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29
Constructing hyperelliptic curves of genus 2 suitable for cryptography
 Math. Comp
, 2003
"... Abstract. In this article we show how to generalize the CMmethod for elliptic curves to genus two. We describe the algorithm in detail and discuss the results of our implementation. 1. ..."
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Abstract. In this article we show how to generalize the CMmethod for elliptic curves to genus two. We describe the algorithm in detail and discuss the results of our implementation. 1.
Analytic Urns
 March
, 2003
"... This article describes a purely analytic approach to urn models of the generalized or extended PólyaEggenberger type, in the case of two types of balls and constant "balance", i.e., constant row sum. (Under such models, an urn may contain balls of either of two colours and a fixed 2 × 2matri ..."
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This article describes a purely analytic approach to urn models of the generalized or extended PólyaEggenberger type, in the case of two types of balls and constant "balance", i.e., constant row sum. (Under such models, an urn may contain balls of either of two colours and a fixed 2 × 2matrix determines the replacement policy when a ball is drawn and its colour is observed.) The treatment starts from a quasilinear firstorder partial differential equation associated with a combinatorial renormalization of the model and bases itself on elementary conformal mapping arguments coupled with singularity analysis techniques. Probabilistic consequences are new representations for the probability distribution of the urn's composition at any time n, structural information on the shape of moments of all orders, estimates of the speed of convergence to the Gaussian limits, and an explicit determination of the associated large deviation function. In the general case, analytic solutions involve Abelian integrals over the Fermat curve x = 1. Several urn models, including a classical one associated with balanced trees (23 trees and fringebalanced search trees) and related to a previous study of Panholzer and Prodinger as well as all urns of balance 1 or 2, are shown to admit of explicit representations in terms of Weierstraß elliptic functions. Other consequences include a unification of earlier studies of these models and the detection of stable laws in certain classes of urns with an offdiagonal entry equal to zero.
The Jacobian and Formal Group of a Curve of Genus 2 over an Arbitrary Ground
 Math. Proc. Cambridge Philos. Soc. 107
, 1990
"... The ability to perform practical computations on particular cases has greatly influenced the theory of elliptic curves. First, it has allowed a rich subbranch of the Mathematics of Computation to develop, devoted to elliptic curves: the search for curves of large rank, large torsion over number fie ..."
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The ability to perform practical computations on particular cases has greatly influenced the theory of elliptic curves. First, it has allowed a rich subbranch of the Mathematics of Computation to develop, devoted to elliptic curves: the search for curves of large rank, large torsion over number fields, and — more recently — the application of
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...
Measures Of Simultaneous Approximation For QuasiPeriods Of Abelian Varieties
 J. Number Theory, 94 (2002), N
, 2002
"... this paper, the functions # i will be assumed to be normalized as above, i.e. so that all secondorder derivatives of # 0 vanish at 0. 3.4. Conclusion ..."
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this paper, the functions # i will be assumed to be normalized as above, i.e. so that all secondorder derivatives of # 0 vanish at 0. 3.4. Conclusion
Efficient Resolution of Singularities of Plane Curves
 In Proceedings 14th conference on foundations of software technology and theoretical computer science
, 1994
"... . We give a new algorithm for resolving singularities of plane curves. The algorithm is polynomial time in the bit complexity model, does not require factorization, and works over Q or finite fields. 1 Introduction Resolving singularities is a central problem in computational algebraic geometry. In ..."
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. We give a new algorithm for resolving singularities of plane curves. The algorithm is polynomial time in the bit complexity model, does not require factorization, and works over Q or finite fields. 1 Introduction Resolving singularities is a central problem in computational algebraic geometry. In this paper we describe a new algorithm for resolving singularities of irreducible plane curves. The algorithm runs in polynomialtime in the bit complexity model, does not require polynomial factorization, and works over Q or any finite field. Classical algorithms for resolving singularities [2, 15, 7] use a combination of methods involving  the Newton polygon, a polygon in Z 2 whose vertices are the exponents of terms in f ;  Puiseux series, power series with fractional exponents. These algorithms take polynomial time if we assume efficient factorization over algebraic extensions of the base field and unittime arithmetic these extensions. Teitelbaum [13] establishes bounds on the d...
