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\section{Introduction}\label{sec:intro}
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Let $E$ be an elliptic curve over a number field $K$. By the Mordell-Weil theorem
$E(K)\simeq E(K)_\tors\times \Z^r$, and by Merel's theorem \cite{merel} the torsion subgroup
$E(K)_\tors$ is bounded in terms of $d=[K:\Q]$ alone. Hence, for every $d\geq 1$ the set
\[
\Phi(d):=\{\,E(K)_\tors \;:\; [K:\Q]=d,\ E/K \text{ an elliptic curve}\,\}
\]
is finite. The set $\Phi(1)$ was determined by Mazur \cite{mazur}, building on work of
Kubert \cite{kubert}, and $\Phi(2)$ by Kamienny \cite{kamienny}, building on work of Kenku and
Momose \cite{KM}. Derickx, Etropolski, van Hoeij, Morrow and Zureick-Brown \cite{DEHMZ}
determined $\Phi(3)$, using earlier work of Jeon, Kim and Schweizer \cite{JKS}, Parent
\cite{parent} and Najman \cite{najman}. Derickx and Najman \cite{DN} determined $\Phi(4)$, using
earlier work of Jeon, Kim and Park \cite{JKP} and Bruin and Najman \cite{bruin_najman2016}.
The author determined $\Phi(5)$ in \cite{N5}, and $\Phi(6)$, $\Phi(7)$, $\Phi(8)$ and $\Phi(9)$ in \cite{N69}. In this
paper we determine $\Phi(10)$.

Write $\Phi^\infty(d)\subseteq\Phi(d)$ for the subset of groups that occur as $E(K)_\tors$ for
infinitely many $\overline\Q$-isomorphism classes of elliptic curves $E$ over number fields $K$
of degree $d$. The sets $\Phi^\infty(d)$ are known for $d\leq 9$: we have $\Phi^\infty(d)=\Phi(d)$ for $d=1,2$, the sets
$\Phi^\infty(3)$ and $\Phi^\infty(4)$ were determined by Jeon, Kim and Schweizer \cite{JKS} and Jeon, Kim and Park
\cite{JKP}, $\Phi^\infty(5)$ and $\Phi^\infty(6)$ by Derickx and Sutherland \cite{DS}, and $\Phi^\infty(7)$,
$\Phi^\infty(8)$ and $\Phi^\infty(9)$ by Najman and Varivoda \cite{NV}. The set $\Phi^\infty(10)$ has not been determined, and we do not determine
it here either.

Closely related is the set $S(d)$ of primes $p$ for which there exists an elliptic curve over a
number field of degree $d$ with a point of order $p$. The sets $S(d)$ for $d\leq 8$ follow from the work of Mazur,
Kamienny, Kenku and Momose, Parent, Derickx, Kamienny, Stein and Stoll, and Khawaja
\cite{mazur,kamienny,KM,parent,DKSS,khawaja}, and Derickx and Stoll \cite{DSt} developed tools that rule out most primes
for any given $d$. The author \cite[Theorem 1.1]{Sd} recently determined $S(d)$ for $9\leq d\leq 22$, and in particular
showed that $S(10)$ is the set of primes up to $37$. This is the starting point of our proof.

Our main result is the following theorem.

\begin{theorem}\label{thm:main}
Let $K$ be a number field of degree $10$ and let $E$ be an elliptic curve over $K$. Then $E(K)_\tors$ is isomorphic to
one of the following groups:
\begin{alignat*}{2}
&\torz{n}, \qquad && n=1,\dots,40,\ 42,\ 45,\ 50,\\
&\tg{2}{2n}, \qquad && n=1,\dots, 15,\\
&\tg{3}{3n}, \qquad && n=1,\dots, 6,\\
&\tg{4}{4n}, \qquad && n=1,2,3,\\
&\tg{6}{6n}, \qquad && n=1,2.
\end{alignat*}
Each of these $69$ groups occurs as the torsion group of an elliptic curve over a number field of degree $10$.
\end{theorem}

