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Theoreme de Dobrowolski-Laurent pour les extensions abeliennes sur une courbe elliptique a multiplication complexe | | Nicolas Ratazzi
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13 Feb 2004 | | Subject: | Number Theory; Algebraic Geometry MSC-class: 11G50; 14G40; 14K22 | math.NT math.AG | | Abstract: | Let E/K be an elliptic curve with complex multiplication and let $K^{ab}$ be the Abelian closure of $K$. We prove in this article that there exists a constant $c(E/K)$ such that : for all point $Pin E(ar{K})-E_{tors}$, we have [hat{h}(P)geqfrac{c(E/K)}{D}(frac{log log 5D}{log 2D})^{13},where $D=[K^{ab}(P):K^{ab}]$. This result extends to the case of elliptic curve s with complex multiplication the previous resultof Amoroso-Zannier cite{AZ} on the analogous problem on the multiplicative group $mathbb{G}_m$, and generalizes to the case of extensions of degree D the result of Baker cite{baker} on the lower bound of the Néron-Tate height of the points defined over an Abelian extension of an elliptic curve with complex multiplication. This result also enables us to simplify the proof of a theorem of Viada cite{viada}. | | Source: | arXiv, math.NT/0402224 | | Services: | Forum | Review | PDF | Favorites | |
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