论文标题
CM通过Shimura互惠属3属3属曲线
Genus 3 hyperelliptic curves with CM via Shimura Reciprocity
论文作者
论文摘要
直到C上的同构，每一个简单的主要极化的Abelian尺寸3种是3属的光滑投射曲线的Jacobian。此外，此曲线要么是高纤维化曲线，要么是平面四分之一。鉴于六氧化cm场K，我们表明，如果存在k的过度椭圆形雅各布式，那么所有主要偏振的阿伯利亚品种与之共轭是过度的。使用Shimura的互惠定律，我们提供了一种用于计算初始曲线不变的近似值的算法，以及它们的Galois结合物。这使我们可以用CM定义并计算3属的双层次曲线的类别多项式。
Up to isomorphism over C, every simple principally polarized abelian variety of dimension 3 is the Jacobian of a smooth projective curve of genus 3. Furthermore, this curve is either a hyperelliptic curve or a plane quartic. Given a sextic CM field K, we show that if there exists a hyperelliptic Jacobian with CM by K, then all principally polarized abelian varieties that are Galois conjugated to it are hyperelliptic. Using Shimura's reciprocity law, we give an algorithm for computing approximations of the invariants of the initial curve, as well as their Galois conjugates. This allows us ton define and compute class polynomials for genus 3 hyperelliptic curves with CM.
