Sato-Tate, cyclicity, and divisibility statistics on average for elliptic curves of small height

Abstract

We obtain asymptotic formulae for the number of primes p x for which the reduction modulo p of the elliptic curvesatisfies certain "natural" properties, on average over integers a and b such that |a| A and |b| B, where A and B are small relative to x. More precisely, we investigate behavior with respect to the Sato-Tate conjecture, cyclicity, and divisibility of the number of points by a fixed integer m.

Elliptic Curves over Finite Fields: Number Theoretic and Cryptographic Aspects

“…On average over p, such results are established for even smaller boxes [18] and also for more general families of elliptic curves [137,138]. In [32] much smaller boxes have been considered.…”

“…Lenstra (4) with fixed ε > 0, using the exponential sum technique, Fouvry and Murty [72] have obtained an asymptotic formula for every pair (a, b) ∈ F 2 p with 4a 3 + 27b 2 = 0. In [18], using bounds of multiplicative character sums, for almost all (a, b), this condition (4) has been relaxed as…”

“…Square-free values of t 2 p −4p have been studied by David and Jiménez Urroz [49] (using some results of [15,18]) on average over primes p and also over the family of curves E a,b with coefficients in |a| ≤ A, |b| ≤ B.…”

Artin's Primitive Root Conjecture – A Survey

“…Recently, work of Banks and Shparlinski [37], combined with work of N. Jones [247], shows that, on average over a sufficiently large two-parameter family of elliptic curves E over Q, the cyclicity asymptotic formula #{p ≤ x : E(F p ) is cyclic} ∼ C E π(x) holds without any additional hypothesis. However, such an average result does not imply that the formula holds for any curve E.…”

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

“…Fouvry and R. M. Murty [13], and C. David and F. Pappalardi [8], have considered the average of Π E (a, x) for the family of curves Y 2 = X 3 + uX + v where the integers u and v satisfy the inequalities |u| ≤ U , |v| ≤ V ; they have shown that if UV ≥ x 3/2+ε and min{U, V } ≥ x 1/2+ε for some fixed positive ε > 0, then, "on average", the Lang-Trotter conjecture holds for such curves. This result has been extended in various directions [1,2,3,4,9,14,15,16]. A. C. Cojocaru and C. Hall [7] have recently considered the one parametric family of curves of the form (1) and established an improved upper bound on the average value of Π E (a, x) over curves of such families when the parameter t runs through the elements of F(T ) with T of the same order of magnitude as x.…”