Extremal primes for elliptic curves with complex multiplication

Fix an elliptic curve E/Q. For each prime p of good reduction, let ap=p+1−#E(Fp). A well-known theorem of Hasse asserts that |ap|≤2p. We say that p is a champion prime for E if ap=−⌊2p⌋, that is, #E(Fp) is as large as allowed by the Hasse bound. Analogously, we call p a trailing prime if ap=⌊2p⌋. In this note, we study the frequency of champion and trailing primes for CM elliptic curves. Our main theorem is that for CM curves, both the champion primes and trailing primes have counting functions ∼23πx3/4/log⁡x, as x→∞. This confirms (in corrected form) a recent conjecture of James-Tran-Trinh-Wertheimer-Zantout.