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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 27 (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
Sato–Tate, cyclicity, and divisibility statistics on average for elliptic curves of small height
, 2008
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On the exponent of the group of points on elliptic curves in extension fields
 Intern. Math. Research Notices
"... Let E be an elliptic curve defined over Fq, a finite field of q elements. Furthermore, we consider ..."
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Cited by 9 (5 self)
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Let E be an elliptic curve defined over Fq, a finite field of q elements. Furthermore, we consider
Cyclicity of CM elliptic curves modulo p
 TRANSACTIONS OF AMERICAN MATHEMATICAL SOCIETY
, 2003
"... Let E be an elliptic curve defined over Q and with complex multiplication. For a prime p of good reduction, let E be the reduction of E modulo p. We find the density of the primes p ≤ x for which E(Fp) is a cyclic group. An asymptotic formula for these primes had been obtained conditionally by J.P. ..."
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Cited by 7 (1 self)
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Let E be an elliptic curve defined over Q and with complex multiplication. For a prime p of good reduction, let E be the reduction of E modulo p. We find the density of the primes p ≤ x for which E(Fp) is a cyclic group. An asymptotic formula for these primes had been obtained conditionally by J.P. Serre in 1976, and unconditionally by Ram Murty in 1979. The aim of this paper is to give a new simpler unconditional proof of this asymptotic formula, and also to provide explicit error terms in the formula.
Small exponent point groups on elliptic curves
 JOURNAL DE THÉORIE DES NOMBRES DE BORDEAUX 18 (2006), 471–476
, 2006
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Centre de Recherches Mathématiques CRM Proceedings and Lecture Notes Volume??, 2004 Questions About the Reductions Modulo Primes of an Elliptic Curve
"... This is largely a survey paper in which we discuss new and old problems about the reductions Ep modulo primes p of a fixed elliptic curve E defined over the field of rational numbers. We investigate, in particular, how the “noncyclic ” part of the group of points of Ep is distributed, thus making pr ..."
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This is largely a survey paper in which we discuss new and old problems about the reductions Ep modulo primes p of a fixed elliptic curve E defined over the field of rational numbers. We investigate, in particular, how the “noncyclic ” part of the group of points of Ep is distributed, thus making progress toward a conjecture of