Elliptic curves with a given number of points over finite fields

Abstract

Abstract. Given an elliptic curve E and a positive integer N , we consider the problem of counting the number of primes p for which the reduction of E modulo p possesses exactly N points over F p . On average (over a family of elliptic curves), we show bounds that are significantly better than what is trivially obtained by the Hasse bound. Under some additional hypotheses, including a conjecture concerning the short interval distribution of primes in arithmetic progressions, we obtain an asymptotic formula for the average.

A Cohen-Lenstra phenomenon for elliptic curves

“…As with our study of M E (N ) in [13], the restriction imposed by the Hasse bound means that any prime counted by M E (G) must lie in a very short interval near N = #G = N 2 1 N 2 . In particular, all of the primes are of size N , lying in an interval of length 4 √ N .…”

“…Apart from the 'arithmetic factor' K(N )N/ϕ(N ), the above result agrees with a naïve probabilistic model for M E (N ) where one supposes that the values #E p (F p ) are uniformly distributed in the interval (N − , N + ). This is explained in detail in [13]. The occurrence of the weight ϕ(N ) appearing in the denominator on the right-hand side of (2) suggested to the authors that perhaps this is another example of phenomena which are governed by the Cohen-Lenstra Heuristics [7,8], which predict that random groups occur with probability inversely proportional to the size of their automorphism groups.…”

“…Given a prime p where E has good reduction, we consider the reduced curve, which we denote by E p . In previous work [13], we studied the arithmetic function…”

“…In [13], we studied the average behavior of M E (N ) taken over a family of elliptic curves. More precisely, given integers a, b, we let E a,b denote the elliptic curve given by the Weierstrass equation y 2 = x 3 + ax + b; we let C denote the multiset defined by…”

“…Even the Riemann Hypothesis does not guarantee the existence of a prime in such a short interval. Thus, our work here, as in [13], requires a conjecture (Conjecture 17) concerning the distribution of primes of size X in intervals of length X η . The case η = 1 corresponds to the classical Barban-Davenport-Halberstam Theorem.…”