Amicable Pairs and Aliquot Cycles for Elliptic Curves

An amicable pair for an elliptic curve E/Q is a pair of primes (p, q) of good reduction for E satisfying #Ẽp(Fp) = q and #Ẽq(Fq) = p. In this paper we study elliptic amicable pairs and analogously defined longer elliptic aliquot cycles. We show that there exist elliptic curves with arbitrarily long aliqout cycles, but that CM elliptic curves (with j 6= 0) have no aliqout cycles of length greater than two. We give conjectural formulas for the frequency of amicable pairs. For CM curves, the derivation of precise conjectural formulas involves a detailed analysis of the values of the Grössencharacter evaluated at primes p in End(E) having the property that #Ẽp(Fp) is prime. This is especially intricate for the family of curves with j = 0. Introduction Let E/Q be an elliptic curve. In this paper we study pairs of primes (p, q) such that E has good reduction at p and q and such that the reductions Ẽp and Ẽq of E at p and q satisfy #Ẽp(Fp) = q and #Ẽq(Fq) = p. By analogy with a classical problem in number theory (cf. Remark 7), we call (p, q) an amicable pair for the elliptic curve E/Q. Example 1. Searching for amicable pairs using primes smaller than 10 on the two elliptic curves E1 : y 2 + y = x − x and E2 : y + y = x + x, yields one amicable pair on the curve E1, (1622311, 1622471), Date: December 9, 2009, Draft #5. 1991 Mathematics Subject Classification. Primary: 11G05; Secondary: 11B37, 11G20, 14G25.