Dobrowolski-Laurent theorem for abelian extensions over an elliptic curve having complex multiplication.
(Théorème de Dobrowolski-Laurent pour les extensions abéliennes sur une courbe elliptique à multiplication complexe.)

*(French)*Zbl 1091.11020Let \(K\) be a number field. Denote by \(\widehat{h}\) the Néron-Tate height on an elliptic curve \(E/K\). The main results of the paper are the following two theorems.

Theorem 1.1. Let \(E/K\) be an elliptic curve with complex multiplication. Then there exists a constant \(c(E/K)>0\) such that \( \widehat{h}(P) \geq \frac{c(E/K)}{D}( \frac{\log \log 5D}{\log 2D})^{13}\) for all \(P \in E(\overline{K}) \setminus E_{\text{tors}}, \) where \(D=[K^{\text{ab}}(P): K^{\text{ab}}]\).

Theorem 1.2. Let \(c_0>0\) and let \(E/K\) be an elliptic curve with complex multiplication. Then there exists a constant \(c(E/K,c_0)>0\) such that \( \widehat{h}(P) \geq \frac{c(E/K,c_0)}{D} (\frac{\log \log 5D}{\log 2D})^{3} \) for every abelian extension \(F/K\) and for every point \(P\in E( \overline{K}) \setminus E_{\text{tors}}\) verifying \(D= [F(P): F]\), provided the number of ramified in \(F\) primes is bounded by \(c_0( \log 2D / \log \log 5D)^2\).

Problems of such kind come from the work of Lehmer who formulated in 1933 the conjecture asserting the existence of a constant \(c>0\) such that \(h(x) \geq \frac{c}{D}\) for all \(x \in \mathbb{G}_m (\overline{\mathbb{Q}})\setminus \mu_\infty\), where \(D=[\mathbb{Q}(x):\mathbb{Q}]\). E. Dobrowolski [Acta Arith. 34, 391–401 (1979; Zbl 0416.12001)] proved that there exists a constant \(c>0\) such that \( h(x) \geq \frac{c}{D} (\frac{\log \log 3D}{\log 2D})^3\) for all \(x \in \mathbb{G}_m (\overline{\mathbb{Q}})\setminus \mu_\infty\), where \(D= [\mathbb{Q}(x):\mathbb{Q}]\). For an arbitrary number field \(K\), M. Amoroso and U. Zannier [Ann. Sc. Norm. Sup. Pisa, Cl. Sci. (4) 29, No. 3, 711–727 (2000; Zbl 1016.11026)] proved the existence of a constant \(c(K)>0\) such that \( h(x) \geq \frac{c(K)}{D} (\frac{\log \log 5D}{\log 2D})^{13}\) for all \(x \in \mathbb{G}_m (\overline{K})\setminus \mu_\infty\), where \(D= [K^{\text{ab}}{x}: K^{\text{ab}}]\). Thus Theorems 1.1 and 1.2 are counterparts for elliptic curves of Amoroso-Zamir’s and Dobrowolski’s results.

In the case of elliptic curves with complex multiplication Theorem 1.1 generalizes a result of M. H. Baker [Int. Math. Res Not. 2003, No. 29, 1571–1589 (2003; Zbl 1114.11058)], and answers David’s conjecture on the lower bound for the Neron-Tate height on abelian varieties, and thus may be considered as a first step towards proving this conjecture. The author shows that Theorem 1.1 allows to simplify the proof of a result of E. Viada [Ann. Sc. Norm. Sup. Pisa Cl. Sci. (5) 2, No. 1, 47–75 (2002; Zbl 1170.11314)] on the intersection of a curve with algebraic subgroups in a product of elliptic curves. As for Theorem 1.2, it generalizes a previous result of M. Laurent [Prog. Math. 38, 137–151 (1983; Zbl 0521.14010)].

The proof of Theorem 1.1 is based on the classical transcendence argument. Using the Siegel lemma the author considers two cases, of many ramifications and of a few ones. In the first case the author follows the method due to F. Amoroso and U. Zannier (loc. cit.) and applies the Frobenius tranformations and the Laurent trick of diluting variables to have more freedom for choices of auxiliary parameters. In the case of not very much ramifications the proof develops the approach used by Dobrowolski in his proof of the lower bound for the heights of algebraic numbers.

