Formes linéaires de logarithmes de points algébriques sur une courbe elliptique. (Linear forms of logarithms of algebraic points on an elliptic curve)
In connection with Siegel's theorem on the finiteness of integral points on a curve of genus \(\geq 1\), \textit{S. Lang} raised the following question [Conjecture 2 of Diophantine approximations on toruses. Am. J. Math. 86, 521-533 (1964; Zbl 0142.29601); = Collected Papers, Vol. I, 1952-1970, Springer (2000; Zbl 0955.01032), p. 313-325]. Given an abelian variety \(A\) of dimension \(1\) over a number field \(K\) and a non-constant rational function \(\phi\) on \(A\) defined over \(K\), there exists a positive constant \(C\) such that the height of points \(P\) in \(A(K)\) which satisfy \(|\phi(P)|\leq H(P)^{-C}\) is bounded. Here, \(H(P)\) is the absolute height of \(P\). Lang also speculated a best possible value for \(C\), but we shall only consider the mere existence of \(C\). Later he extended his conjecture to abelian varieties of arbitrary dimension [p. 673 of Transcendental numbers and diophantine approximations. Bull. Am. Math. Soc. 77, 635-677 (1971; Zbl 0218.10053); = Collected Papers, Vol. II, 1971-1977, Springer (2000; Zbl 0955.01033), p. 1-43; p. 783 of Higher dimensional diophantine problems. Bull. Am. Math. Soc. 80, 779-787 (1974; Zbl 0298.14014); = Collected Papers, Vol. II, 1971-1977, Springer (2000), p. 102-110] Non-trivial effective lower bounds for \(|\phi(P)|\) have been obtained by \textit{D. W. Masser} [Elliptic functions and transcendence. Lecture Notes Math. 437, Springer-Verlag, Berlin-New York (1975; Zbl 0312.10023)] for elliptic curves with complex multiplication, and [Linear forms in algebraic points of Abelian functions. I--III. Math. Proc. Camb. Philos. Soc. 77, 499-513 (1975; Zbl 0306.14018); ibid. 79, 55-70 (1976; Zbl 0318.14011); Proc. Lond. Math. Soc. (3) 33, 549-564 (1976; Zbl 0334.14019)], \textit{S. Lang} [Diophantine approximation on abelian varieties with complex multiplication. Adv. Math. 17, 281-336 (1975; Zbl 0306.14019); = Collected Papers, Vol. II, 1971-1977, Springer (2000), p. 113-168] and also \textit{J. Coates} and \textit{S. Lang} [Diophantine approximation on abelian varieties with complex multiplication. Invent. Math. 34, 129-133 (1976; Zbl 0342.10018); = Collected Papers, Vol. II, 1971-1977, Springer (2000), p. 236-240] for abelian varieties of CM type, by \textit{P. Philippon} and the reviewer [Formes linéaires de logarithmes sur les groupes algébriques commutatifs. Ill. J. Math. 32, 281-314 (1988; Zbl 0651.10023)] for the general case, by \textit{S. David} [Minorations de formes linéaires de logarithmes elliptiques. Mém. Soc. Math. France, Nouv. Sér. 62 (1995; Zbl 0859.11048)] (with completely explicit estimates) for elliptic curves, and finally by \textit{N. Hirata-Kohno} [Formes linéaires de logarithmes de points algébriques sur les groupes algébriques. Invent. Math. 104, 401-433 (1991; Zbl 0716.11035)] for the general case. So far, the best known estimate, valid for the general case, was Hirata-Kohno's one: \[ |\phi(P)|\geq \exp\bigl\{-C\bigl(\log H(P) \bigr) \bigl(\log\log H(P) \bigr)^{g}\bigr\} \] (where \(g\) is the dimension of \(A\)), for any \(P\in A(K)\) of sufficiently large height. In the present paper, for the first time, Lang's above mentioned conjecture is proved, for the special case of elliptic curves with complex multiplication. The proof involves some modification of Hirata-Kohno's argument, together with interpolation polynomials previously introduced by the author and \textit{M. M'Zari} [Polynômes de Lagrange sur les entiers d'un corps quadratique imaginaire. J. Théor. Nombres Bordx. 10, 85-105 (1998; Zbl 0914.11009)]. These polynomials, attached to the ring of endomorphisms of the elliptic curve, generalize the binomial polynomials used by \textit{N. I. Fel'dman} for solving the analogous question for linear toruses [An improvement of the estimate of a linear form in the logarithms of algebraic numbers (Russian), Mat. Sb., Nov. Ser. 77(119), 423-436 (1968); Engl. transl. in Math. USSR, Sb. 6, 393-406 (1968; Zbl 0235.10018)]. Recently, \textit{S. David} and \textit{N. Hirata} obtained a solution of Lang's conjecture for elliptic curves, without any assumption on their endomorphisms, by a suitable adaptation of an argument of G. V. Chudnovsky. Even more recently, \textit{E. Gaudron} succeeded in solving the general case of Lang's conjecture for any abelian variety (and even more generally for commutative algebraic groups), using integrals models and Arakelov's theory.
elliptic curves with complex multiplication
scientific article; zbMATH DE number 1409062