Tamagawa Numbers of Elliptic Curves with an $\ell$-isogeny

TL;DR

This paper investigates the divisibility properties of Tamagawa numbers of elliptic curves over rationals that are related via $ ext{ell}$-isogenies, extending previous work on curves with $ ext{ell}$-torsion points.

Contribution

It introduces a study of Tamagawa number divisibility for elliptic curves connected through $ ext{ell}$-isogenies, broadening understanding beyond curves with explicit $ ext{ell}$-torsion points.

Findings

Characterization of $ ext{ell}$-divisibility of Tamagawa numbers for $ ext{ell}$-isogenous curves.

Extension of divisibility results from curves with $ ext{ell}$-torsion to those related by $ ext{ell}$-isogenies.

New insights into the structure of Tamagawa numbers in the context of isogeny classes.

Abstract

Let $\ell$ be an odd prime, and suppose $E$ is an elliptic curve defined over the rational numbers $\mathbb{Q}$. If $E$ has an $\ell$-torsion point, then there has been significant work done on characterizing the $\ell$-divisibility of the global Tamagawa number of $E$. In this paper, we consider elliptic curves that are $\ell$-isogenous to elliptic curves with an $\ell$-torsion point and study the $\ell$-divisibility of their global Tamagawa numbers.