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arXiv · 2202.06235

Tamagawa numbers of elliptic curves with torsion points

Mentzelos Melistas
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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 $\mathbb{Q}$.

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Mentzelos Melistas. 2022-02-13. Tamagawa numbers of elliptic curves with torsion points. https://arxiv.org/abs/2202.06235

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