Vol. 318, No. 1, 2022

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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 ET of elliptic curves with the property that they parametrize all elliptic curves E 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 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
Authors
Alexander J. Barrios
Department of Mathematics
University of St. Thomas
St. Paul, MN
United States
Manami Roy
Department of Mathematics
Fordham University
Bronx, NY
United States