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 $${\mathbb {Q}}$$ .