Division by 1–ζ on Superelliptic Curves and Jacobians

2020 ◽  
Author(s):  
Vishal Arul
Keyword(s):  
Hyperelliptic Curve ◽  
Theta Divisor ◽  
Intersection Multiplicities ◽  
Superelliptic Curves ◽  
Mumford Representation ◽  
Effective Degree

Abstract Yuri Zarhin gave formulas for “dividing a point on a hyperelliptic curve by 2”. Given a point $P$ on a hyperelliptic curve $\mathcal{C}$ of genus $g$, Zarhin gives the Mumford representation of an effective degree $g$ divisor $D$ satisfying $2(D - g \infty ) \sim P - \infty $. The aim of this paper is to generalize Zarhin’s result to superelliptic curves; instead of dividing by 2, we divide by $1 - \zeta $. There is no Mumford representation for divisors on superelliptic curves, so instead we give formulas for functions that cut out a divisor $D$ satisfying $(1 - \zeta ) D \sim P - \infty $. Additionally, we study the intersection of $(1 - \zeta )^{-1} \mathcal{C}$ and the theta divisor $\Theta $ inside the Jacobian $\mathcal{J}$. We show that the intersection is contained in $\mathcal{J}[1 - \zeta ]$ and compute the intersection multiplicities.

scholarly journals Application of Divisors on a Hyperelliptic Curve in Python

2021 ◽  
Vol 3 ◽  
pp. 11-24
Author(s):  
Denys Boiko
Keyword(s):  
Programming Language ◽  
Hyperelliptic Curve ◽  
Polynomial Function ◽  
Hyperelliptic Curves ◽  
Unique Form ◽  
Python Programming Language ◽  
Definition Of ◽  
Mumford Representation ◽  
Python Programming ◽  
Special Case

The paper studies hyperelliptic curves of the genus g > 1, divisors on them and their applications in Python programming language. The basic necessary definitions and known properties of hyperelliptic curves are demonstrated, as well as the notion of polynomial function, its representation in unique form, also the notion of rational function, norm, degree and conjugate to a polynomial are presented. These facts are needed to calculate the order of points of desirable functions, and thus to quickly and efficiently calculate divisors. The definition of a divisor on a hyperelliptic curve is shown, and the main known properties of a divisor are given. There are also an example of calculating a divisor of a polynomial function, reduced and semi-reduced divisors are described, theorem of the existence of such a not unique semi-reduced divisor, and theorem of the existence of a unique reduced divisor, which is equivalent to the initial one, are proved. In particular, a semi-reduced divisor can be represented as an GCD of divisors of two polynomial functions. It is also demonstrated that each reduced divisor can be represented in unique form by pair of polynomials [a(x), b(x)], which is called Mumford representation, and several examples of its representation calculation are given. There are shown Cantor’s algorithms for calculating the sum of two divisors: its compositional part, by means of which a not unique semi-reduced divisor is formed, and the reduction part, which gives us a unique reduced divisor. In particular, special case of the compositional part of Cantor’s algorithm, doubling of the divisor, is described: it significantly reduces algorithm time complexity. Also the correctness of the algorithms are proved, examples of applications are given. The main result of the work is the implementation of the divisor calculation of a polynomial function, its Mumford representation, and Cantor’s algorithm in Python programming language. Thus, the aim of the work is to demonstrate the possibility of e↵ective use of described algorithms for further work with divisors on the hyperelliptic curve, including the development of cryptosystem, digital signature based on hyperelliptic curves, attacks on such cryptosystems.

Effective degree household network disease model

2012 ◽  
Vol 66 (1-2) ◽  
pp. 75-94 ◽  
Author(s):  
Junling Ma ◽  
P. van den Driessche ◽  
Frederick H. Willeboordse
Keyword(s):  
Disease Model ◽  
Effective Degree

scholarly journals THE HYPERELLIPTIC THETA MAP AND OSCULATING PROJECTIONS

2020 ◽  
pp. 1-23
Author(s):  
MICHELE BOLOGNESI ◽  
NÉSTOR FERNÁNDEZ VARGAS
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Vector Bundles ◽  
Hyperelliptic Curve ◽  
Geometric Description ◽  
Birational Model ◽  
Rank 2 ◽  
Semistable Vector Bundles ◽  
Ramification Locus

Abstract Let C be a hyperelliptic curve of genus $g \geq 3$ . In this paper, we give a new geometric description of the theta map for moduli spaces of rank 2 semistable vector bundles on C with trivial determinant. In order to do this, we describe a fibration of (a birational model of) the moduli space, whose fibers are GIT quotients $(\mathbb {P}^1)^{2g}//\text {PGL(2)}$ . Then, we identify the restriction of the theta map to these GIT quotients with some explicit degree 2 osculating projection. As a corollary of this construction, we obtain a birational inclusion of a fibration in Kummer $(g-1)$ -varieties over $\mathbb {P}^g$ inside the ramification locus of the theta map.

