scholarly journals Pseudoprime Reductions of Elliptic Curves

2012 ◽  
Vol 64 (1) ◽  
pp. 81-101 ◽  
Author(s):  
C. David ◽  
J. Wu
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Complex Multiplication ◽  
Upper Bounds ◽  
Good Reduction ◽  
Elliptic Pseudoprimes

Abstract Let E be an elliptic curve over ℚ without complex multiplication, and for each prime p of good reduction, let nE(p) = |E(𝔽p)|. For any integer b, we consider elliptic pseudoprimes to the base b. More precisely, let QE,b(x) be the number of primes p ⩽ x such that bnE(p) ≡ b (mod nE(p)), and let πpseuE,b (x) be the number of compositive nE(p) such that bnE(p) ≡ b (mod nE(p)) (also called elliptic curve pseudoprimes). Motivated by cryptography applications, we address the problem of finding upper bounds for QE,b(x) and πpseuE,b (x), generalising some of the literature for the classical pseudoprimes to this new setting.

scholarly journals p-adic Eisenstein-Kronecker series for CM elliptic curves and the Kronecker limit formulas

2015 ◽  
Vol 219 ◽  
pp. 269-302
Author(s):  
Kenichi Bannai ◽  
Hidekazu Furusho ◽  
Shinichi Kobayashi
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Theta Function ◽  
Quadratic Field ◽  
Complex Multiplication ◽  
Imaginary Quadratic Field ◽  
Good Reduction ◽  
Ring Of Integers

AbstractConsider an elliptic curve defined over an imaginary quadratic fieldKwith good reduction at the primes abovep≥ 5 and with complex multiplication by the full ring of integersof K. In this paper, we constructp-adic analogues of the Eisenstein-Kronecker series for such an elliptic curve as Coleman functions on the elliptic curve. We then provep-adic analogues of the first and second Kronecker limit formulas by using the distribution relation of the Kronecker theta function.

On the Surjectivity of the Galois Representations Associated to Non-CM Elliptic Curves

2005 ◽  
Vol 48 (1) ◽  
pp. 16-31 ◽  
Author(s):  
Alina Carmen Cojocaru ◽  
Ernst Kani
Keyword(s):  
Positive Integer ◽  
Elliptic Curve ◽  
Elliptic Curves ◽  
Galois Representations ◽  
Complex Multiplication ◽  
Galois Representation ◽  
Upper Bounds

AbstractLet E be an elliptic curve defined over ℚ, of conductor N and without complex multiplication. For any positive integer l, let ϕl be the Galois representation associated to the l-division points of E. From a celebrated 1972 result of Serre we know that ϕl is surjective for any sufficiently large prime l. In this paper we find conditional and unconditional upper bounds in terms of N for the primes l for which ϕl is not surjective.

scholarly journals p-adic Eisenstein-Kronecker series for CM elliptic curves and the Kronecker limit formulas

2015 ◽  
Vol 219 ◽  
pp. 269-302
Author(s):  
Kenichi Bannai ◽  
Hidekazu Furusho ◽  
Shinichi Kobayashi
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Theta Function ◽  
Quadratic Field ◽  
Complex Multiplication ◽  
Imaginary Quadratic Field ◽  
Good Reduction ◽  
Ring Of Integers

AbstractConsider an elliptic curve defined over an imaginary quadratic field K with good reduction at the primes above p ≥ 5 and with complex multiplication by the full ring of integers of K. In this paper, we construct p-adic analogues of the Eisenstein-Kronecker series for such an elliptic curve as Coleman functions on the elliptic curve. We then prove p-adic analogues of the first and second Kronecker limit formulas by using the distribution relation of the Kronecker theta function.

scholarly journals Primitive divisors of sequences associated to elliptic curves with complex multiplication

2021 ◽  
Vol 7 (2) ◽  
Author(s):  
Matteo Verzobio
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Number Field ◽  
Complex Multiplication ◽  
Dedekind Domain ◽  
Torsion Point ◽  
Integral Ideal ◽  
Primitive Divisors ◽  
Primitive Divisor ◽  
The Ideal

AbstractLet P and Q be two points on an elliptic curve defined over a number field K. For $$\alpha \in {\text {End}}(E)$$ α ∈ End ( E ) , define $$B_\alpha $$ B α to be the $$\mathcal {O}_K$$ O K -integral ideal generated by the denominator of $$x(\alpha (P)+Q)$$ x ( α ( P ) + Q ) . Let $$\mathcal {O}$$ O be a subring of $${\text {End}}(E)$$ End ( E ) , that is a Dedekind domain. We will study the sequence $$\{B_\alpha \}_{\alpha \in \mathcal {O}}$$ { B α } α ∈ O . We will show that, for all but finitely many $$\alpha \in \mathcal {O}$$ α ∈ O , the ideal $$B_\alpha $$ B α has a primitive divisor when P is a non-torsion point and there exist two endomorphisms $$g\ne 0$$ g ≠ 0 and f so that $$f(P)= g(Q)$$ f ( P ) = g ( Q ) . This is a generalization of previous results on elliptic divisibility sequences.