The paper ends by explaning that Theorem 1.2 can be proved essentially in the same way and Theorem 1.1 can be applied to simplify the proof of Viada’s theorem.

Theorem 1.1. Let \(E/K\) be an elliptic curve with complex multiplication. Then there exists a constant \(c(E/K)>0\) such that \( \widehat{h}(P) \geq \frac{c(E/K)}{D}( \frac{\log \log 5D}{\log 2D})^{13}\) for all \(P \in E(\overline{K}) \setminus E_{\text{tors}}, \) where \(D=[K^{\text{ab}}(P): K^{\text{ab}}]\).

Theorem 1.2. Let \(c_0>0\) and let \(E/K\) be an elliptic curve with complex multiplication. Then there exists a constant \(c(E/K,c_0)>0\) such that \( \widehat{h}(P) \geq \frac{c(E/K,c_0)}{D} (\frac{\log \log 5D}{\log 2D})^{3} \) for every abelian extension \(F/K\) and for every point \(P\in E( \overline{K}) \setminus E_{\text{tors}}\) verifying \(D= [F(P): F]\), provided the number of ramified in \(F\) primes is bounded by \(c_0( \log 2D / \log \log 5D)^2\).

Problems of such kind come from the work of Lehmer who formulated in 1933 the conjecture asserting the existence of a constant \(c>0\) such that \(h(x) \geq \frac{c}{D}\) for all \(x \in \mathbb{G}_m (\overline{\mathbb{Q}})\setminus \mu_\infty\), where \(D=[\mathbb{Q}(x):\mathbb{Q}]\). E. Dobrowolski [Acta Arith. 34, 391–401 (1979; Zbl 0416.12001)] proved that there exists a constant \(c>0\) such that \( h(x) \geq \frac{c}{D} (\frac{\log \log 3D}{\log 2D})^3\) for all \(x \in \mathbb{G}_m (\overline{\mathbb{Q}})\setminus \mu_\infty\), where \(D= [\mathbb{Q}(x):\mathbb{Q}]\). For an arbitrary number field \(K\), M. Amoroso and U. Zannier [Ann. Sc. Norm. Sup. Pisa, Cl. Sci. (4) 29, No. 3, 711–727 (2000; Zbl 1016.11026)] proved the existence of a constant \(c(K)>0\) such that \( h(x) \geq \frac{c(K)}{D} (\frac{\log \log 5D}{\log 2D})^{13}\) for all \(x \in \mathbb{G}_m (\overline{K})\setminus \mu_\infty\), where \(D= [K^{\text{ab}}{x}: K^{\text{ab}}]\). Thus Theorems 1.1 and 1.2 are counterparts for elliptic curves of Amoroso-Zamir’s and Dobrowolski’s results.

In the case of elliptic curves with complex multiplication Theorem 1.1 generalizes a result of M. H. Baker [Int. Math. Res Not. 2003, No. 29, 1571–1589 (2003; Zbl 1114.11058)], and answers David’s conjecture on the lower bound for the Neron-Tate height on abelian varieties, and thus may be considered as a first step towards proving this conjecture. The author shows that Theorem 1.1 allows to simplify the proof of a result of E. Viada [Ann. Sc. Norm. Sup. Pisa Cl. Sci. (5) 2, No. 1, 47–75 (2002; Zbl 1170.11314)] on the intersection of a curve with algebraic subgroups in a product of elliptic curves. As for Theorem 1.2, it generalizes a previous result of M. Laurent [Prog. Math. 38, 137–151 (1983; Zbl 0521.14010)].

The proof of Theorem 1.1 is based on the classical transcendence argument. Using the Siegel lemma the author considers two cases, of many ramifications and of a few ones. In the first case the author follows the method due to F. Amoroso and U. Zannier (loc. cit.) and applies the Frobenius tranformations and the Laurent trick of diluting variables to have more freedom for choices of auxiliary parameters. In the case of not very much ramifications the proof develops the approach used by Dobrowolski in his proof of the lower bound for the heights of algebraic numbers.

The paper ends by explaning that Theorem 1.2 can be proved essentially in the same way and Theorem 1.1 can be applied to simplify the proof of Viada’s theorem.

Reviewer: Vasyl I. Andriychuk (Lviv)