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首页> 外文会议>International Symposium on Symbolic and Algebraic Computation Aug 3-6, 2003 Philadelphia, Pennsylvania, USA >Determining the Automorphism Group of a Hyperelliptic Curve
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Determining the Automorphism Group of a Hyperelliptic Curve

机译：确定超椭圆曲线的自同构群

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摘要

In this note we discuss techniques for determining the automorphism group of a genus g hyperelliptic curve χ_g defined over an algebraically closed field κ of characteristic zero. The first technique uses the classical GL_2(κ)-invariants of binary forms. This is a practical method for curves of small genus, but has limitations as the genus increases, due to the fact that such invariants are not known for large genus. The second approach, which uses dihedral invariants of hyperelliptic curves, is a very convenient method and works well in all genera. First we define the normal decomposition of a hyperelliptic curve with extra automorphisms. Then dihedral invariants are defined in terms of the coefficients of this normal decomposition. We define such invariants independently of the automorphism group Aut(χ_g). However, to compute such invariants the curve is required to be in its normal form. This requires solving a nonlinear system of equations. We find conditions in terms of classical invariants of binary forms for a curve to have reduced automorphism group A_4, S_4, A_5. As far as we are aware, such results have not appeared before in the literature.
机译：在本说明中，我们讨论了确定特征为零的代数封闭场κ上定义的g类超椭圆曲线χ_g的自同构群的技术。第一种技术使用二进制形式的经典GL_2（κ）不变量。这是用于小属的曲线的实用方法，但是由于这种不变量对于大属而言是未知的，因此随着属的增加而受到限制。第二种方法使用超椭圆曲线的二面体不变量，是一种非常方便的方法，并且在所有属中都适用。首先，我们定义具有额外自同构性的超椭圆曲线的正态分解。然后根据该正态分解的系数定义二面体不变式。我们独立于自同构群Aut（χ_g）定义这样的不变量。但是，要计算此类不变量，则曲线必须为正常形式。这需要求解非线性方程组。我们根据二进制形式的经典不变量找到条件，以使曲线的自同构群A_4，S_4，A_5减少。据我们所知，这种结果在文献中还没有出现过。

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