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 $E{\left(L\right)}_{\mathrm{tors}}=E{\left(\mathcal{K}\right)}_{\mathrm{tors}}$. By not assuming the GRH, this is a generalization of the results by Genao, and Gonález-Jiménez and Najman.

The title and typos are corrected