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

TL;DR

This paper proves that for certain quadratic fields without rational CM, the torsion subgroup of elliptic curves remains stable over extensions of degree with large minimal prime divisors, generalizing previous results without assuming GRH.

Contribution

It establishes a prime-dependent bound ensuring torsion stability of elliptic curves over extensions of quadratic fields without rational CM, extending prior work without GRH.

Findings

Torsion groups do not grow over extensions with degree divisible by large primes.

Existence of a prime threshold depending on the quadratic field.

Generalization of previous results without assuming GRH.

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.