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 $E(L)_{\operatorname {tors}}=E(F)_{\operatorname {tors}}$ 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 $\mathbb {Q}$.