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Heights and the specialization map for families of elliptic curves over $ \mathbb{P}^n$


Author: Wei Pin Wong
Journal: Trans. Amer. Math. Soc. 369 (2017), 3207-3220
MSC (2010): Primary 11G05; Secondary 11G50, 14G40
DOI: https://doi.org/10.1090/tran/6756
Published electronically: July 7, 2016
MathSciNet review: 3605969
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Abstract: For $ n\geq 2$, let $ K=\overline {\mathbb{Q}}(\mathbb{P}^n)=\overline {\mathbb{Q}}(T_1, \ldots , T_n)$. Let $ E/K$ be the elliptic curve defined by a minimal Weierstrass equation $ y^2=x^3+Ax+B$, with $ A,B \in \overline {\mathbb{Q}}[T_1, \ldots , T_n]$. There's a canonical height $ \hat {h}_{E}$ on $ E(K)$ induced by the divisor $ (O)$, where $ O$ is the zero element of $ E(K)$. On the other hand, for each smooth hypersurface $ \Gamma $ in $ \mathbb{P}^n$ such that the reduction mod $ \Gamma $ of $ E$, $ E_{\Gamma } / \overline {\mathbb{Q}}(\Gamma )$ is an elliptic curve with the zero element $ O_\Gamma $, there is also a canonical height $ \hat {h}_{E_{\Gamma }}$ on $ E_{\Gamma }(\overline {\mathbb{Q}}(\Gamma ))$ that is induced by $ (O_\Gamma )$. We prove that for any $ P \in E(K)$, the equality $ \hat {h}_{E_{\Gamma }}(P_\Gamma )/ \deg \Gamma =\hat {h}_{E}(P)$ holds for almost all hypersurfaces in $ \mathbb{P}^n$. As a consequence, we show that for infinitely many $ t \in \mathbb{P}^n(\overline {\mathbb{Q}})$, the specialization map $ \sigma _t : E(K) \rightarrow E_t(\overline {\mathbb{Q}})$ is injective.


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Additional Information

Wei Pin Wong
Affiliation: Department of Mathematics, Brown University, Box 1917, Providence, Rhode Island 02912
Address at time of publication: Engineering Systems and Design, Singapore University of Technology and Design, 8 Somapah Road, 487372 Singapore
Email: weipin_wong@sutd.edu.sg

DOI: https://doi.org/10.1090/tran/6756
Keywords: Height function, family of elliptic curves, function field, higher dimensional, projective space, variety, hypersurface, specialization map
Received by editor(s): September 25, 2014
Received by editor(s) in revised form: April 28, 2015
Published electronically: July 7, 2016
Article copyright: © Copyright 2016 American Mathematical Society