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Elliptic Curves And Primality Proving
 Math. Comp
, 1993
"... The aim of this paper is to describe the theory and implementation of the Elliptic Curve Primality Proving algorithm. ..."
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Cited by 162 (22 self)
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The aim of this paper is to describe the theory and implementation of the Elliptic Curve Primality Proving algorithm.
On the singular values of Weber modular functions
 Department of Mathematics National University of Singapore Kent Ridge, Singapore
, 1997
"... Abstract. The minimal polynomials of the singular values of the classical Weber modular functions give far simpler defining polynomials for the class fields of imaginary quadratic fields than the minimal polynomials of singular moduli of level 1. We describe computations of these polynomials and giv ..."
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Cited by 11 (1 self)
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Abstract. The minimal polynomials of the singular values of the classical Weber modular functions give far simpler defining polynomials for the class fields of imaginary quadratic fields than the minimal polynomials of singular moduli of level 1. We describe computations of these polynomials and give conjectural formulas describing the prime decomposition of their resultants and discriminants, extending the formulas of GrossZagier for the level 1 case.
Computing the cardinality of CM elliptic curves using torsion points
, 2008
"... Let E be an elliptic curve having complex multiplication by a given quadratic order of an imaginary quadratic field K. The field of definition of E is the ring class field Ω of the order. If the prime p splits completely in Ω, then we can reduce E modulo one the factors of p and get a curve E define ..."
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Cited by 10 (1 self)
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Let E be an elliptic curve having complex multiplication by a given quadratic order of an imaginary quadratic field K. The field of definition of E is the ring class field Ω of the order. If the prime p splits completely in Ω, then we can reduce E modulo one the factors of p and get a curve E defined over Fp. The trace of the Frobenius of E is known up to sign and we need a fast way to find this sign. For this, we propose to use the action of the Frobenius on torsion points of small order built with class invariants à la Weber, in a manner reminiscent of the SchoofElkiesAtkin algorithm for computing the cardinality of a given elliptic curve modulo p. We apply our results to the Elliptic Curve Primality Proving algorithm (ECPP).
Hilbert class polynomials and traces of singular moduli
 MATHEMATISCHE ANNALEN, ACCEPTED FOR PUBLICATION
, 2005
"... Let j(z) be the modular function for SL2(Z) defined by ..."
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Cited by 7 (3 self)
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Let j(z) be the modular function for SL2(Z) defined by
Generalised Weber functions
, 2009
"... A generalised Weber function is wN(z) = η(z/N)/η(z) where η(z) is the Dedekind function and N is any integer (the original function corresponds to N = 2). We give the complete classification of cases where some power we N evaluated at some quadratic integer generates the ring class field associated ..."
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Cited by 4 (1 self)
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A generalised Weber function is wN(z) = η(z/N)/η(z) where η(z) is the Dedekind function and N is any integer (the original function corresponds to N = 2). We give the complete classification of cases where some power we N evaluated at some quadratic integer generates the ring class field associated to an order of an imaginary quadratic field. We compare the heights of our invariants by giving a general formula for the degree of the relevant modular equation relating wN(z) and j(z).
ENDOMORPHISM ALGEBRAS OF HYPERELLIPTIC JACOBIANS AND FINITE PROJECTIVE LINES
, 2006
"... Let K be a field with char(K) ̸ = 2. Let us fix an algebraic closure Ka of K. Let us put Gal(K): = Aut(Ka/K). If X is an abelian variety of positive dimension over Ka then we write End(X) for the ring of all its Kaendomorphisms and End 0 (X) for the corresponding (semisimple finitedimensional) Qa ..."
