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\title[Torsion over number fields of degree $10$]
      {Torsion of elliptic curves over number fields of degree $10$}
\author{Filip Najman}
\address{University of Zagreb Faculty of Science, Department of Mathematics,
Bijeni\v cka cesta 30, 10000 Zagreb, Croatia}
\email{fnajman@math.hr}
\thanks{The author was supported by the Croatian Science Foundation under the project no.\ IP-2022-10-5008, by the project ``Implementation of cutting-edge research and its application as part of the Scientific Center of Excellence for Quantum and Complex Systems, and Representations of Lie Algebras'', PK.1.1.10.0004, co-financed by the European Union through the European Regional Development Fund -- Competitiveness and Cohesion Programme 2021--2027, and by the European Union -- NextGenerationEU through the National Recovery and Resilience Plan 2021--2026, via an institutional grant of the University of Zagreb Faculty of Science, IK IA 1.1.3, Impact4Math.}
\subjclass[2020]{11G05, 11G18, 14G05}
\keywords{elliptic curves, torsion, number fields, modular curves, sporadic points}

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\begin{document}

\begin{abstract}
We determine all the groups that appear as the torsion group of an elliptic curve over a number field of degree $10$.
There are $69$ such groups. For $4$ of them, namely $\tg{2}{26}$, $\tg{3}{15}$, $\tg{3}{18}$ and $\tg{6}{12}$, we also
determine all the elliptic curves over number fields of degree $10$ that realize them, and there are only finitely many.
These four groups were not known to occur over number fields of degree $10$.
The proofs combine the global method of Derickx, Hecke sieves, formal immersion criteria at the prime $2$, and
Riemann-Roch computations on modular curves and their quotients over finite fields.
\end{abstract}

\date{\today}
\maketitle

\input{sections/01_intro}
\input{sections/02_notation}
\input{sections/03_occurrences}
\input{sections/04_groups}
\input{sections/05_global}
\input{sections/06_mxn}
\input{sections/07_2x2n}
\input{sections/08_sieve}
\input{sections/08b_sieve}
\input{sections/09_formal}
\input{sections/10_further}
\input{sections/10b_further}
\input{sections/11_sporadic}
\input{sections/12_proof}
\input{sections/13_computations}

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