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Lower bounds for linear forms in elliptic logarithms. (Minorations de formes linéaires de logarithmes elliptiques.) (French) Zbl 0859.11048
It is well known that lower bounds for linear forms in logarithms of algebraic numbers are a basic tool in solving a number of diophantine equations. This requires lower bounds that are completely explicit in terms of the heights of the coefficients of the linear forms, the moduli of the logarithms and the heights of their arguments. For other classes of diophantine equations, explicit lower bounds for linear forms in elliptic logarithms are needed, and this paper fulfills this need. It gives a completely explicit form of the qualitative lower bounds established by N. Hirata-Kohno [Sémin. Théor. Nombres, Paris 1988-89, Prog. Math. 91, 117-140 (1990; Zbl 0716.11033)].
Let \({\mathcal E}_1, \dots, {\mathcal E}_k\) be elliptic curves defined over a number field \(K\) and given in Weierstrass form by \[ {\mathcal E}_i: y^2 = 4x^3-g_{2,i} x-g_{3,i}, \quad g_{2,i}, g_{3,i} \in K, \quad 1\leq i\leq k. \] Put \(D= [K:\mathbb{Q}]\) and, using Weyl’s absolute logarithmic height, define \(h\) as the maximum of 1, the heights of the points \((1,g_{2,i}, g_{3,i})\) and the heights of the \(j\)-invariants of the elliptic curves \({\mathcal E}_i\). Also, for each \(i\), choose a basis \((w_{1,i}, w_{2,i})\) of the period lattice of \({\mathcal E}_i\) such that the ratio \(\tau_i= w_{2,i}/w_{1,i}\) lies in the usual fundamental domain of the Poincaré upper half plane for the action of \(\text{SL}_2(\mathbb{Z})\). Let \({\mathcal L} (\mathbf z) = \beta_0 z_0 + \cdots+ \beta_kz_k\) be a linear form on \(\mathbb{C}^{k+1}\) with coefficients in \(K\) and, for \(i=1, \dots, k\), let \(u_i\) be a complex number whose image \(\gamma_i\) under the usual exponential map of \({\mathcal E}_i\) lies in \({\mathcal E}_i(K)\). Define \({\mathbf v}= (1,u_1, \dots, u_k)\).
The first part of the main theorem provides a (positive) upper bound for \(|{\mathcal L} ({\mathbf v}) |\) which insures \({\mathcal L} ({\mathbf v}) =0\). This upper bound is completely explicit in terms of \(k,h,D\), the heights of the points \(\beta_i\), the Néron-Tate heights of the points \(\gamma_i\), and the numbers \(|u_i |\), \(|w_{1,i} |\) and \(\text{Im} (\tau_i)\). More precisely, it is given explicitly in terms of \(k,h,D\) and of some parameters \(B,E,V_1, \dots, V_k\) which are required to satisfy a family of inequalities involving the previous quantities. These parameters are introduced in order to provide more freedom for the applications.
When \(|{\mathcal L}(\mathbf v)|\) falls below this bound, the second part of the theorem asserts the existence of an algebraic subgroup \(H\) of \({\mathbf G}_a \times {\mathcal E}_1 \times \cdots \times {\mathcal E}_k\) whose tangent space at the origin contains \({\mathbf v}\) and is contained in the kernel of \({\mathcal L}\), with an upper bound for the degree of \(H\) which is completely explicit in terms of \(k,h,D\), the dimension \(\widetilde d\) of \(H\), and the same parameters \(B,E,V_1, \dots, V_k\). As a corollary of this result, the author gives a weaker but explicit version of the isogeny theorem of D. Masser and G. Wüstholz for elliptic curves [Invent. Math. 100, 1-24 (1990; Zbl 0722.14027)]. With the above notations, assume \(k=1\). Put \({\mathcal E}: ={\mathcal E}_1\) and \(h({\mathcal E}):= h\). The corollary says that, if \({\mathcal E}^*\) is an elliptic curve defined over \(K\) and if there exists an isogeny between \({\mathcal E}\) and \({\mathcal E}^*\), then there is such an isogeny of degree at most \(10^{160} D^{20} h({\mathcal E})^{10}\).
Finally, note that Chapter 3 of this paper contains explicit bounds for the growth of the Weierstrass functions, and that Chapter 8 provides an effective control on the translation and derivation formulas. These have independent interest and in fact are used in a recent work of M. Waldschmidt [Approximation diophantienne dans les groupes algébriques commutatifs (submitted)].
Reviewer: D.Roy (Ottawa)

MSC:
11J89 Transcendence theory of elliptic and abelian functions
11J86 Linear forms in logarithms; Baker’s method
11G05 Elliptic curves over global fields
14K02 Isogeny
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