Divisor Class Halving Algorithms for Genus Three Hyperelliptic Curves

2020 ◽  
Vol 29 (1) ◽  
pp. 97-104
Author(s):  
Lin YOU ◽  
Yilin YANG ◽  
Shuhong GAO
Keyword(s):  
Hyperelliptic Curves ◽  
Divisor Class

scholarly journals Improved divisor arithmetic on generic hyperelliptic curves

2020 ◽  
Vol 54 (3) ◽  
pp. 95-99
Author(s):  
Sebastian Lindner ◽  
Laurent Imbert ◽  
Michael J. Jacobson
Keyword(s):  
Algebraic Geometry ◽  
Hyperelliptic Curve ◽  
Class Group ◽  
Finite Abelian Group ◽  
Hyperelliptic Curves ◽  
Computer Algebra Systems ◽  
Divisor Class Group ◽  
Divisor Class ◽  
Open Questions ◽  
Fast Arithmetic

The divisor class group of a hyperelliptic curve defined over a finite field is a finite abelian group at the center of a number of important open questions in algebraic geometry, number theory and cryptography. Many of these problems lend themselves to numerical investigation, and as emphasized by Sutherland [14, 13], fast arithmetic in the divisor class group is crucial for their efficiency. Besides, implementations of these fundamental operations are at the core of the algebraic geometry packages of widely-used computer algebra systems such as Magma and Sage.

scholarly journals Rational Picard Group of Moduli of Pointed Hyperelliptic Curves

2020 ◽  
Vol 2020 (21) ◽  
pp. 8027-8056
Author(s):  
Federico Scavia
Keyword(s):  
Moduli Spaces ◽  
Class Group ◽  
Hyperelliptic Curves ◽  
Picard Group ◽  
Complex Numbers ◽  
Divisor Class Group ◽  
Divisor Class

Abstract We determine the rational divisor class group of the moduli spaces of smooth pointed hyperelliptic curves and of their Deligne–Mumford compactification, over the field of complex numbers.

scholarly journals Coleman integration for even-degree models of hyperelliptic curves

2015 ◽  
Vol 18 (1) ◽  
pp. 258-265 ◽  
Author(s):  
Jennifer S. Balakrishnan
Keyword(s):  
Number Theory ◽  
Computer Science ◽  
Hyperelliptic Curves ◽  
Open Book ◽  
Line Integral ◽  
Book Series ◽  
Numerical Examples ◽  
Lecture Notes ◽  
Integration Algorithms ◽  
Mathematical Sciences

The Coleman integral is a $p$-adic line integral that encapsulates various quantities of number theoretic interest. Building on the work of Harrison [J. Symbolic Comput. 47 (2012) no. 1, 89–101], we extend the Coleman integration algorithms in Balakrishnan et al. [Algorithmic number theory, Lecture Notes in Computer Science 6197 (Springer, 2010) 16–31] and Balakrishnan [ANTS-X: Proceedings of the Tenth Algorithmic Number Theory Symposium, Open Book Series 1 (Mathematical Sciences Publishers, 2013) 41–61] to even-degree models of hyperelliptic curves. We illustrate our methods with numerical examples computed in Sage.

scholarly journals Integral points on hyperelliptic curves

2008 ◽  
Vol 2 (8) ◽  
pp. 859-885 ◽  
Author(s):  
Yann Bugeaud ◽  
Maurice Mignotte ◽  
Samir Siksek ◽  
Michael Stoll ◽  
Szabolcs Tengely
Keyword(s):  
Hyperelliptic Curves ◽  
Integral Points

The surfaces whose prime-sections contain a

1934 ◽  
Vol 30 (2) ◽  
pp. 170-177 ◽  
Author(s):  
J. Bronowski
Keyword(s):  
Hyperelliptic Curves ◽  
Ruled Surfaces ◽  
Minimum Order ◽  
Rational Surfaces ◽  
Pencil Of Conics

The surfaces whose prime-sections are hyperelliptic curves of genus p have been classified by G. Castelnuovo. If p > 1, they are the surfaces which contain a (rational) pencil of conics, which traces the on the prime-sections. Thus, if we exclude ruled surfaces, they are rational surfaces. The supernormal surfaces are of order 4p + 4 and lie in space [3p + 5]. The minimum directrix curve to the pencil of conics—that is, the curve of minimum order which meets each conic in one point—may be of any order k, where 0 ≤ k ≤ p + 1. The prime-sections of these surfaces are conveniently represented on the normal rational ruled surfaces, either by quadric sections, or by quadric sections residual to a generator, according as k is even or odd.

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