Growth of Torsion Groups of Elliptic Curves Over Number Fields without Rationally Defined CM

Abstract

For a quadratic field $\mathcal{K}$ without rationally defined CM, we prove that there exists of a prime $p_{\mathcal{K}}$ depending only on $\mathcal{K}$ such that if $d$ is a positive integer whose minimal prime divisor is greater than $p_{\mathcal{K}}$, then for any extension $L/\mathcal{K}$ of degree d and any elliptic curve $E/\mathcal{K}$, we have $E\left(L\right)_{\operatorname{tors}} = E\left(\mathcal{K}\right)_{\operatorname{tors}}$. By not assuming the GRH, this is a generalization of the results by Genao, and Gon\'alez-Jim\'enez and Najman.