Almost all reductions modulo p of an elliptic curve have a large exponent

2003 ◽  
Vol 337 (11) ◽  
pp. 689-692 ◽  
Author(s):  
William Duke
Keyword(s):  
Elliptic Curve ◽  
Large Exponent ◽  
Modulo P ◽  
Almost All

scholarly journals On the distribution of Atkin and Elkies primes for reductions of elliptic curves on average

2015 ◽  
Vol 18 (1) ◽  
pp. 308-322 ◽  
Author(s):  
Igor E. Shparlinski ◽  
Andrew V. Sutherland
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Riemann Hypothesis ◽  
Complex Multiplication ◽  
Log P ◽  
Point Counting ◽  
Generalized Riemann Hypothesis ◽  
Expected Time ◽  
Almost All ◽  
Counting Algorithm

For an elliptic curve$E/\mathbb{Q}$without complex multiplication we study the distribution of Atkin and Elkies primes$\ell$, on average, over all good reductions of$E$modulo primes$p$. We show that, under the generalized Riemann hypothesis, for almost all primes$p$there are enough small Elkies primes$\ell$to ensure that the Schoof–Elkies–Atkin point-counting algorithm runs in$(\log p)^{4+o(1)}$expected time.

scholarly journals POWER RESIDUES OF FOURIER COEFFICIENTS OF ELLIPTIC CURVES WITH COMPLEX MULTIPLICATION

2009 ◽  
Vol 05 (01) ◽  
pp. 109-124
Author(s):  
TOM WESTON ◽  
ELENA ZAUROVA
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Fourier Coefficient ◽  
Fourier Coefficients ◽  
Complex Multiplication ◽  
Roots Of Unity ◽  
Modulo P

Fix m greater than one and let E be an elliptic curve over Q with complex multiplication. We formulate conjectures on the density of primes p (congruent to one modulo m) for which the pth Fourier coefficient of E is an mth power modulo p; often these densities differ from the naive expectation of 1/m. We also prove our conjectures for m dividing the number of roots of unity lying in the CM field of E; the most involved case is m = 4 and complex multiplication by Q(i).

On the Local Structure of Mahler Systems

2020 ◽  
Author(s):  
Julien Roques
Keyword(s):  
Differential Equations ◽  
Local Structure ◽  
Alternative Proof ◽  
Reduction Modulo ◽  
Modulo P ◽  
Almost All ◽  
The Given ◽  
Algebraic Solutions

Abstract This paper is a 1st step in the direction of a better understanding of the structure of the so-called Mahler systems: we classify these systems over the field $\mathscr{H}$ of Hahn series over $\overline{{\mathbb{Q}}}$ and with value group ${\mathbb{Q}}$. As an application of (a variant of) our main result, we give an alternative proof of the following fact: if, for almost all primes $p$, the reduction modulo $p$ of a given Mahler equation with coefficients in ${\mathbb{Q}}(z)$ has a full set of algebraic solutions over $\mathbb{F}_{p}(z)$, then the given equation has a full set of solutions in $\overline{{\mathbb{Q}}}(z)$ (this is analogous to Grothendieck’s conjecture for differential equations).

scholarly journals Algebraic solutions of differential equations over ℙ1 −{0,1,∞}

2018 ◽  
Vol 14 (05) ◽  
pp. 1427-1457
Author(s):  
Yunqing Tang
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Elliptic Curve ◽  
Convergence Condition ◽  
Differential Galois Group ◽  
Reduction Modulo ◽  
Local Monodromy ◽  
Monodromy Groups ◽  
Almost All ◽  
Algebraic Solutions

The Grothendieck–Katz [Formula: see text]-curvature conjecture predicts that an arithmetic differential equation whose reduction modulo [Formula: see text] has vanishing [Formula: see text]-curvatures for almost all [Formula: see text] has finite monodromy. It is known that it suffices to prove the conjecture for differential equations on [Formula: see text] We prove a variant of this conjecture for [Formula: see text] which asserts that if the equation satisfies a certain convergence condition for all [Formula: see text] then its monodromy is trivial. For those [Formula: see text] for which the [Formula: see text]-curvature makes sense, its vanishing implies our condition. We deduce from this a description of the differential Galois group of the equation in terms of [Formula: see text]-curvatures and certain local monodromy groups. We also prove similar variants of the [Formula: see text]-curvature conjecture for an elliptic curve with [Formula: see text]-invariant [Formula: see text] minus its identity and for [Formula: see text].

scholarly journals REDUCTION modp OF SUBGROUPS OF THE MORDELL–WEIL GROUP OF AN ELLIPTIC CURVE

