Growth of torsion groups of elliptic curves upon base change

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$P$isapointofprimeorder$p$forall$p$suchthat$p\not\equiv 8 \pmod 9$or$\left( \frac{-D}{p}\right)=1$forany$D\in {1,2,7,11,19,43,67,163}$;thisistrueforasetofdensity$\frac{1535}{1536}$ofallprimesandinparticularforall$p<3167$. Using this result, we determine all the possible prime orders of a point $P\in E(K)_{tors}$,where$[K:\mathbb Q]=d$,forall$d\leq 3342296$. 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 $\mathbb{Q}$.