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Local to global trace questions and twists of genus one curves


Authors: Mirela Çiperiani and Ekin Ozman
Journal: Proc. Amer. Math. Soc. 143 (2015), 3815-3826
MSC (2010): Primary 11G05
DOI: https://doi.org/10.1090/proc/12560
Published electronically: May 6, 2015
MathSciNet review: 3359573
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Abstract: Let $ \mathrm {E}$ be an elliptic curve defined over a number field $ \mathrm {F}$ and $ \mathrm {K}/\mathrm {F}$ a quadratic extension. For a point $ P\in \mathrm {E}(\mathrm {F})$ that is a local trace for every completion of $ \mathrm {K}/\mathrm {F}$, we find necessary and sufficient conditions for $ P$ to lie in the image of the global trace map. These conditions can then be used to determine whether a quadratic twist of $ \mathrm {E}$, as a genus one curve, has rational points. In the case of quadratic twists of genus one modular curves $ X_0(N)$ with squarefree $ N$, the existence of rational points corresponds to the existence of $ \mathbb{Q}$-curves of degree $ N$ defined over $ \mathrm {K}$.


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Additional Information

Mirela Çiperiani
Affiliation: Department of Mathematics, RLM 8.100, University of Texas at Austin, 2515 Speedway Stop C1200, Austin, Texas 78712-1202
Email: mirela@math.utexas.edu

Ekin Ozman
Affiliation: Department of Mathematics, RLM 8.100, University of Texas at Austin, 2515 Speedway Stop C1200, Austin, Texas 78712-1202
Email: ozman@math.utexas.edu

DOI: https://doi.org/10.1090/proc/12560
Keywords: Elliptic curve, $\mathbb{Q}$-curves, twists of genus one curves
Received by editor(s): October 28, 2013
Received by editor(s) in revised form: June 10, 2014
Published electronically: May 6, 2015
Additional Notes: The first author was partially supported by an NSA grant during the preparation of this paper
Communicated by: Ken Ono
Article copyright: © Copyright 2015 American Mathematical Society