Cycles of quadratic polynomials and rational points on a genus-$2$ curve
Cycles of quadratic polynomials and rational points on a genus-$2$ curve
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genus-$2$ 曲线上的二次多项式循环和有理点
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发表时间：
1995
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通讯作者：
Edward F. Schaefer
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作者：
E. V. Flynn;B. Poonen;Edward F. Schaefer
证明了当N足够大时，Q[z]中不存在周期为N的有理周期点的二次多项式。莫顿证明没有N=4，通过显示亏格2代数曲线分类周期点的周期4是双有理数X $1 $（16），其合理的点已经计算。我们证明了当N=5时，没有一个。这里的相关曲线有亏格14，但它有亏格2商，我们通过对其雅可比矩阵执行2-下降并应用Chabauty和科尔曼的方法的改进来计算其有理点。我们希望我们的计算将作为一个模型，为其他人谁需要计算合理的点超椭圆曲线。我们还描述了三种可能的Galois稳定的5-圈，并证明了对于无穷多个N，存在Galois稳定的N-圈。此外，我们回答了Morton的一个问题，证明亏格14曲线及其商不是模的。最后，我们给出了N=6时的部分结果。
It has been conjectured that for N sufficiently large, there are no quadratic polynomials in Q[z] with rational periodic points of period N. Morton proved there were none with N=4, by showing that the genus 2 algebraic curve that classifies periodic points of period 4 is birational to X$_1$(16), whose rational points had been previously computed. We prove there are none with N=5. Here the relevant curve has genus 14, but it has a genus 2 quotient, whose rational points we compute by performing a 2-descent on its Jacobian and applying a refinement of the method of Chabauty and Coleman. We hope that our computation will serve as a model for others who need to compute rational points on hyperelliptic curves. We also describe the three possible Galois-stable 5-cycles, and show that there exist Galois-stable N-cycles for infinitely many N. Furthermore, we answer a question of Morton by showing that the genus 14 curve and its quotient are not modular. Finally, we mention some partial results for N=6.