On the field of definition of $$p$$ -torsion points on elliptic curves over the rationals

Abstract

Let \(S_\mathbb Q (d)\) be the set of primes \(p\) for which there exists a number field \(K\) of degree \(\le d\) and an elliptic curve \(E/\mathbb Q \), such that the order of the torsion subgroup of \(E(K)\) is divisible by \(p\). In this article we give bounds for the primes in the set \(S_\mathbb Q (d)\). In particular, we show that, if \(p\ge 11\), \(p\ne 13,37\), and \(p\in S_\mathbb Q (d)\), then \(p\le 2d+1\). Moreover, we determine \(S_\mathbb Q (d)\) for all \(d\le 42\), and give a conjectural formula for all \(d\ge 1\). If Serre’s uniformity problem is answered positively, then our conjectural formula is valid for all sufficiently large \(d\). Under further assumptions on the non-cuspidal points on modular curves that parametrize those \(j\)-invariants associated to Cartan subgroups, the formula is valid for all \(d\ge 1\).