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

TL;DR

The paper investigates which primes can occur as orders of $K$-rational points on elliptic curves over number fields of degree $d$, and analyzes the asymptotics of $S(d)$ as $d\to\infty$ under conjectural sparsity of certain weight-$2$ newforms. It develops a generalized kernel framework for Hecke correspondences on modular curves, leverages gonality bounds to bound rational points, and derives conditional asymptotics that bound $\max S(d)$ by $3d+1$ for large even $d$ and by $o(d)$ for odd $d$. A key portion identifies and analyzes strange primes, including a construction from genus $2$ curves that yields infinitely many such primes under standard conjectures, and provides concrete data on their distribution. The results illustrate how modular-curve geometry, Hecke action, and automorphic $L$-function phenomena constrain torsion structures on elliptic curves over growing-degree number fields, offering conditional pathways toward uniform torsion bounds.

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 $d$.

Theorems & Definitions (62)

• Definition 1.1
• Theorem 1.2
• Proposition 1.3
• Proposition 2.1
• Proposition 2.2
• proof
• Definition 3.1
• Lemma 3.2
• proof
• Theorem 3.3: Yau, Abramovich, Kim-Sarnak