Annegret, Construction of CM Picard curves
 Math. Comp
"... Abstract. In this article we generalize the CM method for elliptic and hyperelliptic curves to Picard curves. We describe the algorithm in detail and discuss the results of our implementation. 1. ..."
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Abstract. In this article we generalize the CM method for elliptic and hyperelliptic curves to Picard curves. We describe the algorithm in detail and discuss the results of our implementation. 1.
”New” Veneziano amplitudes from ”old” Fermat (hyper)surfaces
, 2003
"... The history of the discovery of bosonic string theory is well documented. This theory evolved as an attempt to find a multidimensional analogue of Euler’s beta function to describe the multiparticle Veneziano amplitudes. Such an analogue had in fact been known in mathematics at least in 1922. Its ma ..."
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The history of the discovery of bosonic string theory is well documented. This theory evolved as an attempt to find a multidimensional analogue of Euler’s beta function to describe the multiparticle Veneziano amplitudes. Such an analogue had in fact been known in mathematics at least in 1922. Its mathematical meaning was studied subsequently from different angles by mathematicians such as Selberg, Weil and Deligne among others. The mathematical interpretation of this multidimensional beta function that was developed subsequently is markedly different from that described in physics literature. This work aims to bridge the gap between the mathematical and physical treatments. Using some results of recent publications (e.g. J.Geom.Phys.38 (2001) 81; ibid 43 (2002) 45) new topological, algebrogeometric, numbertheoretic and combinatorial treatment of the multiparticle Veneziano amplitudes is developed. As a result, an entirely new physical meaning of these amplitudes is emerging: they are periods of differential forms associated with homology cycles on Fermat (hyper)surfaces. Such (hyper)surfaces are considered as complex projective varieties of Hodge type. Although the computational formalism developed in this work resembles that used in mirror symmetry calculations, many additional results from mathematics are used along with their suitable physical interpretation. For instance, the Hodge spectrum of the Fermat (hyper)surfaces is in onetoone correspondence with the possible spectrum of particle masses. The formalism also allows us to obtain correlation functions of both conformal field theory and particle physics using the same type of the PicardFuchs equations whose solutions are being interpreted in terms of periods.
Unramified abelian extensions of Galois covers
 Proceedings of Symposia in Pure Mathematics, Part 1 49
, 1989
"... Abstract: We consider a ramified Galois cover ϕ: ˆ X →P 1 x of the Riemann sphere P 1 x, with monodromy group G. The monodromy group over P 1 x of the maximal unramified abelian exponent n cover of ˆ X is an extension n ˜ G of G by the group (Z/nZ) 2g, where g is the genus of ˆ X. Denote the set of ..."
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Abstract: We consider a ramified Galois cover ϕ: ˆ X →P 1 x of the Riemann sphere P 1 x, with monodromy group G. The monodromy group over P 1 x of the maximal unramified abelian exponent n cover of ˆ X is an extension n ˜ G of G by the group (Z/nZ) 2g, where g is the genus of ˆ X. Denote the set of linear equivalence classes of divisors of degree k on ˆ X by Pic k ( ˆ X) = Pic k. This is equipped with a natural G action. We show that the equivalence class of the extension n ˜ G → G is determined by the element of H 1 (G, Pic 0) representing Pic 1 (§2.2). From this we give an effective criterion (involving the Schur multiplier of G) to decide when this group extension splits for all n (§4.2). In particular we easily produce examples from this of cases where ˆ X has G invariant divisor classes of degree 1, but no G invariant divisor of degree 1 (§5.1). The extension n ˜ G → G naturally factors into a sequence n ˜ G → H → G where H is the smallest quotient of n ˜ G giving a frattini cover (§1.1) that fits between n ˜ G and G. Extension of the main result of §4.2 would consider all maximal quotients M of n ˜ G such that M → G splits. We note that the sequence including such an M factors through H, and by example we demonstrate that such maximal quotients M may not be unique (§5.2). INTRODUCTION: We consider a ramified Galois cover ϕ: ˆ X →P 1 x of the Riemann sphere P 1 x, with monodromy group G. Choose an integer n>1. Let ˆ Xn be the maximal unramified abelian exponent n