Of these $69$ groups, the $35$ groups of $\Phi^\infty(1)\cup\Phi^\infty(2)\cup\Phi^\infty(5)$ and the $15$ groups
\begin{gather*}
\torz n,\ n\in\{23,26,27,32,33,34,36\},\qquad \tg{2}{2n},\ n\in\{10,11,12\},\\
\tg39,\quad \tg3{12},\quad \tg48,\quad \tg4{12},\quad \tg66
\end{gather*}
occur for infinitely many $\overline\Q$-isomorphism classes of elliptic curves over number fields of degree $10$
(\Cref{lem:base,prop:families}). The groups $\torz{28}$, $\torz{30}$ and $\tg{2}{18}$ occur because they occur over
quintic fields \cite{N5}, and the groups $\tg{2}{28}$ and $\tg{2}{30}$ because the quintic curves with torsion
$\torz{28}$ and $\torz{30}$ gain full $2$-torsion over a quadratic extension (\Cref{prop:2x28-2x30}). For
$N\in\{29,31,35,37,38,39,40,42,45,50\}$ the group $\torz N$ is the torsion group of an elliptic curve coming from a
point of degree $10$ on $X_1(N)$ found by van Hoeij \cite{vanHoeij} (\Cref{prop:explicit_cyclic}). For the remaining
$4$ groups we also determine all the elliptic curves in question.

\begin{theorem}\label{thm:sporadic}
Let $G$ be one of the groups in the following table, let $K$ be a number field of degree $10$, and let $E/K$ be an
elliptic curve such that $E(K)$ has a subgroup isomorphic to $G$. Then $E(K)_\tors\simeq G$, and up to isomorphism the
pair $(K,E)$ is one of the pairs given in \Cref{sec:sporadic}. The number of these pairs and their fields $K$ are as
follows.
\begin{center}
\begin{tabular}{@{}ccl@{}}
\toprule
$G$ & pairs $(K,E)$ & fields $K$\\
\midrule
$\tg{2}{26}$ & $2$ & \lmfdbnf{10.6.396320664709.1}, \lmfdbnf{10.2.799905449.1}\\
$\tg{3}{15}$ & $1$ & \lmfdbnf{10.0.306454347.1}\\
$\tg{3}{18}$ & $2$ & \lmfdbnf{10.0.2571428187.1}\\
$\tg{6}{12}$ & $1$ & \lmfdbnf{10.0.4678854192.1}\\
\bottomrule
\end{tabular}
\end{center}
In particular, these $4$ groups occur for only finitely many $\overline\Q$-isomorphism classes of elliptic curves over
number fields of degree $10$.
\end{theorem}

The two pairs with torsion group $\tg{3}{18}$ are defined over the same field, and their elliptic curves are
$2$-isogenous. All the curves of \Cref{thm:sporadic} have $j$-invariants of degree $5$ over $\Q$, which are not
algebraic integers, so none of them has complex multiplication.

The elliptic curves of \Cref{thm:sporadic} are new, and they are listed in \cite{x1mn} as the points
\xpoint{2.26.10.a}, \xpoint{2.26.10.b}, \xpoint{3.15.10.a}, \xpoint{3.18.10.a}, \xpoint{3.18.10.b} and \xpoint{6.12.10.a}.
We found them in the course of this work, so the groups $\tg{2}{26}$, $\tg{3}{15}$, $\tg{3}{18}$ and $\tg{6}{12}$ were not
known to occur in degree $10$. The elliptic curves with torsion groups $\tg{2}{28}$ and $\tg{2}{30}$ over decic fields of
\Cref{prop:2x28-2x30} (the points \xpoint{2.28.10.a}, \xpoint{2.30.10.a} and \xpoint{2.30.10.b}), which we obtained from the
quintic curves of van Hoeij with torsion $\torz{28}$ and $\torz{30}$, are also new.

The isomorphism of pairs is defined in \Cref{subsec:trace}. Note that $\torz{37}$ occurs in degrees $6$ and $10$, but
in none of the degrees $7$, $8$ and $9$ \cite[Theorem 1.1]{N69}.

