Heights of torsion points on commutative group varieties. II. (English) Zbl 0686.14046
[For part I see S. P. Cohen, ibid. 52, 427-444 (1986; Zbl 0567.14024).]
We study the arithmetic properties of torsion points on the commutative group variety given by a non-trivial additive extension of an elliptic curve defined over a number field, and show in particular that the Weil height of these points grows as the logarithm of their order, when this order is coprime to a finite set of exceptional primes. The results obtained are applied to derive explicit formulae for the p-adic periods, or Tate constants, of the height one and height two formal groups associated to elliptic curves with good reduction at the prime \(p\geq 5\), defined over a finite extension of the completion of \({\mathbb{Q}}\) at p.
We study the arithmetic properties of torsion points on the commutative group variety given by a non-trivial additive extension of an elliptic curve defined over a number field, and show in particular that the Weil height of these points grows as the logarithm of their order, when this order is coprime to a finite set of exceptional primes. The results obtained are applied to derive explicit formulae for the p-adic periods, or Tate constants, of the height one and height two formal groups associated to elliptic curves with good reduction at the prime \(p\geq 5\), defined over a finite extension of the completion of \({\mathbb{Q}}\) at p.
Reviewer: P.Cohen
MSC:
14K05 | Algebraic theory of abelian varieties |
14G40 | Arithmetic varieties and schemes; Arakelov theory; heights |
14L99 | Algebraic groups |
14H52 | Elliptic curves |
14G25 | Global ground fields in algebraic geometry |