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22
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
Uniform results for Serre’s theorem for elliptic curves
 MR 2189500 ↑1.5
"... A celebrated theorem of Serre from 1972 asserts that if E is an elliptic curve defined over Q and without complex multiplication, then its associated mod ℓ representation is surjective for all sufficiently large primes ℓ. In this paper we address the question of what sufficiently large means in Serr ..."
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Cited by 11 (3 self)
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A celebrated theorem of Serre from 1972 asserts that if E is an elliptic curve defined over Q and without complex multiplication, then its associated mod ℓ representation is surjective for all sufficiently large primes ℓ. In this paper we address the question of what sufficiently large means in Serre’s theorem. More precisely, we obtain a uniform version of Serre’s theorem for nonconstant elliptic curves defined over function fields, and a uniform version of Serre’s theorem for oneparameter families of elliptic curves defined over Q.
The square sieve and the Lang–Trotter conjecture
 Canadian Journal of Mathematics
, 2001
"... 1 Let E be an elliptic curve defined over Q and without complex multiplication. Let K be a fixed imaginary quadratic field. We find nontrivial upper bounds for the number of ordinary primes p ≤ x for which Q(πp) = K, where πp denotes the Frobenius endomorphism of E at p. More precisely, �under a ce ..."
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Cited by 9 (3 self)
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1 Let E be an elliptic curve defined over Q and without complex multiplication. Let K be a fixed imaginary quadratic field. We find nontrivial upper bounds for the number of ordinary primes p ≤ x for which Q(πp) = K, where πp denotes the Frobenius endomorphism of E at p. More precisely, �under a certain generalized Riemann hypothesis we show that this number is OE x 17 18 log x, and unconditionally. We also prove that the number we show that this number is OE,K � 13 x(log log x) 12 (log x) 25 24 of imaginary quadratic fields K, with − disc K ≤ x and of the form K = Q(πp), is ≫E log log log x for x ≥ x0(E). These results represent progress towards a 1976 LangTrotter conjecture. 1
Frobenius fields for elliptic curves
 Amer. J. Math
"... Let E/Q be an elliptic curve over the field of rational numbers, with End ¯ Q(E) = Z. Let K be a fixed imaginary quadratic field over Q, and x a positive real number. For each prime p of good reduction for E, let πp(E) be a root of the characteristic polynomial of the Frobenius endomorphism of E ov ..."
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Cited by 5 (3 self)
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Let E/Q be an elliptic curve over the field of rational numbers, with End ¯ Q(E) = Z. Let K be a fixed imaginary quadratic field over Q, and x a positive real number. For each prime p of good reduction for E, let πp(E) be a root of the characteristic polynomial of the Frobenius endomorphism of E over the finite field Fp. Let ΠE(K; x) be the number of primes p ≤ x such that the field extension Q(πp(E)) is the fixed imaginary quadratic field K. We present upper bounds for ΠE(K; x) obtained using two different approaches. The first one, inspired from work of Serre, is to consider the image of Frobenius in a mixed Galois representation associated to K and to the elliptic curve E. The second one, inspired from work of Cojocaru, Fouvry and Murty, is based on an application of the square sieve. The bounds obtained using the first approach are better, ΠE(K; x) ≪ x 4/5 /(log x) 1/5, and are the best known so far. The bounds obtained using the second approach are weaker, but are ∗ also affiliated with the Institute of Mathematics of the Romanian Academy 1 independent of the number field K, a property which is essential for other applications. All
COUNTING CONGRUENCE SUBGROUPS
"... Abstract. Let Γ denote the modular group SL(2, Z) and Cn(Γ) the number of congruence ..."
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Cited by 4 (2 self)
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Abstract. Let Γ denote the modular group SL(2, Z) and Cn(Γ) the number of congruence
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
Almost prime values of the order of elliptic curves over finite fields
, 2008
"... Abstract. Let E be an elliptic curve over Q without complex multiplication, and which is not isogenous to a curve with nontrivial rational torsion. For each prime p of good reduction, let E(Fp)  be the order of the group of points of the reduced curve over Fp. We prove in this paper that, under t ..."
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Cited by 2 (1 self)
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Abstract. Let E be an elliptic curve over Q without complex multiplication, and which is not isogenous to a curve with nontrivial rational torsion. For each prime p of good reduction, let E(Fp)  be the order of the group of points of the reduced curve over Fp. We prove in this paper that, under the GRH, there are at least 2.778Ctwin E x/(log x)2 primes p such that E(Fp)  has at most 8 prime factors, counted with multiplicity. This improves previous results of Steuding & Weng [18] and Murty & Miri [13]. This is also the first result where the dependence on the conjectural constant Ctwin E appearing in the twin prime conjecture for elliptic curves (also known as Koblitz’s conjecture) is made explicit. This is achieved by sieving a slightly different sequence than the one of [18] and [13]. By sieving the same sequence and using Selberg’s linear sieve, we can also improve the constant of Zywina [22] appearing in the upper bound for the number of primes p such that E(Fp)  is prime. Finally, we remark that our results still hold under an hypothesis weaker than the GRH. 1.
Distribution of Farey Fractions in Residue Classes and Lang–Trotter Conjectures on Average
"... We prove that the set of Farey fractions of order T, that is, the set {α/β ∈ Q: gcd(α, β) = 1, 1 � α, β � T}, is uniformly distributed in residue classes modulo a prime p provided T � p 1/2+ε for any fixed ε> 0. We apply this to obtain upper bounds for the Lang–Trotter conjectures on Frobenius trac ..."
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Cited by 1 (0 self)
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We prove that the set of Farey fractions of order T, that is, the set {α/β ∈ Q: gcd(α, β) = 1, 1 � α, β � T}, is uniformly distributed in residue classes modulo a prime p provided T � p 1/2+ε for any fixed ε> 0. We apply this to obtain upper bounds for the Lang–Trotter conjectures on Frobenius traces and Frobenius fields “on average ” over a oneparametric family of elliptic curves.
THE NUMBER OF HECKE EIGENVALUES OF SAME SIGNS
, 812
"... Abstract. We give the best possible lower bounds in order of magnitude for the number of positive and negative Hecke eigenvalues. This improves upon a recent work of Kohnen, Lau & Shparlinski. Also, we study an analogous problem for short intervals. 1. ..."
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Cited by 1 (1 self)
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Abstract. We give the best possible lower bounds in order of magnitude for the number of positive and negative Hecke eigenvalues. This improves upon a recent work of Kohnen, Lau & Shparlinski. Also, we study an analogous problem for short intervals. 1.