Prime order torsion on elliptic curves over number fields. Part I: Asymptotics

TL;DR

This paper investigates the growth of possible prime orders of rational points on elliptic curves over number fields as the degree increases, providing bounds under certain conjectural assumptions.

Contribution

It establishes asymptotic bounds on the maximum prime order of rational points on elliptic curves over number fields, assuming conjectures on the sparsity of certain modular forms.

Findings

For large even degrees, max prime order ≤ 3d + 1.

For odd degrees, max prime order grows slower than any linear function of d.

Results depend on conjectures about modular forms and analytic ranks.

Abstract

We study the asymptotics of the set $S(d)$ of possible prime orders of $K$-rational points on elliptic curves over number fields $K$ of degree $d$ as $d$ tends to infinity. Assuming some conjectures on the sparsity of newforms of weight $2$ and prime level with unexpectedly high analytic rank, we show that $\max S(d) \le 3d + 1$ for sufficiently large even $d$ and $\max S(d) = o(d)$ for odd $d$.