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Cyclicity of elliptic curves modulo p and elliptic curve analogues of Linnik’s problem
, 2001
"... 1 Let E be an elliptic curve defined over Q and of conductor N. For a prime p ∤ N, we denote by E the reduction of E modulo p. We obtain an asymptotic formula for the number of primes p ≤ x for which E(Fp) is cyclic, assuming a certain generalized Riemann hypothesis. The error terms that we get are ..."
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Cited by 14 (3 self)
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1 Let E be an elliptic curve defined over Q and of conductor N. For a prime p ∤ N, we denote by E the reduction of E modulo p. We obtain an asymptotic formula for the number of primes p ≤ x for which E(Fp) is cyclic, assuming a certain generalized Riemann hypothesis. The error terms that we get are substantial improvements of earlier work of J.P. Serre and M. Ram Murty. We also consider the problem of finding the size of the smallest prime p = pE for which the group E(Fp) is cyclic and we show that, under the generalized Riemann hypothesis, pE = O � (log N) 4+ε � if E is without complex multiplication, and pE = O � (log N) 2+ε � if E is with complex multiplication, for any 0 < ε < 1. 1
Averages of elliptic curve constants
, 711
"... We compute the averages over elliptic curves of the constants occurring in the LangTrotter conjecture, the Koblitz conjecture, and the cyclicity conjecture. The results obtained confirm the consistency of these conjectures with the corresponding “theorems on average ” obtained recently by various a ..."
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Cited by 4 (0 self)
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We compute the averages over elliptic curves of the constants occurring in the LangTrotter conjecture, the Koblitz conjecture, and the cyclicity conjecture. The results obtained confirm the consistency of these conjectures with the corresponding “theorems on average ” obtained recently by various authors. 1
THE AVERAGE EXPONENT OF ELLIPTIC CURVES MODULO p
"... Abstract. Let E be an elliptic curve defined over Q. For a prime p of good reduction for E, denote by ep the exponent of the reduction of E modulo p. Under GRH, we prove that there is a constant CE ∈ (0,1) such that 1 π(x) p�x ep = 1 ..."
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Abstract. Let E be an elliptic curve defined over Q. For a prime p of good reduction for E, denote by ep the exponent of the reduction of E modulo p. Under GRH, we prove that there is a constant CE ∈ (0,1) such that 1 π(x) p�x ep = 1