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 F , we show that for a large set of number fields L L , whose Galois group of their normal closure over F F has certain properties, it will hold that E ( L ) tors = E ( F ) tors E(L)ₜₒᵣₛ=E(F)ₜₒᵣₛ for all elliptic curves E E defined over F F . Our methods turn out to be particularly useful in studying the possible torsion groups E ( K ) tors E(K)ₜₒᵣₛ , where K K is a number field and E E is a base change of an elliptic curve defined over Q Q . Suppose that E E is a base change of an elliptic curve over Q Q for the remainder of the abstract. We prove that E ( K ) tors = E ( Q ) tors