On primitive points of elliptic curves with complex multiplication

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Abstract

Let E be an elliptic curve defined over ℚ and P ∈ E(ℚ) a rational point of infinite order. Suppose that E has complex multiplication by an order in the imaginary quadratic field k. Denote by ME,P the set of rational primes ℓ such that ℓ splits in k, E has good reduction at ℓ, and P is a primitive point modulo ℓ. Under the generalized Riemann hypothesis, we can determine the positivity of the density of the set ME,P explicitly.

Original languageEnglish
Pages (from-to)66-87
Number of pages22
JournalJournal of Number Theory
Volume114
Issue number1
DOIs
StatePublished - Sep 2005

Keywords

  • Complex multiplication
  • Density
  • Elliptic curves
  • Primitive points

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