MinimalDegree Liftings of Hyperelliptic Curves
MinimalDegree Liftings of Hyperelliptic Curves
复制标题
超椭圆曲线的最小度提升
DOI：
--
复制
发表时间：
2004
期刊：
影响因子：
--
通讯作者：
Luís R. A. Finotti
中科院分区：
文献类型：
--
作者：
Luís R. A. Finotti
本文的主要目的是分析特征p > 2的理想域上的超椭圆曲线y 0 = f（x 0）的提升的性质（到Witt向量环上的超椭圆曲线），这些提升的点的坐标函数具有最小度.它表明，当试图最小化x坐标的次数时，第（n+ 1）项，比如Fn，可以被认为是x 0中的多项式，使得（dp n−（d−2））/2 ≤ degFn ≤（dp +（d−2））/2，其中d = deg f（x 0）。除了上界和下界的程度，讨论的其他主题包括一个必要的条件，以实现下界和解除弗罗贝纽斯。计算方面也被认为是椭圆曲线的情况下进行了更详细的分析。证明了椭圆Teichmuller提升的坐标函数导数的一个显式公式，即dFn/dx 0 = 0（p = 2），dFn/dx 0 = A（p−1）/（p− 1）y −1 0 − ∑n−1 i=0 F（pn−i−1）idFi/dx 0（p ≥ 3），其中A是曲线的Hasse不变量.最后，在亏格为2的曲线的情况下，我们建立了最小度提升和望月的“典型提升”理论之间的联系。
The main goal of this paper is to analyze the properties of lifts of hyperelliptic curves y 0 = f(x0) over perfect fields of characteristic p > 2 (to hyperelliptic curves over the ring of Witt vectors) that have lifts of points whose coordinate functions have minimal degrees. It is shown that, when trying to minimize the degrees of the x-coordinate, the (n+ 1)-th entry, say Fn, can be taken to be a polynomial in x0 such that (dp n− (d−2))/2 ≤ degFn ≤ (dp +(d−2))/2, where d = deg f(x0). Besides upper and lower bounds for the degrees, other topics discussed include a necessary condition to achieve the lower bounds and lifting the Frobenius. Computational aspects are also considered and the case of elliptic curves is analyzed in more detail. An explicit formula for derivatives of coordinate functions of the elliptic Teichmuller lift is proved, namely dFn/dx0 = 0, if p = 2, and dFn/dx0 = A (p−1)/(p−1) y −1 0 − ∑n−1 i=0 F (pn−i−1) i dFi/dx0, if p ≥ 3, where A is the Hasse invariant of the curve. Finally, we establish a connection between minimal degree liftings and Mochizuki’s theory of “canonical liftings” in the case of genus 2 curves.