Frobenius distributions in short intervals for CM elliptic curves

For an elliptic curve E / Q , Hasse's theorem asserts that # E ( F p ) = p + 1 − a p , where | a p | ≤ 2 p . Assuming that E has complex multiplication, we establish asymptotics for primes p for which a p is in subintervals of the Hasse interval [ − 2 p , 2 p ] of measure o ( p ) . In particular, given a function f = o ( 1 ) satisfying some mild conditions, we provide counting functions for primes p where | a p | ∈ ( 2 p ( 1 − f ( p ) ) , 2 p ) , and for primes where a p ∈ ( 2 p ( c − f ( p ) ) , 2 c p ) , where c ∈ ( 0 , 1 ) is a constant.