Lang-trotter Revisited
نویسندگان
چکیده
Dedicated to the memory of Serge Lang Table of contents 0. Preface
منابع مشابه
Bounds for the Lang-trotter Conjectures
For a non-CM elliptic curve E/Q, Lang and Trotter made very deep conjectures concerning the number of primes p ≤ x for which ap(E) is a fixed integer (and for which the Frobenius field at p is a fixed imaginary quadratic field). Under GRH, we use a smoothed version of the Chebotarev density theorem to improve the best known Lang-Trotter upper bounds of Murty, Murty and Saradha, and Cojocaru and...
متن کاملTo appear in Acta Arithmetica. AVERAGE FROBENIUS DISTRIBUTIONS FOR ELLIPTIC CURVES WITH NONTRIVIAL RATIONAL TORSION
In this paper we consider the Lang-Trotter conjecture (Conjecture 1 below) for various families of elliptic curves with prescribed torsion structure. We prove that the Lang-Trotter conjecture holds in an average sense for these families of curves (see Theorem 3). Let E/Q denote an elliptic curve and let ∆E denote its discriminant. As usual, let ap(E) = p + 1 − #E(Fp). Then we have the following...
متن کاملAverage Frobenius Distributions for Elliptic Curves with 3-torsion
In this paper, we examine the Lang-Trotter conjecture for elliptic curves which possess rational 3-torsion points. We prove that if one averages over all such elliptic curves then one obtains an asymptotic similar to the one predicted by Lang and Trotter.
متن کاملA Remark on the Conjectures of Lang-trotter and Sato-tate on Average
We obtain new average results on the conjectures of Lang-Trotter and Sato-Tate about elliptic curves. Mathematics Subject Classification (2000): 11G05
متن کاملDistribution 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 ≥ p1/2+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.
متن کامل