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Miura :: Miura

Miura -- A Package for Miura Curve Arithmetic

Description

The Miura package realizes arithmetic on the curves such as hyper-elliptic curves (e.g., y^2=x^5+x+1), C_{ab} curves (e.g., y^3=x^4+2x+1), complete intersection (e.g. {y^2-x^3-1,z^2-x*y-1}). For the Miura form, the pole orders should be specified such as 2 and 3 for x and y of an elliptic curve. Currently, only divisor class group computation is available for the package. For the elliptic curves, [(P)-(O)]+[(Q)-(O)] = [(P+Q)-(O)] for two points P, Q and the point O at infinity. For the general nonsingular curves, any divisor class is uniquely expressed by E-g(O) with E a positive divisor of degree g (genus). This package reduces the divisor class group addition to ideal class group multiplication, and utilizes Groebner basis computation. See http://arxiv.org/pdf/1512.08040v1.pdf for the detail

Author

Version

This documentation describes version 0.2 of Miura.

Source code

The source code from which this documentation is derived is in the file Miura.m2.

Exports

  • Functions and commands
    • add -- Add Reduced Ideals
    • double -- Double Reduced Ideal
    • reduced -- Compute Reduced Ideal
    • scalarMultiplication -- Add Reduced Ideal Multiple Times
    • setPolynomialRing (missing documentation)
    • setQuotientRing (missing documentation)
  • Symbols
    • PR (missing documentation)
    • QR (missing documentation)