Points of order 13 on elliptic curves

TL;DR

This paper investigates the properties of elliptic curves with 13-torsion points, focusing on their behavior over specific number fields and extensions, building on foundational work by Mazur and Tate from 45 years ago.

Contribution

It advances the understanding of 13-torsion in elliptic curves over maximal totally real subfields and quadratic extensions, extending previous research in the area.

Findings

Characterization of 13-torsion points over certain number fields

Analysis of elliptic curves with good reduction at specific primes

New results on the distribution of 13-torsion in various extensions

Abstract

We pick up the study of 13-torsion in elliptic curves where Mazur and Tate left off 45 years ago. We consider various questions concerning elliptic curves defined over the maximal totally real subfield of the 13th cyclotomic field (where J_1(13) acquires everywhere good reduction), and over quadratic extensions.