1,395 research outputs found

    Twisted Jacobi Intersections Curves

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    In this paper, the twisted Jacobi intersections which contains Jacobi intersections as a special case is introduced. We show that every elliptic curve over the prime field with three points of order 22 is isomorphic to a twisted Jacobi intersections curve. Some fast explicit formulae for twisted Jacobi intersections curves in projective coordinates are presented. These explicit formulae for addition and doubling are almost as fast as the Jacobi intersections. In addition, the scalar multiplication can be more effective in twisted Jacobi intersections than in Jacobi intersections. Moreover, we propose new addition formulae which are independent of parameters of curves and more effective in reality than the previous formulae in the literature

    Higher Order Intersections in Low-Dimensional Topology

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    We show how to measure the failure of the Whitney trick in dimension 4 by constructing higher- order intersection invariants of Whitney towers built from iterated Whitney disks on immersed surfaces in 4-manifolds. For Whitney towers on immersed disks in the 4-ball, we identify some of these new invariants with previously known link invariants like Milnor, Sato-Levine and Arf invariants. We also define higher- order Sato-Levine and Arf invariants and show that these invariants detect the obstructions to framing a twisted Whitney tower. Together with Milnor invariants, these higher-order invariants are shown to classify the existence of (twisted) Whitney towers of increasing order in the 4-ball. A conjecture regarding the non- triviality of the higher-order Arf invariants is formulated, and related implications for filtrations of string links and 3-dimensional homology cylinders are described. This article is an announcement and summary of results to be published in several forthcoming papers

    Milnor Invariants and Twisted Whitney Towers

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    This paper describes the relationship between the first non-vanishing Milnor invariants of a classical link and the intersection invariant of a twisted Whitney tower. This is a certain 2-complex in the 4-ball, built from immersed disks bounded by the given link in the 3-sphere together with finitely many `layers' of Whitney disks. The intersection invariant is a higher-order generalization of the intersection number between two immersed disks in the 4-ball, well known to give the linking number of the link on the boundary, which measures intersections among the Whitney disks and the disks bounding the given link, together with information that measures the twists (framing obstructions) of the Whitney disks. This interpretation of Milnor invariants as higher-order intersection invariants plays a key role in the classifications of both the framed and twisted Whitney tower filtrations on link concordance (as sketched in this paper). Here we show how to realize the higher-order Arf invariants, which also play a role in the classifications, and derive new geometric characterizations of links with vanishing Milnor invariants of length less than or equal to 2k.Comment: Typo corrected in statement of Theorem 16; no change to proof needed. Otherwise, this revision conforms with the version published in the Journal of Topology. 36 pages, 23 figure

    Addition law structure of elliptic curves

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    The study of alternative models for elliptic curves has found recent interest from cryptographic applications, once it was recognized that such models provide more efficiently computable algorithms for the group law than the standard Weierstrass model. Examples of such models arise via symmetries induced by a rational torsion structure. We analyze the module structure of the space of sections of the addition morphisms, determine explicit dimension formulas for the spaces of sections and their eigenspaces under the action of torsion groups, and apply this to specific models of elliptic curves with parametrized torsion subgroups

    The Weight Distributions of Cyclic Codes and Elliptic Curves

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    Cyclic codes with two zeros and their dual codes as a practically and theoretically interesting class of linear codes, have been studied for many years. However, the weight distributions of cyclic codes are difficult to determine. From elliptic curves, this paper determines the weight distributions of dual codes of cyclic codes with two zeros for a few more cases

    Whitney tower concordance of classical links

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    This paper computes Whitney tower filtrations of classical links. Whitney towers consist of iterated stages of Whitney disks and allow a tree-valued intersection theory, showing that the associated graded quotients of the filtration are finitely generated abelian groups. Twisted Whitney towers are studied and a new quadratic refinement of the intersection theory is introduced, measuring Whitney disk framing obstructions. It is shown that the filtrations are completely classified by Milnor invariants together with new higher-order Sato-Levine and higher-order Arf invariants, which are obstructions to framing a twisted Whitney tower in the 4-ball bounded by a link in the 3-sphere. Applications include computation of the grope filtration, and new geometric characterizations of Milnor's link invariants.Comment: Only change is the addition of this comment: This paper subsumes the entire preprint "Geometric Filtrations of Classical Link Concordance" (arXiv:1101.3477v2 [math.GT]) and the first six sections of the preprint "Universal Quadratic Forms and Untwisting Whitney Towers" (arXiv:1101.3480v2 [math.GT]

    Witten Genus and String Complete Intersections

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    In this note, we prove that the Witten genus of nonsingular string complete intersections in product of complex projective spaces vanishes. Our result generalizes a known result of Landweber and Stong (cf. [HBJ]).Comment: Some materials and references are adde

    Conifold Transitions in M-theory on Calabi-Yau Fourfolds with Background Fluxes

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    We consider topology changing transitions for M-theory compactifications on Calabi-Yau fourfolds with background G-flux. The local geometry of the transition is generically a genus g curve of conifold singularities, which engineers a 3d gauge theory with four supercharges, near the intersection of Coulomb and Higgs branches. We identify a set of canonical, minimal flux quanta which solve the local quantization condition on G for a given geometry, including new solutions in which the flux is neither of horizontal nor vertical type. A local analysis of the flux superpotential shows that the potential has flat directions for a subset of these fluxes and the topologically different phases can be dynamically connected. For special geometries and background configurations, the local transitions extend to extremal transitions between global fourfold compactifications with flux. By a circle decompactification the M-theory analysis identifies consistent flux configurations in four-dimensional F-theory compactifications and flat directions in the deformation space of branes with bundles.Comment: 93 pages; v2: minor changes and references adde
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