Growth of torsion groups of elliptic curves upon base change

Abstract

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 $E(L{)}_{tors}=E(F{)}_{tors}$ for all elliptic curves $E$ defined over $F$. Our methods turn out to be particularly useful in studying the possible torsion groups $E(K{)}_{tors}$, 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 $E(K{)}_{tors}=E(\mathbb{Q}{)}_{tors}$ 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 $\le 7$. Using this fact, we determine all the possible torsion groups $E(K{)}_{tors}$ over number fields $K$ of prime degree $p\ge 7$. 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 $p\not\equiv 8(\mathrm{mod}9)$ 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 $P\in E(K{)}_{tors}$, where $[K:\mathbb{Q}]=d$, for all $d\le 3342296$. Finally, we determine all the possible groups $E(K{)}_{tors}$, 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 curves $X_1(m,n)$ comes from an elliptic curve defined over $\mathbb{Q}$.

27 pages. The file contains text colored in blue; this text can be clicked on and is a link to the Magma code used to obtain that particular result

References in corpus (3)

• Complete classification of the torsion structures of rational elliptic curves over quintic number fields
• On the torsion of rational elliptic curves over quartic fields
• Gonality of the modular curve $X_0(N)$

Cited by in corpus (15)

• $\ell$-adic images of Galois for elliptic curves over $\mathbb{Q}$
• Complete classification of the torsion structures of rational elliptic curves over quintic number fields
• On the torsion of rational elliptic curves over quartic fields
• Torsion of rational elliptic curves over the maximal abelian extension of Q
• An algorithm for determining torsion growth of elliptic curves
• Arithmetic statistics and diophantine stability for elliptic curves
• Rank Jumps and Growth of Shafarevich--Tate Groups for Elliptic Curves in $\mathbb{Z}/p\mathbb{Z}$-Extensions
• Groups of generalized $G$-type and applications to torsion subgroups of rational elliptic curves over infinite extensions of $\mathbb{Q}$
• Torsion of $\Q$-curves over quadratic fields
• Torsion groups of Mordell curves over number fields of higher degree
• Sporadic points of odd degree on $X_1(N)$ coming from $\mathbb{Q}$-curves
• Squares in arithmetic progression over certain non-primitive quartic number fields
• Integers that are sums of two cubes in the cyclotomic ${\mathbb{Z}}_p$-extension
• Torsion groups of elliptic curves over some infinite abelian extensions of $\mathbb{Q}$
• Cartan images and $\ell$-torsion points of elliptic curves with rational $j$-invariant