Tamagawa Numbers of elliptic curves with $C_{13}$ torsion over quadratic fields

Filip Najman

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arXiv · 1605.05512

Tamagawa Numbers of elliptic curves with $C_{13}$ torsion over quadratic fields

Filip Najman

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Abstract

Let $E$ be an elliptic curve over a number field $K$, $c_v$ the Tamagawa number of $E$ at $v$, and let $c_E=\prod_{v}c_v$. Lorenzini proved that $v_{13}(c_E)$ is postive for all elliptic curves over quadratic fields with a point of order $13$. Krumm conjectured, based on extensive computation, that the $13$-adic valuation of $c_E$ is even for all such elliptic curves. In this note we prove this conjecture and furhtermore prove that there is an unique such curve satisfying $v_{13}(c_E)=2$.

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Filip Najman. 2016-10-26. Tamagawa Numbers of elliptic curves with $C_{13}$ torsion over quadratic fields. https://arxiv.org/abs/1605.05512

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