The authors conjecture a positive natural density for primes p of good reduction such that |E1(F_p)| and |E2(F_p)| are coprime for non-CM elliptic curves E1, E2 over Q, prove the associated series converges with an almost Euler product, and establish a moments result for Serre pairs.
504, Springer-Verlag, Berlin-New York, 1976, Distribution of Frobenius automorphisms in GL2-extensions of the rational numbers
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On the Density of Coprime Reductions of Elliptic Curves
The authors conjecture a positive natural density for primes p of good reduction such that |E1(F_p)| and |E2(F_p)| are coprime for non-CM elliptic curves E1, E2 over Q, prove the associated series converges with an almost Euler product, and establish a moments result for Serre pairs.