Computing local constants for CM elliptic curves

Abstract

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.

Arithmetic local constants for abelian varieties with extra endomorphisms

“…Thus δ(E, M i /K) ≡ 0 for i ≥ 2. By Theorem 2.8 of [4], condition (2) along with K ⊂ k gives δ(v, E, M 1 /K) ≡ (1, 1), and so δ(E, M 1 /K) ≡ m.…”

Arithmetic local constants for abelian varieties with extra endomorphisms

“…Thus δ(E, M i /K) ≡ 0 for i ≥ 2. By Theorem 2.8 of [4], condition (2) along with K ⊂ k gives δ(v, E, M 1 /K) ≡ (1, 1), and so δ(E, M 1 /K) ≡ m. Using Theorem 6.9, we combine the calculations to see that r p (A L /K, O L ) ≡ r p (E/K, O) + m (mod 2) .…”