论文标题
二次多项式和理性点的动力学曲线$ 4 $
Dynamics of quadratic polynomials and rational points on a curve of genus $4$
论文作者
论文摘要
令$ f_t（z）= z^2+t $。对于任何$ z \ in \ mathbb {q} $，令$ s_z $为$ t \ in \ mathbb {q} $的集合，这样$ z $对于$ f_t $来说是preperiodic。在本文中，假设Flynn，Poonen和Schaefer的众所周知的猜想，我们证明了$ s_z $超过$ z \ in \ mathbb {q} $的$ s_z $的统一结果。为了证明这一点，我们需要确定特定的非hyperelliptic曲线$ c $ of属$ 4 $定义的$ \ mathbb {q} $的理性点集。我们使用Chabauty的方法，它要求我们确定$ c $的Jacobian $ j $的Mordell-Weil等级。我们给出了两个证据，即排名为$ 1 $：一种分析证明，这是基于BSD等级的猜想，$ j $和L系列的一些标准猜想，以及一个无条件的代数证明，但分别依赖于两个数量的$ 12 $ $ 12 $和$ 24 $的班级组的计算。我们最终将两个证明中获得的信息结合在一起，以提供$ J $的强BSD猜想的数值验证。
Let $f_t(z)=z^2+t$. For any $z\in\mathbb{Q}$, let $S_z$ be the collection of $t\in\mathbb{Q}$ such that $z$ is preperiodic for $f_t$. In this article, assuming a well-known conjecture of Flynn, Poonen, and Schaefer, we prove a uniform result regarding the size of $S_z$ over $z\in\mathbb{Q}$. In order to prove it, we need to determine the set of rational points on a specific non-hyperelliptic curve $C$ of genus $4$ defined over $\mathbb{Q}$. We use Chabauty's method, which requires us to determine the Mordell-Weil rank of the Jacobian $J$ of $C$. We give two proofs that the rank is $1$: an analytic proof, which is conditional on the BSD rank conjecture for $J$ and some standard conjectures on L-series, and an algebraic proof, which is unconditional, but relies on the computation of the class groups of two number fields of degree $12$ and degree $24$, respectively. We finally combine the information obtained from both proofs to provide a numerical verification of the strong BSD conjecture for $J$.