scholarly journals Two algebraic deformations of a K3 surface

1981 ◽  
Vol 82 ◽  
pp. 1-26
Author(s):  
Daniel Comenetz
Keyword(s):  
Hyperelliptic Curve ◽  
Double Cover ◽  
Rational Curves ◽  
K3 Surface ◽  
Quartic Surface ◽  
Quadric Cone ◽  
Birational Model ◽  
Affine Line ◽  
Characteristic 2 ◽  
General Member

Let X be a nonsingular algebraic K3 surface carrying a nonsingular hyperelliptic curve of genus 3 and no rational curves. Our purpose is to study two algebraic deformations of X, viz. one specialization and one generalization. We assume the characteristic ≠ 2. The generalization of X is a nonsingular quartic surface Q in P3 : we wish to show in § 1 that there is an irreducible algebraic family of surfaces over the affine line, in which X is a member and in which Q is a general member. The specialization of X is a surface Y having a birational model which is a ramified double cover of a quadric cone in P3.

scholarly journals ON SELMER RANK PARITY OF TWISTS

2016 ◽  
Vol 102 (3) ◽  
pp. 316-330 ◽  
Author(s):  
MAJID HADIAN ◽  
MATTHEW WEIDNER
Keyword(s):  
Elliptic Curves ◽  
Abelian Variety ◽  
Number Field ◽  
Arbitrary Number ◽  
Hyperelliptic Curve ◽  
Abelian Varieties ◽  
Hyperelliptic Curves ◽  
Torsion Subgroup ◽  
Tate Conjecture ◽  
Quadratic Twists

In this paper we study the variation of the $p$-Selmer rank parities of $p$-twists of a principally polarized Abelian variety over an arbitrary number field $K$ and show, under certain assumptions, that this parity is periodic with an explicit period. Our result applies in particular to principally polarized Abelian varieties with full $K$-rational $p$-torsion subgroup, arbitrary elliptic curves, and Jacobians of hyperelliptic curves. Assuming the Shafarevich–Tate conjecture, our result allows one to classify the rank parities of all quadratic twists of an elliptic or hyperelliptic curve after a finite calculation.

Random Dieudonné modules, random -divisible groups, and random curves over finite fields

2013 ◽  
Vol 12 (3) ◽  
pp. 651-676 ◽  
Author(s):  
Bryden Cais ◽  
Jordan S. Ellenberg ◽  
David Zureick-Brown
Keyword(s):  
Probability Distribution ◽  
Finite Field ◽  
Random Matrix ◽  
Hyperelliptic Curve ◽  
Hyperelliptic Curves ◽  
Plane Curves ◽  
Random Curves ◽  
Isomorphism Classes ◽  
Curves Over Finite Fields ◽  
Divisible Groups

AbstractWe describe a probability distribution on isomorphism classes of principally quasi-polarized $p$-divisible groups over a finite field $k$ of characteristic $p$ which can reasonably be thought of as a ‘uniform distribution’, and we compute the distribution of various statistics ($p$-corank, $a$-number, etc.) of $p$-divisible groups drawn from this distribution. It is then natural to ask to what extent the $p$-divisible groups attached to a randomly chosen hyperelliptic curve (respectively, curve; respectively, abelian variety) over $k$ are uniformly distributed in this sense. This heuristic is analogous to conjectures of Cohen–Lenstra type for $\text{char~} k\not = p$, in which case the random $p$-divisible group is defined by a random matrix recording the action of Frobenius. Extensive numerical investigation reveals some cases of agreement with the heuristic and some interesting discrepancies. For example, plane curves over ${\mathbf{F} }_{3} $ appear substantially less likely to be ordinary than hyperelliptic curves over ${\mathbf{F} }_{3} $.

scholarly journals Cryptographic aspects of real hyperelliptic curves

2010 ◽  
Vol 47 (1) ◽  
pp. 31-65 ◽  
Author(s):  
Michael J. Jacobson ◽  
Renate Scheidler ◽  
Andreas Stein
Keyword(s):  
Hyperelliptic Curve ◽  
Discrete Logarithm ◽  
Mathematical Structure ◽  
Hyperelliptic Curves ◽  
Cryptographic Protocols ◽  
Discrete Logarithm Problem ◽  
Explicit Formulas ◽  
Genus 2

Abstract In this paper, we give an overview of cryptographic applications using real hyperelliptic curves. We review previously proposed cryptographic protocols and discuss the infrastructure of a real hyperelliptic curve, the mathematical structure underlying all these protocols. We then describe recent improvements to infrastructure arithmetic, including explicit formulas for divisor arithmetic in genus 2, and advances in solving the infrastructure discrete logarithm problem, whose presumed intractability is the basis of security for the related cryptographic protocols.

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