Abstract. Let E/k be an elliptic curve with CM by O. We determine a formula for (a generalization of) the arithmetic local constant of [5] at almost all primes of good reduction. We apply this formula to the CM curves defined over Q and are able to describe extensions F/Q over which the O-rank of E grows.

Let E/k be an elliptic curve with CM by O. We determine a formula for (a generalization of) the arithmetic local constant of [5] at almost all primes of good reduction. We apply this formula to the CM curves defined over Q and are able to describe extensions F/Q over which the O-rank of E grows.