Average Frobenius distribution of elliptic curves

James, Kevin; Yu, Gang

doi:10.4064/aa124-1-7

Cited by 11 publications

(8 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It is natural to try to combine our approach with recent work of James and Yu [33] which studies, on average over |a| A and |b| B, the number of primes p x for which p + 1 − E a,b (F p ) is a perfect k-th power. In some cases, it may be possible to lower the threshold on A and B.…”

Section: Further Applicationsmentioning

confidence: 99%

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

Banks

Shparlinski

2009

Isr. J. Math.

View full text Add to dashboard Cite

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.

show abstract

Section: Further Applicationsmentioning

confidence: 99%

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

Banks

Shparlinski

2009

Isr. J. Math.

View full text Add to dashboard Cite

show abstract

“…Several more interesting questions on elliptic curves have been studied "on average" for similar families of curves in [1,3,5,7,8,18,20,21].…”

Section: Applicationsmentioning

confidence: 99%

Distribution of modular inverses and multiples of small integers and the Sato-Tate conjecture on average

Shparlinski¹

2008

Michigan Math. J.

View full text Add to dashboard Cite

We show that, for sufficiently large integers m and X, for almost all a = 1, . . . , m the ratios a/x and the products ax, where |x| X, are very uniformly distributed in the residue ring modulo m. This extends some recent results of Garaev and Karatsuba. We apply this result to show that on average over r and s, ranging over relatively short intervals, the distribution of Kloosterman sums K r,s (p) = p−1 n=1 exp(2πi(rn + sn −1 )/p), for primes p T is in accordance with the Sato-Tate conjecture.

show abstract

“…8. Counting elliptic curves whose reductions are isomorphic over F p In this section, we will prove the estimates of Lemma 1 by extending standard character sum techniques as in [8] or [14]. The main difference is that we have to extend the domain of our characters to O K .…”

mentioning

confidence: 99%

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

James

Smith

2011

Math. Proc. Camb. Phil. Soc.

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Average Frobenius distribution of elliptic curves

Cited by 11 publications

References 10 publications

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

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

Distribution of modular inverses and multiples of small integers and the Sato-Tate conjecture on average

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

Contact Info

Product

Resources

About