The Square Sieve and the Lang–Trotter Conjecture

Cojocaru, Alina Carmen; Fouvry, Étienne; Murty, M. Ram

doi:10.4153/cjm-2005-045-7

Cited by 27 publications

(40 citation statements)

References 6 publications

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“…These results improve upon earlier work of J.-P. Serre [19] and A. C. Cojocaru,É. Fouvry and M. R. Murty [6]. In the case of a = 0, better bounds are known: unconditionally, N. Elkies [10,11] has shown that, if E is without CM, there are infinitely primes p such that a p = 0 (for explicit lower bounds, see [11] and [13]).…”

Section: ) E(t) : Y 2 = X 3 + A(t)x + B(t) Over Q(t) Where A(t) B(supporting

confidence: 82%

Distribution of Farey fractions in residue classes and Lang–Trotter conjectures on average

Cojocaru

Shparlinski

2008

Proc. Amer. Math. Soc.

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Abstract. 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 one-parametric family of elliptic curves.

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Section: ) E(t) : Y 2 = X 3 + A(t)x + B(t) Over Q(t) Where A(t) B(supporting

confidence: 82%

Distribution of Farey fractions in residue classes and Lang–Trotter conjectures on average

Cojocaru

Shparlinski

2008

Proc. Amer. Math. Soc.

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“…In particular, using [CoFoMu,theorem 2·4] together with [CoFoMu,lemma 2·7] (see also [CoFoMu,lemma 2·8]), we obtain: LEMMA 3. For any elliptic curve E/Q without complex multiplication and any sufficiently large positive real number x, the estimate…”

Section: Preliminariesmentioning

confidence: 99%

Pseudoprime reductions of elliptic curves

Cojocaru

Luca

Shparlinski

2008

Math. Proc. Camb. Phil. Soc.

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“…We first recall some group theoretical results from [2]. We will then apply versions of the Chebotarev density theorem (under GRH or without) and obtain our results.…”

Section: Square-sieve Approachmentioning

confidence: 99%

Applications of the Square Sieve to a Conjecture of Lang and Trotter for a Pair of Elliptic Curves Over the Rationals

Baier

Patankar

2018

Springer Proceedings in Mathematics &Amp; Statistics

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Let E be an elliptic curve over Q. Let p be a prime of good reduction for E. Then, for a prime p = ℓ, the Frobenius automorphism associated to p (unique up to conjugation) acts on the ℓ-adic Tate module of E. The characteristic polynomial of the Frobenius automorphism has rational integer coefficients and is independent of ℓ. Its splitting field is called the Frobenius field of E at p. Let E 1 and E 2 be two elliptic curves defined over Q that are non-isogenous over Q and also without complex multiplication over Q. In analogy with the well-known Lang-Trotter conjecture for a single elliptic curve, it is natural to consider the asymptotic behaviour of the function that counts the number of primes p ≤ x such that the Frobenius fields of E 1 and E 2 at p coincide. In this short note, using Heath-Brown's square sieve, we provide both conditional (upon the Generalized Riemann Hypothesis) and unconditional upper bounds. To V. Kumar Murty: On the occasion of his sixtieth birthdayConjecture 5. 1 [5, Conjecture 1] (Pair of elliptic curves and equal Frobenius fields) Let E 1 and E 2 be two elliptic curves over the rationals, and both without complex multiplication over Q. Then, E 1 is not isogenous to E 2 over Q if and only ifRemark 3. Suppose E 1 and E 2 are isogenous over Q, hence over some number field L. By extending L if necessary, we can assume that L is Galois over Q. Then, for any prime p ∤ N 1 N 2 and that splits in L, a p (E 1 ) = a p (E 2 ). Thus, from the Chebotarev density theorem, it follows that S(E 1 , E 2 ; x) grows at least as much as

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The Square Sieve and the Lang–Trotter Conjecture

Cited by 27 publications

References 6 publications

Distribution of Farey fractions in residue classes and Lang–Trotter conjectures on average

Distribution of Farey fractions in residue classes and Lang–Trotter conjectures on average

Pseudoprime reductions of elliptic curves

Applications of the Square Sieve to a Conjecture of Lang and Trotter for a Pair of Elliptic Curves Over the Rationals

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