2009 ◽  
Vol 05 (03) ◽  
pp. 465-487 ◽  
Author(s):  
AMIR AKBARY ◽  
V. KUMAR MURTY
Keyword(s):  
Finite Field ◽  
Elliptic Curve ◽  
Free Subgroup ◽  
Good Reduction ◽  
P Elements ◽  
Density Zero ◽  
Weil Group ◽  
Modulo P

Let E be an elliptic curve defined over ℚ. Let Γ be a free subgroup of rank r of E(ℚ). For any prime p of good reduction, let Γp be the reduction of Γ modulo p and Ep be the reduction of E modulo p. We prove that if E has CM then for all but o(x/ log x) of primes p ≤ x, [Formula: see text] where ∊(p) is any function of p such that ∊(p) → 0 as p → ∞. This is a consequence of two other results. Denote by Np the cardinality of Ep(𝔽p), where 𝔽p is a finite field of p elements. Then for any δ > 0, the set of primes p for which Np has a divisor in the range (pδ - ∊(p), pδ + ∊(p)) has density zero. Moreover, the set of primes p for which [Formula: see text] has density zero.

scholarly journals Finding elliptic curves with a subgroup of prescribed size

2016 ◽  
Vol 13 (01) ◽  
pp. 133-152
Author(s):  
Igor E. Shparlinski ◽  
Andrew V. Sutherland
Keyword(s):  
Positive Integer ◽  
Finite Field ◽  
Elliptic Curve ◽  
Elliptic Curves ◽  
Riemann Hypothesis ◽  
Time Complexity ◽  
Rational Functions ◽  
Deterministic Algorithm ◽  
Probabilistic Algorithm ◽  
Almost All

Assuming the Generalized Riemann Hypothesis, we design a deterministic algorithm that, given a prime [Formula: see text] and positive integer [Formula: see text], outputs an elliptic curve [Formula: see text] over the finite field [Formula: see text] for which the cardinality of [Formula: see text] is divisible by [Formula: see text]. The running time of the algorithm is [Formula: see text], and this leads to more efficient constructions of rational functions over [Formula: see text] whose image is small relative to [Formula: see text]. We also give an unconditional version of the algorithm that works for almost all primes [Formula: see text], and give a probabilistic algorithm with subexponential time complexity.

scholarly journals The Frequency of Elliptic Curve Groups over Prime Finite Fields

2016 ◽  
Vol 68 (4) ◽  
pp. 721-761 ◽  
Author(s):  
Vorrapan Chandee ◽  
Chantal David ◽  
Dimitris Koukoulopoulos ◽  
Ethan Smith
Keyword(s):  
Asymptotic Formula ◽  
Finite Field ◽  
Elliptic Curve ◽  
Elliptic Curves ◽  
Finite Fields ◽  
Arithmetic Progressions ◽  
Constant Multiple ◽  
Group Order ◽  
Almost All ◽  
Candidate Group

AbstractLetting p vary over all primes and E vary over all elliptic curves over the finite field 𝔽p, we study the frequency to which a given group G arises as a group of points E(𝔽p). It is well known that the only permissible groups are of the form Gm,k:=ℤ/mℤ×ℤ/mkℤ. Given such a candidate group, we let M(Gm,k) be the frequency to which the group Gm,karises in this way. Previously, C.David and E. Smith determined an asymptotic formula for M(Gm,k) assuming a conjecture about primes in short arithmetic progressions. In this paper, we prove several unconditional bounds for M(Gm,k), pointwise and on average. In particular, we show thatM(Gm,k) is bounded above by a constant multiple of the expected quantity when m ≤ kA and that the conjectured asymptotic for M(Gm,k) holds for almost all groups Gm,k when m ≤ k1/4-∈. We also apply our methods to study the frequency to which a given integer N arises as a group order #E(𝔽p).

scholarly journals Average normalisations of elliptic curves

2002 ◽  
Vol 66 (3) ◽  
pp. 353-358 ◽  
Author(s):  
William D. Banks ◽  
Igor E. Shparlinski
Keyword(s):  
Finite Field ◽  
Elliptic Curve ◽  
Elliptic Curves ◽  
Almost All

Ciet, Quisquater, and Sica have recently shown that every elliptic curve E over a finite field 𝔽p is isomorphic to a curve y2 = x3 + ax + b with a and b of size O (p¾). In this paper, we show that almost all elliptic curves satisfy the stronger bound O (p⅔). The problem is motivated by cryptographic considerations.

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