\subsection*{Outline of the proof and of the paper}
As in \cite{DN}, \cite{N5} and \cite{N69}, the classification reduces to showing that certain modular curves
$X_1(m,n)$ have no non-cuspidal points over number fields of degree $10$. We introduce the notation and collect the
general tools from \cite{N69} in \Cref{sec:prelim}. In \Cref{sec:occur} we show that the groups of \Cref{thm:main}
occur, apart from the $4$ groups of \Cref{thm:sporadic}, and in \Cref{sec:groups} we show that $110$ groups have to be
ruled out. Over a number field $K$ of degree $10$ with $\tg mn\subseteq E(K)$ we have $m\in\{1,2,3,4,6,11,22\}$, and for
$m\in\{3,4,6\}$ the problem becomes one about points of degree $5$ over $\Q(\zeta_m)$.

The global method of Derickx \cite[Proposition 4.3]{DN}, combined with arguments with diamond operators, eliminates
the groups $\torz n$ of $\mathcal M_{10}$ with $n\geq 95$ apart from $n\in\{99,100,105,117\}$ (\Cref{sec:global}). The groups
$\tg mn$ with $m\geq3$ are treated in \Cref{sec:mxn}, and the groups $\tg{2}{2n}$, together with the cyclic groups
that \Cref{lem:isogeny} reduces to them, in \Cref{sec:2x2n}. The remaining cyclic groups are treated in
\Cref{sec:sieve,sec:formal,sec:further}: with the Hecke sieve of \cite[Proposition 5.5]{DN} and computations with
the Picard groups of modular curves over finite fields, often after pushing the divisors forward to a curve of
smaller level (\Cref{sec:sieve}), with formal immersion criteria at the prime $2$ (\Cref{sec:formal}), and, together with $\tg{3}{39}$, with
combinations of these methods on quotients of $X_1(n)$ (\Cref{sec:further}). In \Cref{sec:sporadic} we construct the
elliptic curves with torsion groups $\tg{2}{26}$, $\tg{3}{15}$, $\tg{3}{18}$ and $\tg{6}{12}$ and determine all the
points of the relevant degree on the corresponding modular curves, which also rules out the groups properly containing
them. We prove \Cref{thm:main,thm:sporadic} in \Cref{sec:proof}, and the computations are described in
\Cref{sec:computations}.

\subsection*{Data availability and reproducibility}
The code, the stored logs and the data of the computations described in this paper are available at
\begin{center}
\url{https://github.com/F-Najman/deg10_torsion}
\end{center}
This paper refers to the revision \texttt{\gitrevshort} of this repository, and the links in the paper point to this
revision. The \texttt{README} file there lists, statement by statement, which scripts establish each claim and which
logs they produced, and also the checks that the proofs do not need (see \Cref{sec:computations}). In the digital form
of this article, the words reporting a computation, such as ``we \gitlink{computations/witness_C29.m}{compute}'', are
hyperlinks to the script that carries it out.

\subsection*{AI disclosure and comments}
AI models were extensively used in the work leading to this paper. After completing \cite{DN} (with Derickx) and
\cite{N5} and giving talks about this work, I was asked several times whether we could do such a classification
for number fields of degree $d>5$. My answer was that I thought it was not possible as it would require explicit
computations that did not seem feasible to me. To check whether this was correct, in mid-October 2026 I asked AI
whether this was possible, and whether it could do it. To my surprise, it was able to complete the
classifications of possible torsion groups of elliptic curves over number fields, not only in degree 6, but later
also in degrees 7, 8, 9 and finally 10. After some initial hesitation about releasing primarily AI-generated mathematics, I
decided that the results were simply too interesting and the mathematics too beautiful to just stay on my
computer.

OpenAI's GPT-6.1 Sol produced almost all of the mathematical content of the first version of the proof and
carried out most of the computations. Anthropic's Claude Opus 5.5 reviewed that version, independently recomputed some of the
computations (see \Cref{sec:computations}) and the explicit examples, wrote the draft of the present manuscript and
organized the code repository.

%\section*{Acknowledgements}