Tamagawa numbers of elliptic curves with torsion points

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 $\mathbb{Q}$.

arXiv comment9 pages. Final version. To appear in Archiv der Mathematik