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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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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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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
Bounded gaps between primes with a given primitive root, II
"... Let m be a natural number, and letQ be a set containing at least exp(Cm) primes. We show that one can find infinitely many strings of m consecutive primes each of which has some q ∈ Q as a primitive root, all lying in an interval of length OQ(exp(C ′m)). This is a bounded gaps variant of a theorem o ..."
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Let m be a natural number, and letQ be a set containing at least exp(Cm) primes. We show that one can find infinitely many strings of m consecutive primes each of which has some q ∈ Q as a primitive root, all lying in an interval of length OQ(exp(C ′m)). This is a bounded gaps variant of a theorem of Gupta and Ram Murty. We also prove a result on an elliptic analogue of Artin’s conjecture. Let E/Q be an elliptic curve with an irrational 2torsion point. Assume GRH. Then for every m, there are infinitely many strings of m consecutive primes p for which E(Fp) is cyclic, all lying an interval of length OE(exp(C ′′m)). If E has CM, then the GRH assumption can be removed. Here C, C ′, and C ′ ′ are absolute constants.
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