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59
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 201 (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.
ARITHMETIC QUANTUM UNIQUE ERGODICITY FOR SYMPLECTIC LINEAR MAPS OF THE MULTIDIMENSIONAL TORUS
, 2006
"... We look at the expectation values for quantized linear symplectic maps on the multidimensional torus and their distribution in the semiclassical limit. We construct superscars that are stable under the arithmetic symmetries of the system and localize around invariant manifolds. We show that these ..."
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Cited by 22 (3 self)
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We look at the expectation values for quantized linear symplectic maps on the multidimensional torus and their distribution in the semiclassical limit. We construct superscars that are stable under the arithmetic symmetries of the system and localize around invariant manifolds. We show that these superscars exist only when there are isotropic rational subspaces, invariant under the linear map. In the case where there are no such scars, we compute the variance of the fluctuations of the matrix elements for the desymmetrized system, and present a conjecture for their limiting distributions.
Planar coincidences for Nfold symmetry
, 2005
"... The coincidence problem for planar patterns with Nfold symmetry is considered. For the Nfold symmetric module with N < 46, all isometries of the plane are classified that result in coincidences of finite index. This is done by reformulating the problem in terms of algebraic number fields and us ..."
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Cited by 20 (11 self)
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The coincidence problem for planar patterns with Nfold symmetry is considered. For the Nfold symmetric module with N < 46, all isometries of the plane are classified that result in coincidences of finite index. This is done by reformulating the problem in terms of algebraic number fields and using prime factorization. The more complicated case N ≥ 46 is briefly discussed and N = 46 is described explicitly. The results of the coincidence problem also solve the problem of colour lattices in two dimensions and its natural generalization to colour modules.
Class Field Theory in Characteristic p, its Origin and Development
 the Proceedings of the International Conference on Class Field Theory (Tokyo
, 2002
"... Today's notion of "global field" comprises number fields (algebraic, of finite degree) and function fields (algebraic, of dimension 1, finite base field). They have many similar arithmetic properties. The systematic study of these similarities seems to have been started by Dedekind (1 ..."
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Cited by 17 (5 self)
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Today's notion of "global field" comprises number fields (algebraic, of finite degree) and function fields (algebraic, of dimension 1, finite base field). They have many similar arithmetic properties. The systematic study of these similarities seems to have been started by Dedekind (1857). A new impetus was given by the seminal thesis of E. Artin (1921, published in 1924). In this exposition I shall report on the development during the twenties and thirties of our century, with emphasis on the emergence of class field theory for function elds. The names of F.K.Schmidt, H. Hasse, E. Witt, C. Chevalley (among others) are closely connected with that development.
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 10 (4 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).
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 implem ..."
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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.
On the exponent of the group of points on elliptic curves in extension fields
 Intern. Math. Research Notices
"... Let E be an elliptic curve defined over Fq, a finite field of q elements. Furthermore, we consider ..."
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Cited by 9 (5 self)
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Let E be an elliptic curve defined over Fq, a finite field of q elements. Furthermore, we consider
Heegner points, Heegner cycles, and congruences
, 2000
"... We define certain objects associated to a modular elliptic curve E and a discriminant D satisfying suitable conditions. These objects interpolate special values of the complex Lfunctions associated to E over the quadratic field Q( p D), in the same way that Bernouilli numbers interpolate specia ..."
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Cited by 9 (6 self)
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We define certain objects associated to a modular elliptic curve E and a discriminant D satisfying suitable conditions. These objects interpolate special values of the complex Lfunctions associated to E over the quadratic field Q( p D), in the same way that Bernouilli numbers interpolate special values of Dirichlet Lseries. Following an approach of Mazur and Tate [MT], one can make conjectures about congruences satised by these objects which are resonant with the usual Birch and SwinnertonDyer conjectures. These conjectures exhibit some surprising features not apparent in the classical case.
Topological Conjugacy of Linear Endomorphisms of the 2Torus
 Trans. Amer. Math. Soc
, 1997
"... . We describe two complete sets of numerical invariants of topological conjugacy for linear endomorphisms of the twodimensional torus, i.e., continuous maps from the torus to itself which are covered by linear maps of the plane. The trace and determinant are part of both complete sets, and two can ..."
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Cited by 9 (1 self)
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. We describe two complete sets of numerical invariants of topological conjugacy for linear endomorphisms of the twodimensional torus, i.e., continuous maps from the torus to itself which are covered by linear maps of the plane. The trace and determinant are part of both complete sets, and two candidates are proposed for a third (and last) invariant which, in both cases, can be understood from the topological point of view. One of our invariants is in fact the ideal class of the LatimerMacDuffeeTaussky theory, reformulated in more elementary terms and interpreted as describing some topology. Merely, one has to look at how closed curves on the torus intersect their image under the endomorphism. Part of the intersection information (the intersection number counted with multiplicity) can be captured by a binary quadratic form associated to the map, so that we can use the classical theories initiated by Lagrange and Gauss. To go beyond the intersection number, and shortcut the classifi...