Division polynomials and canonical local heights on hyperelliptic Jacobians

Uchida, Yukihiro

doi:10.1007/s00229-010-0394-9

Cited by 8 publications

(4 citation statements)

References 21 publications

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“…Apart from the powers of C, the latter is the same as Kanayama's phi-function introduced in [22] in genus 2, and considered for hyperelliptic curves of arbitrary genus in [30]. The sequence (φ n ) is a natural companion to (τ n ): it satisfies the same Somos-6 recurrence (1.1) and produces the same values of the first integrals K 1 , K 2 .…”

Section: Solution Of the Initial Value Problemmentioning

confidence: 99%

Sigma-function solution to the general Somos-6 recurrence via hyperelliptic Prym varieties

Fedorov¹,

Hone²

2015

J Integrable Syst

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We construct the explicit solution of the initial value problem for sequences generated by the general Somos-6 recurrence relation, in terms of the Kleinian sigmafunction of genus two. For each sequence there is an associated genus two curve X, such that iteration of the recurrence corresponds to translation by a fixed vector in the Jacobian of X. The construction is based on a Lax pair with a spectral curve S of genus four admitting an involution σ with two fixed points, and the Jacobian of X arises as the Prym variety Prym(S, σ).

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Section: Solution Of the Initial Value Problemmentioning

confidence: 99%

Sigma-function solution to the general Somos-6 recurrence via hyperelliptic Prym varieties

Fedorov¹,

Hone²

2015

J Integrable Syst

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“…The explicit realisation of addition law should also have applications to the theory of heights on hyperelliptic Jacobians. In [10] ℘ functions are used to produce formulas for division polynomials on hyperelliptic curves of low genera which was later applied in [5] to compute canonical heights on genus 2 curves. Though the question of division polynomials isn't treated explicitly in the present paper it is strongly related to the reduction algorithm we propose as it is essentially equivalent to reduced divisors of the form nP on the curve.…”

Section: Introductionmentioning

confidence: 99%

Addition of Divisors on Hyperelliptic Curves via Interpolation Polynomials

Bernatska¹,

Kopeliovich²

2020

SIGMA

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Two problems are addressed: reduction of an arbitrary degree non-special divisor to the equivalent divisor of the degree equal to genus of a curve, and addition of divisors of arbitrary degrees. The hyperelliptic case is considered as the simplest model. Explicit formulas defining reduced divisors for some particular cases are found. The reduced divisors are obtained in the form of solution of the Jacobi inversion problem which provides the way of computing Abelian functions on arbitrary non-special divisors. An effective reduction algorithm is proposed, which has the advantage that it involves only arithmetic operations on polynomials. The proposed addition algorithm contains more details comparing with the known in cryptography, and is extended to divisors of arbitrary degrees comparing with the known in the theory of hyperelliptic functions.

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“…Though high genera are considered as nonsecure and have a limited application in cryptology, they are still interesting. One application of such algorithms can be finding conditions on torsion points that exist in Jacobians similar to the mentioned in [22] and [23,24]. Those conditions were used in [25] to give new algorithms to compute Falting's invariants for hyperelliptic curves.…”

Section: Introductionmentioning

confidence: 99%

Addition of divisors on trigonal curves

Bernatska¹,

Kopeliovich²

2020

Preprint

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In this paper we present a realisation of addition of divisors on the basis of uniformisation of Jacobi varieties of plain algebraic curves. Addition is realised through consecutive reduction of a degree g + 1 divisor to a divisor of degree g. Representation of these divisors does not go beyond the functions which give a solution of the Jacobi inversion problem -the functions of the smallest order. We focus on the simplest non-hyperelliptic family which is the family of trigonal curves. The method can be generalised to a curve of arbitrary gonality.

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Division polynomials and canonical local heights on hyperelliptic Jacobians

Cited by 8 publications

References 21 publications

Sigma-function solution to the general Somos-6 recurrence via hyperelliptic Prym varieties

Sigma-function solution to the general Somos-6 recurrence via hyperelliptic Prym varieties

Addition of Divisors on Hyperelliptic Curves via Interpolation Polynomials

Addition of divisors on trigonal curves

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