The distribution of the number of points modulo an integer on elliptic curves over finite fields

Castryck, Wouter; Hubrechts, Hendrik

doi:10.1007/s11139-012-9444-0

Cited by 15 publications

(21 citation statements)

References 22 publications

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“…Then Frob p (E) corresponds to a pair (F E , T E ) where F E is a conjugacy class in GL 2 (Z/N ′ Z) of determinant p and T E ∈ (Z/p e Z) * . The following equidistribution theorem was proved by Castryck and Hubrechts in [8]. We state their result only when N ≤ p 1/4 .…”

Section: Links With Equidistribution In Groups Of Matricesmentioning

confidence: 91%

Sums of Euler products and statistics of elliptic curves

2016

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Abstract. We present several results related to statistics for elliptic curves over a finite field F p as corollaries of a general theorem about averages of Euler products that we demonstrate. In this general framework, we can reprove known results such as the average LangTrotter conjecture, the average Koblitz conjecture, and the vertical Sato-Tate conjecture, even for very short intervals, not accessible by previous methods. We also compute statistics for new questions, such as the problem of amicable pairs and aliquot cycles, first introduced by Silverman and Stange. Our technique is rather flexible and should be easily applicable to a wide range of similar problems. The starting point of our results is a theorem of Gekeler which gives a reinterpretation of Deuring's theorem in terms of an Euler product involving random matrices, thus making a direct connection between the (conjectural) horizontal distributions and the vertical distributions. Our main technical result then shows that, under certain conditions, a weighted average of Euler products is asymptotic to the Euler product of the average factors.

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Section: Links With Equidistribution In Groups Of Matricesmentioning

confidence: 91%

Sums of Euler products and statistics of elliptic curves

2016

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“…To proceed, we again divide the δ ∈ A(d) into two classes. This time, we say δ is good for d if δ d(log x) 4 and bad otherwise. Using π(x; δ, 1) x/ δ and #A(d) τ (d 2 ), we see that the bad δ make a contribution to the double sum that is…”

Section: •3 Exploiting Symmetrymentioning

confidence: 99%

A Titchmarsh divisor problem for elliptic curves

Pollack¹

2015

Math. Proc. Camb. Phil. Soc.

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Let E/Q be an elliptic curve with complex multiplication. We study the average size of τ(#E(Fp)) as p varies over primes of good ordinary reduction. We work out in detail the case of E: y2 = x3 − x, where we prove that $$\begin{equation} \sum_{\substack{p \leq x \\p \equiv 1\pmod{4}}} \tau(\#E({\bf{F}}_p)) \sim \left(\frac{5\pi}{16} \prod_{p > 2} \frac{p^4-\chi(p)}{p^2(p^2-1)}\right)x, \quad\text{as $x\to\infty$}. \end{equation}$$ Here χ is the nontrivial Dirichlet character modulo 4. The proof uses number field analogues of the Brun–Titchmarsh and Bombieri–Vinogradov theorems, along with a theorem of Wirsing on mean values of nonnegative multiplicative functions.Now suppose that E/Q is a non-CM elliptic curve. We conjecture that the sum of τ(#E(Fp)), taken over p ⩽ x of good reduction, is ~cEx for some cE > 0, and we give a heuristic argument suggesting the precise value of cE. Assuming the Generalized Riemann Hypothesis for Dedekind zeta functions, we prove that this sum is ≍Ex. The proof uses combinatorial ideas of Erdős.

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“…There is an extensive literature where the authors study the distribution of isomorphism and isogeny classes of elliptic curves in these and several other related families, see [4,27,[62][63][64]83] and references therein.…”

Section: Isogeny and Isomorphism Classes In Various Familiesmentioning

confidence: 99%