Existence and non-existence of rational elliptic curves with prescribed torsion subgroups over quadratic fields

TL;DR

This paper investigates the existence of rational elliptic curves with specific torsion subgroups over quadratic fields, establishing infinite families of primes with and without such curves, and extends torsion classification results in related number fields.

Contribution

It proves the existence of infinitely many primes with no such elliptic curves over quadratic fields and, conditionally, infinitely many with these curves, extending torsion classification over various number field extensions.

Findings

Infinitely many primes with no elliptic curves having specified torsion over quadratic fields.

Infinitely many primes with infinitely many such elliptic curves, assuming the parity conjecture.

Refined torsion classification over Kummer and composite extensions of cyclotomic fields.

Abstract

Let $K=\mathbb{Q}(\sqrt{-p})$ be a quadratic field for an odd prime $p$. We show that there exist infinitely many primes $p$ for which no elliptic curve $E/\mathbb{Q}$ has torsion subgroup $\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2N\mathbb{Z}$ over $K$ for $N=5,6$. We also prove that there exist infinitely many primes $p$ for which there are infinitely many elliptic curves $E/\mathbb{Q}$ with this torsion structure, conditional on the parity conjecture. Using these results, we obtain new torsion classification results over Kummer extensions of cyclotomic fields and over composites of $\mathbb{Z}_p$-extensions of number fields, refining and extending our previous work.