Torsion groups of elliptic curves that appear infinitely often over septic fields

TL;DR

The paper determines the set $\Phi_infty_7$, the torsion groups of elliptic curves that occur infinitely often over degree-7 number fields. It combines modular-curve gonality bounds, density arguments, and explicit computations on curves $X_1(2,2n)$ to separate the $(1,n)$ and $(2,2n)$ cases, leveraging both theoretical bounds and computational methods. The main result is $\Phi_infty_7 = \{(1,n): 1 \le n \le 30, n \notin \{25,29\}\} \cup \{(2,2n): 1 \le n \le 10\}$. These findings sharpen our understanding of torsion structures in high-degree number fields and illustrate how gonality, reductions modulo primes, and modular-curve computations constrain possible torsion.

Abstract

In this short note we determine the set $Φ^\infty(7)$ of Abelian groups that appear as torsion groups of infinitely many elliptic curves (up to $\overline \mathbb Q$-isomorphism) over number fields of degree 7.

Theorems & Definitions (14)

• Theorem 1.1
• Proposition 2.1
• Theorem 2.2
• Theorem 2.3
• Lemma 2.4
• proof
• Lemma 3.1
• proof
• Lemma 3.2
• proof