Conjectured Diophantine Estimates on Elliptic Curves

  • Serge Lang
Part of the Progress in Mathematics book series (PM, volume 35)


Let A be an elliptic curve defined over the rational numbers Q. Mordell’s theorem asserts that the group of points A(Q) is finitely generated. Say {P 1,..., P r } is a basis of A(Q) modulo torsion. Explicit upper bounds for the heights of elements in such a basis are not known. The purpose of this note is to conjecture such bounds for a suitable basis. Indeed, RA(Q) is a vector space over R with a positive definite quadratic form given by the Néron-Tate height: if A is defined by the equation
$${y^2} = {x^3} + ax + b,a,b \in {\text{Z}}$$
, and P = (x, y) is a rational point with x = c/d written as a fraction in lowest form, then one defines the x-height
$${h_x} = {\text{log max}}\left( {\left| c \right|,\left| d \right|} \right)$$


Zeta Function Elliptic Curve Elliptic Curf Riemann Zeta Function Riemann Hypothesis 
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Copyright information

© Springer Science+Business Media New York 1983

Authors and Affiliations

  • Serge Lang
    • 1
  1. 1.Department of MathematicsYale UniversityNew HavenUSA

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