Local data of rational elliptic curves with nontrivial torsion

Alexander J. Barrios and Manami Roy

Vol. 318 (2022), No. 1, 1–42

DOI: 10.2140/pjm.2022.318.1

Abstract

By Mazur’s torsion theorem, there are fourteen possibilities for the nontrivial torsion subgroup $T$ of a rational elliptic curve. For each $T$, such that $E$ may have additive reduction at a prime $p$, we consider a parametrized family $E_T$ of elliptic curves with the property that they parametrize all elliptic curves $E∕\mathbb{Q}$ which contain $T$ in their torsion subgroup. Using these parametrized families, we explicitly classify the Kodaira–Néron-type, the conductor exponent and the local Tamagawa number at each prime $p$ where $E∕\mathbb{Q}$ has additive reduction. As a consequence, we find all rational elliptic curves with a $2$- or $3$-torsion point that have global Tamagawa number $1$.

Keywords

elliptic curves, Tamagawa numbers, Kodaira–Néron-types, Tate's algorithm

Mathematical Subject Classification

Primary: 11G05, 11G07, 11G40, 14H52

Supplementary material

Kodaira--N'{e}ron-type and conductor exponent

Milestones

Received: 11 December 2020

Revised: 14 November 2021

Accepted: 5 March 2022

Published: 1 August 2022