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Some consequences of Masser’s counting theorem on elliptic curves. (English) Zbl 1381.11049

In this article, it is studied the Néron-Tate height on non-torsion point of elliptic curves and more specially a lower bound on this height in some special cases, this problem was stated and known as the elliptic Lehmer problem. In particular, this lower bound will depend on the degree of the point. In this article, the D. W. Masser’s counting theorem result [Bull. Soc. Math. Fr. 117, No. 2, 247–265 (1989; Zbl 0723.14026)] is used to prove a lower bound for the canonical height in powers of elliptic curves. The elliptic Lehmer problem is also proved in the Galois case using Kummer theory and Masser’s result with bounds on the rank and torsion of rational points on an elliptic curve, a refinement of this bound is proven in the CM case.

MSC:

11G05 Elliptic curves over global fields
11G50 Heights
14H52 Elliptic curves

Citations:

Zbl 0723.14026
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Full Text: DOI

References:

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