Local data of rational elliptic curves with non-trivial torsion

TL;DR

This paper classifies local properties of rational elliptic curves with specific torsion subgroups, explicitly determining reduction types, conductors, and Tamagawa numbers, and identifies those with global Tamagawa number 1.

Contribution

It provides explicit classifications for all rational elliptic curves with certain torsion points, detailing their local reduction types and Tamagawa numbers.

Findings

Classified Kodaira-Néron types for all torsion subgroups.

Determined conductor exponents and Tamagawa numbers at primes.

Identified all curves with global Tamagawa number 1.

Abstract

By Mazur's Torsion Theorem, there are fourteen possibilities for the non-trivial 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 parameterized family $E_T$ of elliptic curves with the property that they parameterize all elliptic curves $E/\mathbb{Q}$ which contain $T$ in their torsion subgroup. Using these parameterized families, we explicitly classify the Kodaira-N\'{e}ron 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 $2$-torsion or a $3$-torsion point that have global Tamagawa number $1$.