Hyperelliptic Curves With Quaternion Multiplication And Unsw Sydney

Hyperelliptic curves with quaternion multiplication, and - UNSW Sydney

Jun 14, 2023 Hyperelliptic curves with quaternion multiplication, and their arithmetic. Wednesday, 14 June 2023. Abstract. For each even genus g, we study explicitly the family of hyperelliptic curves with a specific action by the quaternion group of order 8.

Counting points on hyperelliptic curves - UNSW Sydney

Some algorithms: Naive: try every pair x; y 2 Fp. Complexity: O(p2+"). Slightly better: try every x 2 Fp, test if x3 + ax + b is a square modulo p. Complexity: O(p1+"). Shanks{Mestre: use group structure on X(Fp), \baby-step, giant-step" algorithm, and the Hasse bound. j#X(Fp) (p + 1)j. Complexity: O(p1=4+"). p.

improvements-point-counting-hyperelliptic-curves-genus-two - UNSW Sydney

Motivated by the fact that Jacobians of hyperelliptic curves of genus two have recently been found to be good alternatives to elliptic curves in cryptography, we investigate the possibility of applying the improvements of Elkies and Atkin to Pila's point counting algorithm for such varieties.

Unexpected quadratic points on random hyperelliptic curves - UNSW Sydney

But for a positive proportion of genus g odd hyperelliptic curves over QQ, we give a bound on the number of quadratic points not arising in this way. The proof uses tropical geometry work of Park, as well as that of Bhargava and Gross on average ranks of hyperelliptic Jacobians.

Counting points on hyperelliptic curves over nite elds - UNSW Sites

in detail for the special case of a hyperelliptic curve over a prime eld. In particu- lar, we explain how to obtain square-root time and average polynomial time

Isogenies for point counting on genus two hyperelliptic curves with

Milio have computed analogues of modular polynomials for genus-2 curves whose Jacobians have real multiplication by maximal orders of small discriminant. In this article, we prove Atkin-style results for genus-2 Jacobians with real multiplication Sean Ballentine Department of Mathematics, University of Maryland, 4176 Campus Dr., College Park ...

Computing $L$-series of geometrically hyperelliptic curves of genus

Computing -series of geometrically hyperelliptic curves of genus three - Volume 19 Issue A ... Sydney NSW 2052, Australia email [email protected]. Andrew V. Sutherland Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge MA 02139, USA email [email protected] ...

Hyperelliptic Curves over Q and Number Fields

A hyperelliptic curve, which is taken to include the genus one case, is given by a nonsingular generalized Weierstrass equation y 2 + h (x)y = f (x), where h (x) and f (x) are polynomials over a field K. The curve is viewed as embedded in a weighted projective space, with weights 1, g + 1, and 1, in which the points at infinity are nonsingular.

Speeding Up Point Multiplication on Hyperelliptic Curves with

Jan 1, 2002 For this special family of curves, a speedup of up to 55 (59) % can be achieved over the best general methods for a 160-bit point multiplication in case of genus g =2 (3). Keywords. Point Multiplication. Elliptic Curve. Characteristic Polynomial. Elliptic Curf. Hyperelliptic Curve. These keywords were added by machine and not by the authors.

Speeding up Scalar Multiplication in Genus 2 Hyperelliptic Curves with

This paper proposes an efficient scalar multiplication algorithm for hyperelliptic curves, which is based on the idea that efficient endomorphisms can be used to speed up scalar multiplication. We first present a new Frobenius expansion method for special hyperelliptic curves that have Gallant-Lambert-Vanstone (GLV) endomorphisms.

Hyperelliptic Curves with Many Automorphisms - arXiv.org

We determine all complex hyperelliptic curves with many automorphisms and decide which of their jacobians have complex multiplication. MSC classification: 14H45 (14H37, 14K22) 1 Introduction. Let X be a smooth connected projective algebraic curve of genus g 2 over the field of complex numbers.

EXPLICIT HYPERELLIPTIC CURVES WITH REAL MULTIPLICATION AND PERMUTATION

The aim of this paper is to present a very explicit construction of one parameter families of hyperelliptic curves C of genus (p 1 )/ 2, for any odd prime number/?, with the property that the endomorphism algebra of the jacobian of C contains the real subfield Q(2COS(2TT/p)) of the cyclotomic field Q(e2ni/p).

[PDF] Complex Multiplication Formulae for Hyperelliptic Curves of Genus

Dec 1, 1998 [PDF] Complex Multiplication Formulae for Hyperelliptic Curves of Genus Three | Semantic Scholar. DOI: 10.3836/TJM/1270041822. Corpus ID: 16725639. Complex Multiplication Formulae for Hyperelliptic Curves of Genus Three. Y. nishi. Published 1 December 1998. Mathematics. Tokyo Journal of Mathematics. TLDR.

Rational points on AtkinLehner quotients of geometrically

Sep 1, 2023 1. Introduction. It is an important problem to study the modular curves X 0 ( N) and their rational points. These curves are the coarse moduli spaces for elliptic curves with a 0 ( N) -level structure and an understanding of their rational points leads to a classification of elliptic curves equipped with an isogeny (cf. [29] ).

Hyperelliptic curve - Wikipedia

In algebraic geometry, a hyperelliptic curve is an algebraic curve of genus g > 1, given by an equation of the form. where f ( x) is a polynomial of degree n = 2 g + 1 > 4 or n = 2 g + 2 > 4 with n distinct roots, and h ( x) is a polynomial of degree < g + 2 (if the characteristic of the ground field is not 2, one can take h ( x) = 0).

Counting points on geometrically hyperelliptic curves of genus three

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Isogenous hyperelliptic and non-hyperelliptic Jacobians with maximal

Apr 11, 2021 Citations (1) References (59) Abstract. We analyze complex multiplication for Jacobians of curves of genus 3, as well as the resulting Shimura class groups and their subgroups corresponding to...

Hyperelliptic supersingular curves | Semantic Scholar

Hyperelliptic supersingular curves. F. Oort. Published 1991. Mathematics. Consider algebraic curves of genus 3 in positive characteristic which are hyperelliptic, and whose Jacobian is a supersingular abelian variety. The main result of this paper is: every component of this locus in the moduli space has dimension one (cf. Theorem (1.12)).

Hyperelliptic Curve -- from Wolfram MathWorld

Mar 15, 2024 A hyperelliptic curve is an algebraic curve given by an equation of the form y^2=f (x), where f (x) is a polynomial of degree n>4 with n distinct roots. If f (x) is a cubic or quartic polynomial, then the curve is called an elliptic curve. The genus of a hyperelliptic curve is related to the degree of the polynomial.

A Hyperelliptic Curve with Real Multiplication of Degree Two

A Hyperelliptic Curve with Real Multiplication of Degree Two. Harvey Cohn. Chapter. 493 Accesses. Abstract. The analogue of complex multiplication in an elliptic curve is illustrated for a hyperelliptic curve with real multiplication of degree two (over C).

[PDF] Quaternions , polarizations and class numbers - Semantic Scholar

Published 2002. Mathematics. We study abelian varieties A with multiplication by a totally indefinite quaternion algebra over a totally real number field and give a criterion for the existence of principal polarizations on them in pure arithmetic terms.

Hyperelliptic curves over a finite field - Elliptic curves - SageMath

Hyperelliptic org

hyperelliptic.org. Explicit-Formulas Database. Handbook of Elliptic and Hyperelliptic Curve Cryptography. Tanja Lange's homepage.

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