admissible for $C'$ if each closed cusp occurring in $C'$ is the
reduction of a $\Q(\zeta_m)$-rational cuspidal divisor of the same degree. As $A_q$
commutes with the diamond operators, and more generally with the level automorphisms defined over
$\Q(\zeta_m)$, it suffices to test one $D'$ in each orbit of these automorphisms.

\begin{lemma}[{\cite[Lemma 5.6]{N69}}]\label{lem:classtest}
Let $X$ be a curve over a number field $F$ with an $F$-rational cusp $c_\ast$, let $\mathfrak p$ be a
prime of good reduction of $X$ with residue field $k$, and let $G\subseteq J(k)$ be a subgroup
containing the reduction of $J(F)$. Let $D\in X^{(d)}(F)$, and write its reduction modulo
$\mathfrak p$ as $D'+C'$, where $D'$ is effective of degree $e$ with non-cuspidal support and $C'$ is
effective and cuspidal. Then
\[
[D'+C'-d\,\overline{c_\ast}]\in G.
\]
\end{lemma}

We apply \Cref{lem:classtest} on $X$ itself or after pushing the trace divisor forward to
another curve, as in \cite[Section 6.1]{N69}. Let $m\leq2$, let $\pi:X\to Y$ be a morphism of modular curves
over $\Q$ which maps cusps to cusps and non-cusps to non-cusps, let $p$ be a prime of good reduction of both
curves such that $\pi$ extends over $\Z_{(p)}$, and let $y_0$ be a rational cusp of $Y$. Assume that $J_Y(\Q)$
is finite, and let $G\subseteq J_Y(\F_p)$ be a subgroup containing the reduction of $J_Y(\Q)$. If the
reduction of $D$ is $D'+C'$ as above, with $\deg D'=e$, then \Cref{lem:classtest}, applied to $\pi_*D$, gives
\begin{equation}\label{eq:pushclass}
[\pi_*D'-e\,\overline{y_0}]\in G+\mathcal S_{10-e},
\end{equation}
where $\mathcal S_j$ is the set of the classes $[j\overline{y_0}-B]$ for the effective cuspidal
divisors $B$ of degree $j$ on $Y_{\F_p}$. Note that we allow all the cuspidal divisors
$B$ over $\F_p$, also those that are not reductions of rational cuspidal divisors. If every closed
cusp of $Y_{\F_p}$ is the reduction of a rational cuspidal divisor of the same degree, then
$\mathcal S_j\subseteq G$, and \eqref{eq:pushclass} reduces to $[\pi_*D'-e\,\overline{y_0}]\in G$.

\subsection{Formal immersion at the prime \texorpdfstring{$2$}{2}}\label{subsec:formal}

Reduction modulo $2$ need not be injective on the torsion of the Jacobian. The following lemma shows
that a suitable Hecke operator kills the torsion points in the kernel of reduction.

\begin{lemma}[{\cite[Lemma 7.1]{N69}}]\label{lem:dyadic}
Let $A$ be an abelian scheme over $\Z_2$, and let $T$ and $\delta$ be endomorphisms of $A$ such
that $T=F+\delta V$ on the special fibre $A_{\F_2}$, where $F$ and $V$ are the Frobenius and the
Verschiebung of $A_{\F_2}$. If $z\in A(\Q_2)$ satisfies $2z=0$ and reduces to $0$ modulo $2$, then
$(T-\delta)z=0$.
\end{lemma}

Let $n$ be odd, let $H\subseteq\Delta(n)$ be a subgroup, and put $Y=X_1(n)/H$. We assume that $H$ has
odd order, or that $H$ acts freely on the geometric points of $X_1(n)$ in characteristic $2$, cusps
included. In both cases $Y$ is the generic fibre of a smooth proper curve over $\Z_{(2)}$, the
quotient of the smooth model of $X_1(n)$, and its cuspidal subscheme is finite \'etale over
$\Z_{(2)}$. We write $\Cart$ for the Cartier operator on the differentials of $Y_{\F_2}$, and
$\mathbb T_Y$ for the ring of endomorphisms of $J_Y$ generated by the Hecke operators $T_\ell$ for primes
$\ell\nmid n$, the operators $U_\ell$ for primes $\ell\mid n$, and the diamond operators. For $\psi\in\mathbb T_Y$
we write $\psi^*$ for its action on the cotangent space $H^0(Y_{\F_2},\Omega^1)$ of the special fibre of the
N\'eron model of $J_Y$ over $\Z_2$. By the Eichler-Shimura relation we have
$(T_2-\dm2)^*=(\Cart-1)\dm2^*$ on $H^0(Y_{\F_2},\Omega^1)$ (see \cite[Section 7]{N69}).

\begin{proposition}[{\cite[Proposition 7.3]{N69}}]\label{prop:cartier}
Let $Y=X_1(n)/H$ be as above, let $\psi_1,\dots,\psi_r\in\mathbb T_Y$ be such that each
$\psi_i\bigl(J_Y(\Q)\bigr)$ is finite, and put
\[
U:=(\Cart-1)\Bigl(\sum_{i=1}^r\psi_i^*H^0(Y_{\F_2},\Omega^1)\Bigr).
\]
Let $\overline C$ be an effective cuspidal divisor of degree $d$ on $Y_{\F_2}$ such that the
restriction map $U\to H^0(\overline C,\Omega^1|_{\overline C})$ is surjective, i.e.\ such that
$\dim U-\dim\bigl(U\cap\Omega(\overline C)\bigr)=d$. Then no effective rational divisor of degree
$d$ on $Y$ with non-cuspidal support reduces to $\overline C$ modulo $2$.
\end{proposition}

When $J_Y(\Q)$ is finite we take $r=1$ and $\psi_1=1$, and then $U=(\Cart-1)H^0(Y_{\F_2},\Omega^1)$.