Local to global trace questions and twists of genus one curves
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- by Mirela Çiperiani and Ekin Ozman PDF
- Proc. Amer. Math. Soc. 143 (2015), 3815-3826 Request permission
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 $\mathrm {K}$.References
- Roberto Dvornicich and Umberto Zannier, Local-global divisibility of rational points in some commutative algebraic groups, Bull. Soc. Math. France 129 (2001), no. 3, 317–338 (English, with English and French summaries). MR 1881198, DOI 10.24033/bsmf.2399
- Jordan S. Ellenberg, $\Bbb Q$-curves and Galois representations, Modular curves and abelian varieties, Progr. Math., vol. 224, Birkhäuser, Basel, 2004, pp. 93–103. MR 2058645
- Anthony W. Knapp, Elliptic curves, Mathematical Notes, vol. 40, Princeton University Press, Princeton, NJ, 1992. MR 1193029
- B. Mazur and K. Rubin, Ranks of twists of elliptic curves and Hilbert’s tenth problem, Invent. Math. 181 (2010), no. 3, 541–575. MR 2660452, DOI 10.1007/s00222-010-0252-0
- Ekin Ozman, Points on quadratic twists of $X_0(N)$, Acta Arith. 152 (2012), no. 4, 323–348. MR 2890545, DOI 10.4064/aa152-4-1
- Joseph H. Silverman, The arithmetic of elliptic curves, 2nd ed., Graduate Texts in Mathematics, vol. 106, Springer, Dordrecht, 2009. MR 2514094, DOI 10.1007/978-0-387-09494-6
Additional Information
- Mirela Çiperiani
- Affiliation: Department of Mathematics, RLM 8.100, University of Texas at Austin, 2515 Speedway Stop C1200, Austin, Texas 78712-1202
- MR Author ID: 838646
- 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
- MR Author ID: 955558
- Email: ozman@math.utexas.edu
- 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
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 143 (2015), 3815-3826
- MSC (2010): Primary 11G05
- DOI: https://doi.org/10.1090/proc/12560
- MathSciNet review: 3359573