scholarly journals Denominators of Igusa class polynomials

10.5802/pmb.6 ◽  
2015 ◽  
pp. 5-29 ◽  
Author(s):  
Kristin Lauter ◽  
Bianca Viray
Keyword(s):  
Class Polynomials

scholarly journals Constructing supersingular elliptic curves with a given endomorphism ring

2014 ◽  
Vol 17 (A) ◽  
pp. 71-91 ◽  
Author(s):  
Ilya Chevyrev ◽  
Steven D. Galbraith
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Endomorphism Ring ◽  
Quaternion Algebra ◽  
Maximal Order ◽  
Running Time ◽  
Successive Minima ◽  
Supersingular Elliptic Curves ◽  
Class Polynomials ◽  
Theoretical Results

AbstractLet $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathcal{O}$ be a maximal order in the quaternion algebra $B_p$ over $\mathbb{Q}$ ramified at $p$ and $\infty $. The paper is about the computational problem: construct a supersingular elliptic curve $E$ over $\mathbb{F}_p$ such that ${\rm End}(E) \cong \mathcal{O}$. We present an algorithm that solves this problem by taking gcds of the reductions modulo $p$ of Hilbert class polynomials.New theoretical results are required to determine the complexity of our algorithm. Our main result is that, under certain conditions on a rank three sublattice $\mathcal{O}^T$ of $\mathcal{O}$, the order $\mathcal{O}$ is effectively characterized by the three successive minima and two other short vectors of $\mathcal{O}^T\! .$ The desired conditions turn out to hold whenever the $j$-invariant $j(E)$, of the elliptic curve with ${\rm End}(E) \cong \mathcal{O}$, lies in $\mathbb{F}_p$. We can then prove that our algorithm terminates with running time $O(p^{1+\varepsilon })$ under the aforementioned conditions.As a further application we present an algorithm to simultaneously match all maximal order types with their associated $j$-invariants. Our algorithm has running time $O(p^{2.5 + \varepsilon })$ operations and is more efficient than Cerviño’s algorithm for the same problem.

scholarly journals Computing Class Polynomials for Abelian Surfaces

2014 ◽  
Vol 23 (2) ◽  
pp. 129-145 ◽  
Author(s):  
Andreas Enge ◽  
Emmanuel Thomé
Keyword(s):  
Abelian Surfaces ◽  
Class Polynomials

scholarly journals Twists of Shimura Curves

2014 ◽  
Vol 66 (4) ◽  
pp. 924-960 ◽  
Author(s):  
James Stankewicz
Keyword(s):  
Local Field ◽  
Rational Numbers ◽  
Shimura Curves ◽  
Shimura Curve ◽  
Modulo P ◽  
First Time ◽  
Class Polynomials

AbstractConsider a Shimura curve XD0(N) over the rational numbers. We determine criteria for the twist by an Atkin–Lenher involution to have points over a local field. As a corollary we give a new proof of the theorem of Jordan and Livné on Qp points when p |D and for the first time give criteria for Qp points when p |N. We also give congruence conditions for roots modulo p of Hilbert class polynomials.

Supersingular j-invariants and the class number of ℚ(−p)

2021 ◽  
pp. 1-14
Author(s):  
Guanju Xiao ◽  
Lixia Luo ◽  
Yingpu Deng
Keyword(s):  
Class Number ◽  
Deterministic Algorithm ◽  
Common Root ◽  
Class Polynomials ◽  
Quadratic Order

Let [Formula: see text] be a prime. Let [Formula: see text] be the discriminant of an imaginary quadratic order. Assume that [Formula: see text] and [Formula: see text]. We compute the number of [Formula: see text]-roots of the class polynomials [Formula: see text]. Suppose [Formula: see text], we prove that two class polynomials [Formula: see text] and [Formula: see text] have a common root in [Formula: see text] if and only if [Formula: see text] is a perfect square. Furthermore, any three class polynomials do not have a common root in [Formula: see text]. As an application, we propose a deterministic algorithm for computing the class number of [Formula: see text].

Constructing Elliptic Curves over Ramanujan's Class Invariants

2014 ◽  
Vol 915-916 ◽  
pp. 1336-1340
Author(s):  
Jian Jun Hu
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Finite Fields ◽  
Quadratic Field ◽  
Complex Multiplication ◽  
Imaginary Quadratic Field ◽  
Hilbert Polynomials ◽  
Class Invariants ◽  
Curves Over Finite Fields ◽  
Class Polynomials

The Complex Multiplication (CM) method is a widely used technique for constructing elliptic curves over finite fields. The key point in this method is parameter generation of the elliptic curve and root compution of a special type of class polynomials. However, there are several class polynomials which can be used in the CM method, having much smaller coefficients, and fulfilling the prerequisite that their roots can be easily transformed to the roots of the corresponding Hilbert polynomials.In this paper, we provide a method which can construct elliptic curves by Ramanujan's class invariants. We described the algorithm for the construction of elliptic curves (ECs) over imaginary quadratic field and given the transformation from their roots to the roots of the corresponding Hilbert polynomials. We compared the efficiency in the use of this method and other methods.

scholarly journals Traces of singular values of Hauptmoduln

2015 ◽  
Vol 11 (03) ◽  
pp. 1027-1048 ◽  
Author(s):  
Lea Beneish ◽  
Hannah Larson
Keyword(s):  
Modular Forms ◽  
Generating Functions ◽  
Singular Values ◽  
Important Paper ◽  
Half Integral Weight ◽  
Integral Weight ◽  
Class Polynomials

In an important paper, Zagier proved that certain half-integral weight modular forms are generating functions for traces of polynomials in the j-function. It turns out that Zagier's work makes it possible to algorithmically compute Hilbert class polynomials using a canonical family of modular forms of weight [Formula: see text]. We generalize these results and consider Hauptmoduln for levels 1, 2, 3, 5, 7, and 13. We show that traces of singular values of polynomials in Hauptmoduln are again described by coefficients of half-integral weight modular forms. This realization makes it possible to algorithmically compute class polynomials.

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