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14
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
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 3 (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.
Reductions of an elliptic curve with almost prime orders
"... 1 Let E be an elliptic curve over Q. For a prime p of good reduction, let Ep be the reduction of E modulo p. We investigate Koblitz’s Conjecture about the number of primes p for which Ep(Fp) has prime order. More precisely, our main result is that if E is with Complex Multiplication, then there exis ..."
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Cited by 3 (0 self)
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1 Let E be an elliptic curve over Q. For a prime p of good reduction, let Ep be the reduction of E modulo p. We investigate Koblitz’s Conjecture about the number of primes p for which Ep(Fp) has prime order. More precisely, our main result is that if E is with Complex Multiplication, then there exist infinitely many primes p for which #Ep(Fp) has at most 5 prime factors. We also obtain upper bounds for the number of primes p ≤ x for which #Ep(Fp) is a prime. 1
On the greatest prime divisor of Np
 J. Ramanujan Math. Soc
"... Let E be an elliptic curve defined over Q. For any prime p of good reduction, let Ep be the reduction of E mod p. Denote by Np the cardinality of Ep(Fp), where Fp is the finite field of p elements. Let P (Np) be the greatest prime divisor of Np. We prove that if E has CM then for all but o(x / log x ..."
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Cited by 2 (1 self)
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Let E be an elliptic curve defined over Q. For any prime p of good reduction, let Ep be the reduction of E mod p. Denote by Np the cardinality of Ep(Fp), where Fp is the finite field of p elements. Let P (Np) be the greatest prime divisor of Np. We prove that if E has CM then for all but o(x / log x) of primes p ≤ x, P (Np)> p ϑ(p), where ϑ(p) is any function of p such that ϑ(p) → 0 as p → ∞. Moreover we show that for such E there is a positive proportion of primes p ≤ x for which P (Np)> p ϑ, where ϑ is any number less than ϑ0 = 1 − 1 2 prove the following. Let Γ be a free subgroup of rank r ≥ 2 of the group of rational points E(Q), and Γp be the reduction of Γ mod p, then for a positive proportion of primes p ≤ x, we have where ɛ> 0. e− 1 4 = 0.6105 · · ·. As an application of this result we Γp > p ϑ0−ɛ Keywords: Reduction mod p of elliptic curves, Elliptic curves over finite fields, BrunTitchmarsh inequality in number fields, BombieriVinogradov theorem in number fields, Abelian extensions of imaginary quadratic number fields. 2000 Mathematics Subject Classification. Primary 11G20, Secondary 11N37. 1
Integral points on punctured abelian surfaces. in Algorithmic number theory
 Lecture Notes in Comput. Sci. 2369
, 2002
"... Abstract. We study the density of integral points on punctured abelian surfaces. Linear growth rates are observed experimentally. 1 ..."
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Abstract. We study the density of integral points on punctured abelian surfaces. Linear growth rates are observed experimentally. 1
Towards LangTrotter for Elliptic Curves over Function Fields
"... Introduction Let K be a global field of char p and let F q K denote the algebraic closure of F p in K. We fix an elliptic curve E/K with nonconstant jinvariant and a torsionfree subgroup E(K) of rank r > 0. We write V for the open set of places v of K such that the special fiber E v is a ..."
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Introduction Let K be a global field of char p and let F q K denote the algebraic closure of F p in K. We fix an elliptic curve E/K with nonconstant jinvariant and a torsionfree subgroup E(K) of rank r > 0. We write V for the open set of places v of K such that the special fiber E v is an elliptic curve and, for v in V , we let # v E v (k v ) be the image of # under reduction modulo v, where k v is the residue field of K at v. We fix a finite set of (rational) prime numbers S which is large enough to include the exceptional primes which we will define explicitly in section 2.4 and section 3), and we let G(#, S) denote the subset of v V such that # v contains the primetoS part of E v (k v ). For every n > 0, we write V n for the subset of v V such that deg(v) = n and let G n (#, S) = V n G(#, S). Theorem 1. Suppose r 6. There exists constants a, b satisfying 0 1 and depending only on r and S such that, for each n 1 there exists # n (#, S), depending on r, S an
Primitive points on CM elliptic curves
, 2011
"... Let E be a CM elliptic curve over a number field F, with CM by the full ring of integers OK of an imaginary quadratic field K of class number 1, of rank ≥ 1 and let a ∈ E(F) be a point of infinite order. In this paper, under GRH, we find the density of the finite places ℘ of F of good reduction for ..."
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Let E be a CM elliptic curve over a number field F, with CM by the full ring of integers OK of an imaginary quadratic field K of class number 1, of rank ≥ 1 and let a ∈ E(F) be a point of infinite order. In this paper, under GRH, we find the density of the finite places ℘ of F of good reduction for E, such that ℘ splits in KF, and N ℘ ≤ x, for which Ē(F℘) is generated by a (modulo ℘).
Primitive points on CM Drinfeld modules of rank 2
, 2011
"... Let A = Fq[T], where Fq is a finite field, let Q = Fq(T), and let F be a finite extension of Q. Let φ be a CM Drinfeld Amodule over F of rank 2, with CM by the full ring of integers OK of an imaginary quadratic extension K/Q of class number 1. If ℘ is a prime of F, we denote by F℘ the residue field ..."
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Let A = Fq[T], where Fq is a finite field, let Q = Fq(T), and let F be a finite extension of Q. Let φ be a CM Drinfeld Amodule over F of rank 2, with CM by the full ring of integers OK of an imaginary quadratic extension K/Q of class number 1. If ℘ is a prime of F, we denote by F℘ the residue field at ℘. If φ has good reduction at ℘, let φ ⊗ F ℘ denote the reduction of φ at ℘ and let φ(F℘): = (φ ⊗ F℘)(F℘). Assume that a ∈ F is point of infinite order for φ. In this paper we obtain an asymptotic formula for the number of primes ℘ of F of degree x, which split in KF, for which φ(F℘) is generated by a (modulo ℘) as an Amodule (given by the action of φ).
9 SpringerVerlag 1984 A remark on Artin's conjecture
"... A famous conjecture of E. Artin [t] states that for any integer a 4 = + _ I or a perfect square, there are infinitely many primes p for which a is a primitive root (modp). This conjecture was shown to be true if one assumes the generalized Riemann hypothesis by Hooley [5]. The purpose of this note i ..."
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A famous conjecture of E. Artin [t] states that for any integer a 4 = + _ I or a perfect square, there are infinitely many primes p for which a is a primitive root (modp). This conjecture was shown to be true if one assumes the generalized Riemann hypothesis by Hooley [5]. The purpose of this note is to exhibit a finite set S such that for some a eS, a is a primitive root (modp) for an infinity ofprimesp. To this end, let q, r and s denote three distinct primes. Define the following set: