Abstract
For an elliptic curve E/Q, we define an extremal prime for E to be a prime p of good reduction such that the trace of Frobenius of E at p is ±⌊2p⌋, i.e., maximal or minimal in the Hasse interval. Conditional on the Riemann Hypothesis for certain Hecke L-functions, we prove that if End(E)=OK, where K is an imaginary quadratic field of discriminant ≠−3,−4, then the number of extremal primes ≤X for E is asymptotic to X3/4/logX. We give heuristics for related conjectures.
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