Abstract
Let E be an elliptic curve defined over the rationals. For any prime p of good reduction, let Ep be the elliptic curve over Fp obtained by reducing E mod p. Let ap(E) be the trace of the Frobenius morphism of E/Fp. Then, #E(Fp) = p+1−ap(E), and |ap(E)| ≤ 2√p. The case where ap(E) = 0 corresponds to supersingular reduction mod p. For a fixed r ∈ Z, what can be said about the number of primes p such that ap(E) = r? If E has complex multiplication, Deuring showed that half of the primes are primes of supersingular reduction (see [3]). More precisely, let
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