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50
MordellWeil lattices in characteristic 2: I. Construction and first properties, Internat
 Invent. Math
, 1997
"... In a famous 1967 paper [ˇST], ˇSafarevič and Tate constructed elliptic curves of arbitrarily ..."
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Cited by 11 (2 self)
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In a famous 1967 paper [ˇST], ˇSafarevič and Tate constructed elliptic curves of arbitrarily
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...
Isogenies Of Supersingular Elliptic Curves Over Finite Fields And Operations In Elliptic Cohomology
"... . In this paper we investigate stable operations in supersingular elliptic cohomology using isogenies of supersingular elliptic curves over finite fields. Our main results provide a framework in which we give a conceptually simple new proof of an elliptic cohomology version of the Morava change of r ..."
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Cited by 6 (3 self)
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. In this paper we investigate stable operations in supersingular elliptic cohomology using isogenies of supersingular elliptic curves over finite fields. Our main results provide a framework in which we give a conceptually simple new proof of an elliptic cohomology version of the Morava change of rings theorem and also gives models for explicit stable operations in terms of isogenies and morphisms in certain enlarged isogeny categories. We are particularly inspired by number theoretic work of G. Robert, whose work we reformulate and generalize in our setting. Introduction In previous work we investigated supersingular reductions of elliptic cohomology [5], stable operations and cooperations in elliptic cohomology [3, 4, 6, 8] and in [9, 10] gave some applications to the Adams spectral sequence based on elliptic (co)homology. In this paper we investigate stable operations in supersingular elliptic cohomology using isogenies of supersingular elliptic curves over finite fields; this is ...
The Klein quartic in number theory
, 1999
"... Abstract. We describe the Klein quartic � and highlight some of its remarkable properties that are of particular interest in number theory. These include extremal properties in characteristics 2, 3, and 7, the primes dividing the order of the automorphism group of �; an explicit identification of � ..."
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Cited by 6 (0 self)
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Abstract. We describe the Klein quartic � and highlight some of its remarkable properties that are of particular interest in number theory. These include extremal properties in characteristics 2, 3, and 7, the primes dividing the order of the automorphism group of �; an explicit identification of � with the modular curve X(7); and applications to the class number 1 problem and the case n = 7 of Fermat.
Elliptic Curves, Primality Proving And Some Titanic Primes
, 1989
"... We describe how to generate large primes using the primality proving algorithm of Atkin. Figure 1: The Titanic . 1. Introduction. During the last ten years, primality testing evolved at great speed. Motivated by the RSA cryptosystem [3], the first deterministic primality proving algorithm was de ..."
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Cited by 5 (3 self)
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We describe how to generate large primes using the primality proving algorithm of Atkin. Figure 1: The Titanic . 1. Introduction. During the last ten years, primality testing evolved at great speed. Motivated by the RSA cryptosystem [3], the first deterministic primality proving algorithm was designed by Adleman, Pomerance and Rumely [2] and made practical by Cohen, H. W. Lenstra and A. K. Lenstra (see [9, 10] and more recently [5]). It was then proved that the time needed to test an arbitrary integer N for primality is O((log N) c log log log N ) for some positive constant c ? 0. When implemented on a huge computer, the algorithm was able to test 200 digit numbers in about 10 minutes of CPU time. A few years ago, Goldwasser and Kilian [11], used the theory of elliptic curves over finite fields to give the first primality proving algorithm whose running time is polynomial in log N (assuming a plausible conjecture in number theory). Atkin [4] used the theory of complex multiplicat...
Iwasawa theory for elliptic curves
 Lecture Notes in Math. 1716
, 1999
"... The topics that we will discuss have their origin in Mazur’s synthesis of the theory of elliptic curves and Iwasawa’s theory of Zpextensions in the early 1970s. We first recall some results from Iwasawa’s theory. Suppose that F is a finite extension of Q and that F ∞ is a Galois extension of F such ..."
