Tamagawa numbers of elliptic curves with torsion points

TL;DR

This paper investigates the divisibility of Tamagawa numbers by a prime for elliptic curves with rational torsion points, connecting to the Birch and Swinnerton-Dyer conjecture, and extends some results to abelian varieties.

Contribution

It provides new results on the frequency of Tamagawa number divisibility by a prime for elliptic curves with rational torsion points, and extends some findings to higher-dimensional abelian varieties.

Findings

Analyzes the divisibility of Tamagawa numbers by prime p.

Establishes results for elliptic curves over global fields.

Includes a partial extension to abelian varieties over Q.

Abstract

Let $K$ be a global field and let $E/K$ be an elliptic curve with a $K$-rational point of prime order $p$. In this paper we are interested in how often the (global) Tamagawa number $c(E/K)$ of $E/K$ is divisible by $p$. This is a natural question to consider in view of the fact that the fraction $c(E/K)/ |E(K)_{\text{tors}}|$ appears in the second part of the Birch and Swinnerton-Dyer Conjecture. We focus on elliptic curves defined over global fields, but we also prove a result for higher dimensional abelian varieties defined over $\mathbb{Q}$.