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Cited by 1 (1 self)
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Let K be a field with char(K) ̸ = 2. Let us fix an algebraic closure Ka of K. Let us put Gal(K): = Aut(Ka/K). If X is an abelian variety of positive dimension over Ka then we write End(X) for the ring of all its Kaendomorphisms and End 0 (X) for the corresponding (semisimple finitedimensional) Qalgebra End(X) ⊗ Q. We write EndK(X) for the ring of all Kendomorphisms of X and End 0 K(X) for the corresponding (semisimple finitedimensional) Qalgebra EndK(X) ⊗ Q. The absolute Galois group Gal(K) of K acts on End(X) (and therefore on End 0 (X)) by ring (resp. algebra) automorphisms and EndK(X) = End(X) Gal(K) , End 0 K(X) = End 0 (X) Gal(K), since every endomorphism of X is defined over a finite separable extension of K. If n is a positive integer that is not divisible by char(K) then we write Xn for the kernel of multiplication by n in X(Ka). It is wellknown [21] that Xn is a free Z/nZmodule of rank 2dim(X). In particular, if n = ℓ is a prime then Xℓ is an Fℓvector space of dimension 2dim(X). If X is defined over K then Xn is a Galois submodule in X(Ka). It is known that all points of Xn are defined over a finite separable extension of K. We write ¯ρn,X,K: Gal(K) → Aut Z/nZ(Xn) for the corresponding homomorphism defining the structure of the Galois module on Xn, ˜Gn,X,K ⊂ Aut Z/nZ(Xn) for its image ¯ρn,X,K(Gal(K)) and K(Xn) for the field of definition of all points of Xn. Clearly, K(Xn) is a finite Galois extension of K with Galois group Gal(K(Xn)/K) = ˜Gn,X,K. If n = ℓ then we get a natural faithful linear representation ˜Gℓ,X,K ⊂ AutFℓ (Xℓ) of ˜ Gℓ,X,K in the Fℓvector space Xℓ. Recall [29] that all endomorphisms of X are defined over K(X4); this gives rise to the natural homomorphism
INRIA Bordeaux–SudOuest
, 2009
"... A generalised Weber function is given by wN(z) = η(z/N)/η(z), where η(z) is the Dedekind function and N is any integer; the original function corresponds to N = 2. We classify the cases where some power we N evaluated at some quadratic integer generates the ring class field associated to an order o ..."
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A generalised Weber function is given by wN(z) = η(z/N)/η(z), where η(z) is the Dedekind function and N is any integer; the original function corresponds to N = 2. We classify the cases where some power we N evaluated at some quadratic integer generates the ring class field associated to an order of an imaginary quadratic field. We compare the heights of our invariants by giving a general formula for the degree of the modular equation relating wN(z) and j(z). 1
Journal de Théorie des Nombres
"... Computing the cardinality of CM elliptic curves using torsion points par François MORAIN ∗ Résumé. Soit E/Q une courbe elliptique avec multiplications complexes par un ordre d’un corps quadratique imaginaire K. Le corps de définition de E est le corps de classe de rayon Ω associé à l’ordre. Si le no ..."
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Computing the cardinality of CM elliptic curves using torsion points par François MORAIN ∗ Résumé. Soit E/Q une courbe elliptique avec multiplications complexes par un ordre d’un corps quadratique imaginaire K. Le corps de définition de E est le corps de classe de rayon Ω associé à l’ordre. Si le nombre premier p est scindé dans Ω, on peut réduire E modulo un des facteurs de p et obtenir une courbe E définie sur Fp. La trace du Frobenius de E est connue au signe près et nous cherchons à déterminer ce signe de la manière la plus rapide possible, avec comme application l’algorithme de primalité ECPP. Dans ce but, nous expliquons comment utiliser l’action du Frobenius sur des points de torsion d’ordre petit obtenus à partir d’invariants de classes qui généralisent les fonctions de Weber. Abstract. Let E/Q be an elliptic curve having complex multiplication by a given quadratic order of an imaginary quadratic field K. The field of definition of E is the ring class field Ω of the order. If the prime p splits completely in Ω, then we can reduce E modulo one the factors of p and get a curve E defined over Fp. The trace of the Frobenius of E is known up to sign and we need a fast way to find this sign, in the context of the Elliptic Curve Primality Proving algorithm (ECPP). For this purpose, we propose to use the action of the Frobenius on torsion points of small order built with class invariants generalizing the classical Weber functions. 1.