Isomorphism classes of elliptic curves over a finite field in some thin families

Abstract

For a prime p and a given square box, B, we consider all elliptic curves E r,s : Y 2 = X 3 + rX + s defined over a field F p of p elements with coefficients (r, s) ∈ B. We obtain a nontrivial upper bound for the number of such curves which are isomorphic to a given one over F p , in terms of the size of B. We also give an optimal lower bound on the number of distinct isomorphic classes represented.

On Bilinear Exponential and Character Sums With Reciprocals of Polynomials

“…Our results are closely related to various questions about congruences with reciprocals and multiplicative congruences with polynomials of the types considered in [4] and [6,7,9,10,11,18], respectively. We also use one of the results from [8] which we extend to more generic settings; we hope this may find further applications.…”

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. Let I (R, S; M) be the number of nonisomorphic curves E r,s over F p with coefficients r, s in a box (r, s)…”

“…The method is related to several recent results on the distribution of solutions to polynomial congruences in very small boxes, see [30][31][32] for further references. Both the bound of [32] on I (R, S; M) and some estimates of [30][31][32] on the density of solutions of polynomial congruences have been improved and generalised in [28].…”

“…Using the method of [30], which in turn has been further developed in [31], in [32] a lower bound of an asymptotically correct order of magnitude is given on I (R, S; M). The method is related to several recent results on the distribution of solutions to polynomial congruences in very small boxes, see [30][31][32] for further references.…”

Additive Combinatorics over Finite Fields: New Results and Applications

“…Several more results of this type, that apply to large sets and are also based on the method of Garaev [96] have been given by Cilleruelo, Garaev, Ostafe & Shparlinski [68]. The ideas of [68] have found application in the study of the distribution of some families of algebraic curves in isomorphism classes, see [63,69,70].…”