Vol. 2, No. 2, 2008

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Divisibility sequences for elliptic curves with complex multiplication

Marco Streng

Vol. 2 (2008), No. 2, 183–208
Abstract

Elliptic divisibility sequences arise as sequences of denominators of the integer multiples of a rational point on an elliptic curve. Silverman proved that almost every term of such a sequence has a primitive divisor (that is, a prime divisor that has not appeared as a divisor of earlier terms in the sequence). If the elliptic curve has complex multiplication, then we show how the endomorphism ring can be used to index a similar sequence and we prove that this sequence also has primitive divisors. The original proof fails in this context and will be replaced by an inclusion-exclusion argument and sharper diophantine estimates.

Keywords
complex multiplication, divisibility sequence, elliptic curve, endomorphism, primitive divisor, Zsigmondy
Mathematical Subject Classification 2000
Primary: 14H52
Secondary: 14K22
Milestones
Received: 30 July 2007
Revised: 12 November 2007
Accepted: 25 December 2007
Published: 15 March 2008
Authors
Marco Streng
Mathematisch Instituut
Universiteit Leiden
Postbus 9512
2300 RA Leiden
The Netherlands
http://www.math.leidenuniv.nl/~streng