Orders of reductions of elliptic curves with many and few prime factors

Abstract

In this paper, we investigate extreme values of [Formula: see text], where [Formula: see text] is an elliptic curve with complex multiplication and [Formula: see text] is the number-of-distinct-prime-divisors function. For fixed [Formula: see text], we prove an asymptotic formula for the quantity [Formula: see text]. The same result holds for the quantity [Formula: see text] when [Formula: see text]. This asymptotic formula matches what one might expect, based on a result of Delange concerning extreme values of [Formula: see text]. The argument is worked out in detail for the curve [Formula: see text], and we discuss how the method can be adapted for other CM elliptic curves.