1,395 research outputs found
Twisted Jacobi Intersections Curves
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
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
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
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
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
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
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
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
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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