Divisibility sequences for elliptic curves with complex multiplication

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.