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19
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.
Reliability of CalderbankShorSteane codes and security of quantum key distribution
 J. Phys. A: Math. Gen
, 2004
"... Abstract. After Mayers (1996, 2001) gave a proof of the security of the Bennett ..."
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Cited by 16 (7 self)
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Abstract. After Mayers (1996, 2001) gave a proof of the security of the Bennett
Symplectic Geometry And The Verlinde Formulas
, 1998
"... The purpose of this paper is to give a proof of the Verlinde formulas by applying the RiemannRochKawasaki theorem to the moduli space of flat Gbundles on a Riemann surface # with marked points, when G is a connected simply connected compact Lie group G. Conditions are given for the moduli spa ..."
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Cited by 11 (1 self)
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The purpose of this paper is to give a proof of the Verlinde formulas by applying the RiemannRochKawasaki theorem to the moduli space of flat Gbundles on a Riemann surface # with marked points, when G is a connected simply connected compact Lie group G. Conditions are given for the moduli space to be an orbifold, and the strata are described as moduli spaces for semisimple centralizers in G. The contribution of the strata are evaluated using the formulas of Witten for the symplectic volume, methods of symplectic geometry, including formulas of WittenJeffreyKirwan, and residue formulas.
LowDimensional Lattices IV: The Mass Formula
 Proc. Royal Soc. London, A
, 1988
"... The mass formula expresses the sum of the reciprocals of the group orders of the lattices in a genus in terms of the properties of any of them. We restate the formula so as to make it easier to compute. In particular we give a simple and reliable way to evaluate the 2adic contribution. Our version, ..."
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Cited by 7 (1 self)
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The mass formula expresses the sum of the reciprocals of the group orders of the lattices in a genus in terms of the properties of any of them. We restate the formula so as to make it easier to compute. In particular we give a simple and reliable way to evaluate the 2adic contribution. Our version, unlike earlier ones, is visibly invariant under scale changes and dualizing. We use the formula to check the enumeration of lattices of determinant d 25 given in the first paper in this series. We also give tables of the "standard mass", the Lseries S (n / m)m  s (m odd), and genera of lattices of determinant d 25. 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...
Hyperelliptic loop solitons with genus g: investigations of a quantized elastica
 J. Geom. and Phys
"... In the previous work (J. Geom. Phys. 39 (2001) 5061), the closed loop solitons in a plane, i.e., loops whose curvatures obey the modified Kortewegde Vries equations, were investigated for the case related to algebraic curves with genera one and two. This article is a generalization of the previous ..."
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Cited by 2 (0 self)
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In the previous work (J. Geom. Phys. 39 (2001) 5061), the closed loop solitons in a plane, i.e., loops whose curvatures obey the modified Kortewegde Vries equations, were investigated for the case related to algebraic curves with genera one and two. This article is a generalization of the previous article to those of hyperelliptic curves with general genera. It was proved that the tangential angle of loop soliton is expressed by the Weierstrass hyperelliptic al function for a given hyperelliptic curve y 2 = f(x) with genus g.
Higher Derivative Couplings and Heterotic–Type I Duality in Eight Dimensions
, 1999
"... We calculate F 4 and R 4 T 4g−4 couplings in d = 8 heterotic and type I string vacua (with gauge and graviphoton field strengths F, T, and Riemann curvature R). The holomorphic piece Fg of the heterotic one–loop coupling R 4 T 4g−4 is given by a polylogarithm of index 5 − 4g and encodes the counting ..."
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We calculate F 4 and R 4 T 4g−4 couplings in d = 8 heterotic and type I string vacua (with gauge and graviphoton field strengths F, T, and Riemann curvature R). The holomorphic piece Fg of the heterotic one–loop coupling R 4 T 4g−4 is given by a polylogarithm of index 5 − 4g and encodes the counting of genus g curves with g nodes on the K3 of the dual F–theory side. We present closed expressions for world–sheet τ–integrals with an arbitrary number of lattice vector insertions. Furthermore we verify that the corresponding heterotic one–loop couplings sum up perturbative open string and nonperturbative D–string contributions on the type I side. Finally we discuss a type I one–loop correction to the R 2 term. 12/98
ON THE TOPOLOGY AND THE GEOMETRY OF SO(3)MANIFOLDS
"... Abstract. Consider the nonstandard embedding of SO(3) into SO(5) given by the 5dimensional irreducible representation of SO(3), henceforth called SO(3)ir. In this note, we study the topology and the differential geometry of 5dimensional Riemannian manifolds carrying such an SO(3)ir structure, i.e. ..."
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Abstract. Consider the nonstandard embedding of SO(3) into SO(5) given by the 5dimensional irreducible representation of SO(3), henceforth called SO(3)ir. In this note, we study the topology and the differential geometry of 5dimensional Riemannian manifolds carrying such an SO(3)ir structure, i.e. with a reduction of the frame bundle to SO(3)ir. 1.
Rational surfaces with a large group of automorphisms
 J. Amer. Math. Soc
"... 2. Halphen surfaces 867 3. Coble surfaces 875 4. Gizatullin’s Theorem and Cremona special point sets of nine points 883 ..."
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2. Halphen surfaces 867 3. Coble surfaces 875 4. Gizatullin’s Theorem and Cremona special point sets of nine points 883
Congruent Numbers, Elliptic Curves and Modular Forms
"... An integer n ≥ 1 is called congruent if it is the area of a right triangle whose three sides have rational lenghts. Equivalently, n is congruent if there exists a rational number x such that x 2 − n and x 2 + n are both squares of rational numbers. The very old problem 1 of determining all congruent ..."
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An integer n ≥ 1 is called congruent if it is the area of a right triangle whose three sides have rational lenghts. Equivalently, n is congruent if there exists a rational number x such that x 2 − n and x 2 + n are both squares of rational numbers. The very old problem 1 of determining all congruent integers has only recently been answered satisfactorily, and nowadays we know all congruent numbers less than 1000 (Table 1). 2 In fact, thanks to J.B. Tunnell (1983) we are now in possession of a very simple criterion for verifying that a given integer is not congruent. Justifying this criterion will take us on a journey into the forest of results and conjectures in the modern arithmetic theory of elliptic curves and modular forms that we have already met in the preceding articles. Tunnell’s criterion can be formulated as follows: Let n ≥ 1 be an odd squarefree integer (that is, not divisible by the square of an integer greater than 1). Consider the following conditions: