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Growth of torsion groups of elliptic curves upon base change

Authors: Enrique González–Jiménez and Filip Najman
Journal: Math. Comp. 89 (2020), 1457-1485
MSC (2010): Primary 11G05
Published electronically: October 28, 2019
MathSciNet review: 4063324
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We study how the torsion of elliptic curves over number fields grows upon base change, and in particular prove various necessary conditions for torsion growth. For a number field $F$, we show that for a large set of number fields $L$, whose Galois group of their normal closure over $F$ has certain properties, it will hold that $E(L)_{\operatorname {tors}}=E(F)_{\operatorname {tors}}$ for all elliptic curves $E$ defined over $F$.

Our methods turn out to be particularly useful in studying the possible torsion groups $E(K)_{\operatorname {tors}}$, where $K$ is a number field and $E$ is a base change of an elliptic curve defined over $\mathbb {Q}$. Suppose that $E$ is a base change of an elliptic curve over $\mathbb {Q}$ for the remainder of the abstract. We prove that $E(K)_{\operatorname {tors}}=E(\mathbb {Q})_{\operatorname {tors}}$ for all elliptic curves $E$ defined over $\mathbb {Q}$ and all number fields $K$ of degree $d$, where $d$ is not divisible by a prime $\leq 7$. Using this fact, we determine all the possible torsion groups $E(K)_{\operatorname {tors}}$ over number fields $K$ of prime degree $p\geq 7$. We determine all the possible degrees of $[\mathbb {Q}(P):\mathbb {Q}]$, where $P$ is a point of prime order $p$ for all $p$ such that $p\not \equiv 8 \pmod 9$ or $\left ( \frac {-D}{p}\right )=1$ for any $D\in \{1,2,7,11,19,43,67,163\}$; this is true for a set of density $\frac {1535}{1536}$ of all primes and in particular for all $p<3167$. Using this result, we determine all the possible prime orders of a point $P\in E(K)_{\operatorname {tors}}$, where $[K:\mathbb {Q}]=d$ for all $d\leq 3342296$. Finally, we determine all the possible groups $E(K)_{\operatorname {tors}}$, where $K$ is a quartic number field and $E$ is an elliptic curve defined over $\mathbb {Q}$ and show that no quartic sporadic point on a modular curve $X_1(m,n)$ comes from an elliptic curve defined over $\mathbb {Q}$.

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Additional Information

Enrique González–Jiménez
Affiliation: Departamento de Matemáticas, Universidad Autónoma de Madrid, Madrid, Spain
MR Author ID: 703386

Filip Najman
Affiliation: Department of Mathematics, University of Zagreb, Bijenička cesta 30, 10000 Zagreb, Croatia
MR Author ID: 886852

Keywords: Elliptic curves, torsion over number fields
Received by editor(s): February 1, 2019
Received by editor(s) in revised form: May 22, 2019, and June 6, 2019
Published electronically: October 28, 2019
Additional Notes: The first author was partially supported by the grant MTM2015–68524–P
The second author gratefully acknowledges support from the QuantiXLie Center of Excellence and by the Croatian Science Foundation under the project no. IP-2018-01-1313.
Article copyright: © Copyright 2019 American Mathematical Society