Tamagawa Numbers of elliptic curves with $C_{13}$ torsion over quadratic fields
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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 $v_{13}(c_E)=2$.References
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Additional Information
- Filip Najman
- Affiliation: Department of Mathematics, University of Zagreb, Bijenička cesta 30, 10000 Zagreb, Croatia
- MR Author ID: 886852
- Email: fnajman@math.hr
- 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
- © Copyright 2017 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 145 (2017), 3747-3753
- MSC (2010): Primary 11G05, 11G15, 11G18
- DOI: https://doi.org/10.1090/proc/13553
- MathSciNet review: 3665029