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18
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.
Computing modular polynomials in quasilinear time
 Mathematics of Computation
"... Abstract. We analyse and compare the complexity of several algorithms for computing modular polynomials. Under the assumption that rounding errors do not influence the correctness of the result, which appears to be satisfied in practice, we show that an algorithm relying on floating point evaluation ..."
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Cited by 17 (3 self)
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Abstract. We analyse and compare the complexity of several algorithms for computing modular polynomials. Under the assumption that rounding errors do not influence the correctness of the result, which appears to be satisfied in practice, we show that an algorithm relying on floating point evaluation of modular functions and on interpolation has a complexity that is up to logarithmic factors linear in the size of the computed polynomials. In particular, it obtains the classical modular polynomial Φℓ of prime level ℓ in time O ( ℓ 2 log 3 ℓM(ℓ) ) ⊆ O ( ℓ 3 log 4+ε ℓ), where M(ℓ) is the time needed to multiply two ℓbit numbers. Besides treating modular polynomials for Γ0 (ℓ), which are an important ingredient in many algorithms dealing with isogenies of elliptic curves, the algorithm is easily adapted to more general situations. Composite levels are handled just as easily as prime levels, as well as polynomials between a modular function and its transform of prime level, such as the Schläfli polynomials and their generalisations.
Implementation Of The AtkinGoldwasserKilian Primality Testing Algorithm
 Rapport de Recherche 911, INRIA, Octobre
, 1988
"... . We describe a primality testing algorithm, due essentially to Atkin, that uses elliptic curves over finite fields and the theory of complex multiplication. In particular, we explain how the use of class fields and genus fields can speed up certain phases of the algorithm. We sketch the actual impl ..."
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Cited by 9 (7 self)
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. We describe a primality testing algorithm, due essentially to Atkin, that uses elliptic curves over finite fields and the theory of complex multiplication. In particular, we explain how the use of class fields and genus fields can speed up certain phases of the algorithm. We sketch the actual implementation of this test and its use on testing large primes, the records being two numbers of more than 550 decimal digits. Finally, we give a precise answer to the question of the reliability of our computations, providing a certificate of primality for a prime number. IMPLEMENTATION DU TEST DE PRIMALITE D' ATKIN, GOLDWASSER, ET KILIAN R'esum'e. Nous d'ecrivons un algorithme de primalit'e, principalement du `a Atkin, qui utilise les propri'et'es des courbes elliptiques sur les corps finis et la th'eorie de la multiplication complexe. En particulier, nous expliquons comment l'utilisation du corps de classe et du corps de genre permet d'acc'el'erer les calculs. Nous esquissons l'impl'ementati...
Arithmetic of singular moduli and class polynomials
, 2005
"... We investigate divisibility properties of the traces and Hecke traces of singular moduli. In particular we prove that, if p is prime, these traces satisfy many congruences modulo powers of p which are described in terms of the factorization of p in imaginary quadratic fields. We also study generaliz ..."
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Cited by 8 (1 self)
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We investigate divisibility properties of the traces and Hecke traces of singular moduli. In particular we prove that, if p is prime, these traces satisfy many congruences modulo powers of p which are described in terms of the factorization of p in imaginary quadratic fields. We also study generalizations of Lehner’s classical congruences j(z)Up ≡ 744 (mod p) (wherep � 11 and j(z) is the usual modular invariant), and we investigate connections between class polynomials and supersingular polynomials in characteristic p.
ALGEBRAIC THETA FUNCTIONS AND THE pADIC INTERPOLATION OF EISENSTEINKRONECKER NUMBERS
, 2007
"... ABSTRACT. We study the properties of EisensteinKronecker numbers, which are related to special values of Hecke Lfunction of imaginary quadratic fields. We prove that the generating function of these numbers is a reduced (normalized or canonical in some literature) theta function associated to the ..."
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Cited by 6 (3 self)
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ABSTRACT. We study the properties of EisensteinKronecker numbers, which are related to special values of Hecke Lfunction of imaginary quadratic fields. We prove that the generating function of these numbers is a reduced (normalized or canonical in some literature) theta function associated to the Poincaré bundle of an elliptic curve. We introduce general methods to study the algebraic and padic properties of reduced theta functions for CM abelian varieties. As a corollary, when the prime p is ordinary, we give a new construction of the twovariable padic measure interpolating special values of Hecke Lfunctions of imaginary quadratic fields, originally constructed by ManinVishik and Katz. Our method via theta functions also gives insight for the case when p is supersingular. The method of this paper will be used in subsequent papers in constructing certain twovariable padic distribution for supersingular p interpolating EisensteinKronecker numbers in twovaribales, as well as explicit calculation in twovariables of the padic elliptic polylogarithms for CM elliptic curves.
Ramanujan and the modular jinvariant
 Canad. Math. Bull
, 1999
"... Abstract. A new infinite product tn was introduced by S. Ramanujan on the last page of his third notebook. In this paper, we prove Ramanujan’s assertions about tn by establishing new connections between the modular j−invariant and Ramanujan’s cubic theory of elliptic functions to alternative bases. ..."
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Cited by 5 (3 self)
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Abstract. A new infinite product tn was introduced by S. Ramanujan on the last page of his third notebook. In this paper, we prove Ramanujan’s assertions about tn by establishing new connections between the modular j−invariant and Ramanujan’s cubic theory of elliptic functions to alternative bases. We also show that for certain integers n, tn generates the Hilbert class field of � ( √ −n). This shows that tn is a new class invariant according to H. Weber’s definition of class invariants. 1.
Construction Of Hilbert Class Fields Of Imaginary Quadratic Fields And Dihedral Equations Modulo p
, 1989
"... . The implementation of the AtkinGoldwasserKilian primality testing algorithm requires the construction of the Hilbert class field of an imaginary quadratic field. We describe the computation of a defining equation for this field in terms of Weber's class invariants. The polynomial we obtain, note ..."
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Cited by 4 (3 self)
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. The implementation of the AtkinGoldwasserKilian primality testing algorithm requires the construction of the Hilbert class field of an imaginary quadratic field. We describe the computation of a defining equation for this field in terms of Weber's class invariants. The polynomial we obtain, noted W(X), has a solvable Galois group. When this group is dihedral, we show how to express the roots of this polynomial in terms of radicals. We then use these expressions to solve the equation W(X) j 0 mod p, where p is a prime. 1 Hilbert polynomials 1.1 Weber's functions We first introduce some functions. Let z be any complex number and put q = exp(2ißz). Dedekind's j function is defined by [21, x24 p. 85] j(z) = j(q) = q 1=24 Y m1 (1 \Gamma q m ): (1) We can expand j as [21, x34 p. 112] j(q) = q 1=24 0 @ 1 + X n1 (\Gamma1) n (q n(3n\Gamma1)=2 + q n(3n+1)=2 ) 1 A : (2) The Weber's functions are [21, x34 p. 114] f(z) = e \Gammaiß=24 j( z+1 2 ) j(z) ; (3) f 1 (z) = j...
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).
POINT COUNTING ON REDUCTIONS OF CM ELLIPTIC CURVES
"... Abstract. We give explicit formulas for the number of points on reductions of elliptic curves with complex multiplication by any imaginary quadratic field. We also find models for CM Qcurves in certain cases. This generalizes earlier results of Gross, Stark, and others. 1. ..."
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Abstract. We give explicit formulas for the number of points on reductions of elliptic curves with complex multiplication by any imaginary quadratic field. We also find models for CM Qcurves in certain cases. This generalizes earlier results of Gross, Stark, and others. 1.