Divisibility sequences for elliptic curves with complex multiplication

Streng, Marco

doi:10.2140/ant.2008.2.183

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Divisibility sequences for elliptic curves with complex multiplication

Marco Streng

Vol. 2 (2008), No. 2, 183–208

DOI: 10.2140/ant.2008.2.183

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

Vol. 2, No. 2, 2008