On Lehmer’s conjecture for elliptic curves. (English) Zbl 0741.14013

Hindry, Marc; Silverman, Joseph H.

Sémin. Théor. Nombres, Paris/Fr. 1988-89, Prog. Math. 91, 103-116 (1990).

[For the entire collection see Zbl 0711.00009.]
The Lehmer conjecture for an elliptic curve \(E\), defined over a number field \(k\), states that the canonical Néron-Tate height \(h(Q)\) of an algebraic point \(Q\) of infinite order should be greater than \(constant\times d(Q)^{-1}\) (where \(d(Q):=[k(Q):k]\) is the degree of the field generated by \(Q\)). This diophantine inequality generalises naturally the classical Lehmer conjecture on the multiplicative group (infinite order is the analog of not being a root of unity). We improve known lower bounds in the case the \(j\)-invariant of the curve \(E\) is not integral to get \(h(Q)\geq cd^{-2}\log d(Q)^{-2}\), where \(c\) is an explicit constant depending on the curve. A close to best possible bound \(h(Q)\geq cd^{-1}\{\log d(Q)/\log\log d(Q)\}^{-3}\) is known for complex multiplication curves which of course have integral \(j\)-invariant [M. Laurent, Théorie des nombres, Sémin. Delange-Pisot-Poitou, Paris 1981/82, Prog. Math. 38, 137-152 (1983; Zbl 0521.14010)]. When the curve is not CM but the \(j\)-invariant is integral, we cannot improve the general bound \(h(Q)\geq cd^{-3}\log d(Q)^{-2}\) of D. W. Masser [Bull. Soc. Math. Fr. 117, No. 2, 247-265 (1989; Zbl 0723.14026)].
Like the last mentioned paper, we obtain the lower bound by actually giving a bound for the number of points of small height, but whereas Laurent and Masser use transcendance methods, we rely on the decomposition of \(h\) as a sum of local Néron heights, using techniques developed by the authors in a previous paper [Invent. Math. 93, No. 2, 419-450 (1988; Zbl 0657.14018)] and a lemma due to N. Elkies on the average of Green functions [see the book by S. Lang: “Introduction to Arakelov theory” (1988; Zbl 0667.14001), theorem 5.1, p. 150].

Reviewer: M.Hindry

MSC:

14G40	Arithmetic varieties and schemes; Arakelov theory; heights
14H52	Elliptic curves
11G05	Elliptic curves over global fields

Keywords:

Lehmer conjecture for an elliptic curve; Néron-Tate height

Citations:

Zbl 0711.00009; Zbl 0521.14010; Zbl 0723.14026; Zbl 0657.14018; Zbl 0667.14001