Tamagawa numbers of elliptic curves with prescribed torsion subgroup or isogeny

TL;DR

This paper investigates Tamagawa numbers of elliptic curves with specific torsion subgroups over cubic fields and with certain isogenies over 9, revealing divisibility properties and exceptions related to their reduction types.

Contribution

It provides new divisibility results for Tamagawa numbers of elliptic curves with prescribed torsion or isogenies, extending understanding of their arithmetic properties.

Findings

Tamagawa numbers for curves with torsion Z/2ZZ/14Z over cubic fields are divisible by 14^2.

Elliptic curves with 18-isogeny have Tamagawa numbers divisible by 4.

Most other n-isogeny curves have Tamagawa numbers divisible by 2, with some exceptions.

Abstract

We study Tamagawa numbers of elliptic curves with torsion $\mathbb{Z}/2\mathbb{Z}\oplus \mathbb{Z}/14\mathbb{Z}$ over cubic fields and of elliptic curves with an $n-$isogeny over $\mathbb{Q}$, for $n\in\{6,8,10,12,14,16,17,18,19,37,43,67,163\}$. Bruin and Najman proved that every elliptic curve with torsion $\mathbb{Z}/2\mathbb{Z}\oplus \mathbb{Z}/14\mathbb{Z}$ over a cubic field is a base change of an elliptic curve defined over $\mathbb{Q}$. We find that Tamagawa numbers of elliptic curves defined over $\mathbb{Q}$ with torsion $\mathbb{Z}/2\mathbb{Z}\oplus \mathbb{Z}/14\mathbb{Z}$ over a cubic field are always divisible by $14^2$, with each factor $14$ coming from a rational prime with split multiplicative reduction of type $I_{14k},$ one of which is always $p=2.$ The only exception is the curve 1922.e2, with $c_E=c_2=14.$ The same curves defined over cubic fields over which they have torsion subgroup $\mathbb{Z}/2\mathbb{Z}\oplus \mathbb{Z}/14\mathbb{Z}$ turn out to have the Tamagawa number divisible by $14^3$. As for $n-$isogenies, Tamagawa numbers of elliptic curves with an $18-$isogeny must be divisible by 4, while elliptic curves with an $n-$isogeny for the remaining $n$ from the mentioned set must have Tamagawa numbers divisible by 2, except for finite sets of specified curves.