Average Frobenius distribution for elliptic curves defined over finite Galois extensions of the rationals

Abstract

Abstract. Let K be a fixed number field, assumed to be Galois over Q. Let r and f be fixed integers with f positive. Given an elliptic curve E, defined over K, we consider the problem of counting the number of degree f prime ideals of K with trace of Frobenius equal to r. Except in the case f = 2, we show that "on average," the number of such prime ideals with norm less than or equal to x satisfies an asymptotic identity that is in accordance with standard heuristics. This work is related to the classical Lang-Trotter conjecture and extends the work of several authors.

Average Frobenius distribution for the degree two primes of a number field

“…We refer to this expression as the average order of π r, f E (x) over C. In order to provide support for Conjecture 1, several authors have proven results about the average order of π r, f E (x) over various families of elliptic curves. See [1,2,4,5,6,9,11,12]. In each case, the results have been found to be in accordance with Conjecture 1.…”

“…We note that the required growth rate min(C ) ≥ √ x for Theorems 4, 7, 8 can be relaxed to min(C ) ≥ √ x/ log x. The key piece of information necessary for making the improvement is to observe that ( 14) (see page 9) can be improved to H(T ) ≪ T 2 log T , where H(T ) is the sum defined by (12). Indeed, the techniques used to prove Propositions 16 and 17 below can be used to show that H(T ) is asymptotic to some constant multiple of T 2 log T .…”

“…Unfortunately, at present, it is necessary to take C to be a family of curves that must "grow" at some specified rate with respect to the variable x. The authors of the works [9,1,12] put a great deal of effort into keeping the average as "short" as possible. This seems like a difficult task for the cases of the average Lang-Trotter problem that we will consider here.…”

“…It turns out that their methods were actually sufficient to handle some non-Abelian Galois extensions as well in the case when f = 2. In [12], the results of [4] were extended to the setting of any Galois extension K /Q except in the case that f = 2. In this paper, we consider the case when f = 2 and K /Q is an arbitrary Galois extension.…”

Distribution of Squarefree Values of Sequences Associated with Elliptic Curves

“…This has been done by various authors originating with the work of Fouvry and Murty [14] for the number of supersingular primes (i.e., the fixed trace Lang-Trotter Conjecture for t = 0). See [10], [11], [17], [4], [18], and [5] for other averages regarding the fixed trace Lang-Trotter Conjecture. The average order for the Koblitz Conjecture was considered in [2].…”