Conjectural asymptotics of prime orders of points on elliptic curves over number fields

Define, for a positive integer $d$, $S(d)$ to be the set of all primes $p$ that occur as the order of a point $P \in E(K)$ on an elliptic curve $E$ defined over a number field $K$ of degree $d$. We discuss how some plausible conjectures on the sparsity of newforms with certain properties would allow us to deduce a fairly precise result on the asymptotic behavior of $\max S(d)$ as $d$ tends to infinity.

This is joint work with Maarten Derickx.