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Cited by 4 (2 self)
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The topics that we will discuss have their origin in Mazur’s synthesis of the theory of elliptic curves and Iwasawa’s theory of Zpextensions in the early 1970s. We first recall some results from Iwasawa’s theory. Suppose that F is a finite extension of Q and that F ∞ is a Galois extension of F such that
A Supersingular Congruence For Modular Forms
 ACTA ARITHMETICA
, 1998
"... Let p ? 3 be a prime. In the ring of modular forms with qexpansions defined over Z (p) , the Eisenstein function Ep+1 is shown to satisfy (Ep+1) p\Gamma1 j \Gamma ` \Gamma1 p ' \Delta (p 2 \Gamma1)=12 mod (p; Ep\Gamma1 ): This is equivalent to a result conjectured by de Shalit on the po ..."
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Cited by 3 (2 self)
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Let p ? 3 be a prime. In the ring of modular forms with qexpansions defined over Z (p) , the Eisenstein function Ep+1 is shown to satisfy (Ep+1) p\Gamma1 j \Gamma ` \Gamma1 p ' \Delta (p 2 \Gamma1)=12 mod (p; Ep\Gamma1 ): This is equivalent to a result conjectured by de Shalit on the polynomial satisfied by all the jinvariants of supersingular elliptic curves over F p . It is also closely related to a result of Gross and Landweber used to define a topological version of elliptic cohomology.
Elliptic Curves with Good Reduction away from 3
, 1984
"... . We list the elliptic curves defined over Q( p \Gamma3) with good reduction away from the prime dividing 3. 0. Introduction. In this paper we list the elliptic curves defined over Q( p \Gamma3) with good reduction away from the prime dividing 3. As in [8] and [9] a discriminant estimate is u ..."
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Cited by 3 (1 self)
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. We list the elliptic curves defined over Q( p \Gamma3) with good reduction away from the prime dividing 3. 0. Introduction. In this paper we list the elliptic curves defined over Q( p \Gamma3) with good reduction away from the prime dividing 3. As in [8] and [9] a discriminant estimate is used to show that such a curve must have a subgroup of order 3 defined over Q( p \Gamma3). 1. The subgroup of order 3. An elliptic curve over a field K can be put in Weierstrass form Y 2 + a 1 XY + a 3 X = X 3 + a 2 X 2 + a 4 X + a 6 : (1:1 In the notation of [8] (1.3), Tate [14] Section 1 or [15], or Silverman [11] Chapter III, we define auxiliary coefficients b 2 = a 2 1 + 4a 2 b 4 = a 1 a 3 + 2a 4 b 6 = a 2 3 + 4a 6 b 8 = a 2 1 \Gamma a 1 a 3 a 4 + 4a 2 a 6 + a 2 a 2 3 \Gamma a 2 4 ; satisfying the relation 4b 8 = b 2 b 6 \Gamma b 2 4 , covariants c 4 = b 2 2 \Gamma 24b 4 c 6 = \Gammab 3 2 + 36b 4 \Gamma 216b 6 ; discriminant \Delta = \Gammab 2 2 b 8 \...
Lfunctions with large analytic rank and abelian varieties with large algebraic rank over function fields
"... The goal of this paper is to explain how a simple but apparently new fact of linear algebra together with the cohomological interpretation of Lfunctions allows one to produce many examples of Lfunctions over function fields vanishing to high order at the center point of their functional equation. ..."
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Cited by 3 (2 self)
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The goal of this paper is to explain how a simple but apparently new fact of linear algebra together with the cohomological interpretation of Lfunctions allows one to produce many examples of Lfunctions over function fields vanishing to high order at the center point of their functional equation. Conjectures of Birch
Atkin's test: news from the front
 In Advances in Cryptology
, 1990
"... We make an attempt to compare the speed of eeme primality testing algorithms for certifying loodigit prime numbers. 1. Introduction. The ..."
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Cited by 2 (2 self)
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We make an attempt to compare the speed of eeme primality testing algorithms for certifying loodigit prime numbers. 1. Introduction. The