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

Author: Filip Najman
Journal: Proc. Amer. Math. Soc. 145 (2017), 3747-3753
MSC (2010): Primary 11G05, 11G15, 11G18
Published electronically: May 4, 2017
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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 positive 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 furthermore prove that there is a unique such curve satisfying $ v_{13}(c_E)=2$.

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Additional Information

Filip Najman
Affiliation: Department of Mathematics, University of Zagreb, Bijenička cesta 30, 10000 Zagreb, Croatia

Received by editor(s): June 2, 2016
Received by editor(s) in revised form: September 14, 2016, September 19, 2016, and October 11, 2016
Published electronically: May 4, 2017
Additional Notes: The author gratefully acknowledges support from the QuantiXLie Center of Excellence.
Communicated by: Mathew A. Papanikolas
Article copyright: © Copyright 2017 American Mathematical Society

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