Explicit Hyperelliptic Curves With Real Multiplication and Permutation Polynomials
Explicit Hyperelliptic Curves With Real Multiplication and Permutation Polynomials
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具有实数乘法和置换多项式的显式超椭圆曲线
DOI：
10.4153/cjm-1991-061-x
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发表时间：
1991
期刊：
影响因子：
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通讯作者：
A. Verberkmoes
中科院分区：
文献类型：
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作者：
Walter Tautz;Jaap Top;A. Verberkmoes
摘要本文给出了对于任意奇素数p属(p−1)/ 2的超椭圆曲线C的一个参数族的非常显式的构造，并证明了C的雅可比矩阵的自同态代数包含环切场Q（e 2π i/p）的实子域Q（2cos (2π/p)）。本文将给出两个证明，证明所构造的曲线具有这种性质。一种方法是在自同构群中提供具有统一的p根的双覆盖。另一种是通过显式地写出C x C中的对应方程，它定义了C的雅可比矩阵上乘以2cos（2π/ p）。作为副产物，我们得到了多项式，它定义了某些同余类中所有素数的双射映射F→F。
Abstract 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 p, with the property that the endomorphism algebra of the jacobian of C contains the real subfield Q(2 cos(2π/p)) of the cyclotomic field Q(e 2π i/p ). Two proofs of the fact that the constructed curves have this property will be given. One is by providing a double cover with the pth roots of unity in its automorphism group. The other is by explicitly writing down equations of a correspondence in C x C which defines multiplication by 2cos(2π/ p) on the jacobian of C. As a byproduct we obtain polynomials which define bijective maps F ℓ → F ℓ for all prime numbers in certain congruence classes.