scholarly journals Constructing elliptic curves from Galois representations

2018 ◽  
Vol 154 (10) ◽  
pp. 2045-2054
Author(s):  
Andrew Snowden ◽  
Jacob Tsimerman
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Galois Representations ◽  
Good Reduction ◽  
Arithmetic Surface

Given a non-isotrivial elliptic curve over an arithmetic surface, one obtains a lisse $\ell$-adic sheaf of rank two over the surface. This lisse sheaf has a number of straightforward properties: cyclotomic determinant, finite ramification, rational traces of Frobenius elements, and somewhere not potentially good reduction. We prove that any lisse sheaf of rank two possessing these properties comes from an elliptic curve.

scholarly journals No Singular Modulus Is a Unit

2018 ◽  
Vol 2020 (24) ◽  
pp. 10005-10041 ◽  
Author(s):  
Yuri Bilu ◽  
Philipp Habegger ◽  
Lars Kühne
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Endomorphism Ring ◽  
Complex Multiplication ◽  
Computer Assisted ◽  
Algebraic Unit ◽  
Special Points ◽  
Effective Proof ◽  
Rule Out

Abstract A result of the 2nd-named author states that there are only finitely many complex multiplication (CM)-elliptic curves over $\mathbb{C}$ whose $j$-invariant is an algebraic unit. His proof depends on Duke’s equidistribution theorem and is hence noneffective. In this article, we give a completely effective proof of this result. To be precise, we show that every singular modulus that is an algebraic unit is associated with a CM-elliptic curve whose endomorphism ring has discriminant less than $10^{15}$. Through further refinements and computer-assisted arguments, we eventually rule out all remaining cases, showing that no singular modulus is an algebraic unit. This allows us to exhibit classes of subvarieties in ${\mathbb{C}}^n$ not containing any special points.

A Titchmarsh divisor problem for elliptic curves

2015 ◽  
Vol 160 (1) ◽  
pp. 167-189 ◽  
Author(s):  
PAUL POLLACK
Keyword(s):  
Elliptic Curve ◽  
Complex Multiplication ◽  
Zeta Functions ◽  
Dirichlet Character ◽  
Good Reduction ◽  
Heuristic Argument ◽  
Mean Values ◽  
Average Size ◽  
Image Position ◽  
Ordinary Reduction

AbstractLet 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.

scholarly journals INFINITUDE OF ELLIPTIC CARMICHAEL NUMBERS

2012 ◽  
Vol 92 (1) ◽  
pp. 45-60 ◽  
Author(s):  
AARON EKSTROM ◽  
CARL POMERANCE ◽  
DINESH S. THAKUR
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Residue Class ◽  
Complex Multiplication ◽  
Weak Form ◽  
Arithmetic Progressions ◽  
Carmichael Numbers ◽  
The Third ◽  
Fermat’S Little Theorem

AbstractIn 1987, Gordon gave an integer primality condition similar to the familiar test based on Fermat’s little theorem, but based instead on the arithmetic of elliptic curves with complex multiplication. We prove the existence of infinitely many composite numbers simultaneously passing all elliptic curve primality tests assuming a weak form of a standard conjecture on the bound on the least prime in (special) arithmetic progressions. Our results are somewhat more general than both the 1999 dissertation of the first author (written under the direction of the third author) and a 2010 paper on Carmichael numbers in a residue class written by Banks and the second author.

scholarly journals Selmer Groups of Elliptic Curves with Complex Multiplication

2004 ◽  
Vol 56 (1) ◽  
pp. 194-208
Author(s):  
A. Saikia
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Number Field ◽  
Simple Formula ◽  
Quadratic Field ◽  
Complex Multiplication ◽  
Selmer Groups ◽  
Selmer Group ◽  
Finitely Generated ◽  
Ring Of Integers

AbstractSuppose K is an imaginary quadratic field and E is an elliptic curve over a number field F with complex multiplication by the ring of integers in K. Let p be a rational prime that splits as in K. Let Epn denote the pn-division points on E. Assume that F(Epn) is abelian over K for all n ≥ 0. This paper proves that the Pontrjagin dual of the -Selmer group of E over F(Ep∞) is a finitely generated free Λ-module, where Λ is the Iwasawa algebra of . It also gives a simple formula for the rank of the Pontrjagin dual as a Λ-module.

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.

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