论文标题
二次图的模量空间：算术和几何形状
Moduli spaces of quadratic maps: arithmetic and geometry
论文作者
论文摘要
我们在复杂动态中两个长期的开放问题之间建立了一个含义。 $ n $ - gleason多项式$ g_n \ in \ mathbb {q} [c} [c] $的根源包括$ 0 $ - 二维模量二次多项式的空间，带有$ n $ n $ periodic的关键点。 $ \ mathrm {per} _n（0）$是$ 1 $ -Dimensional Moduli在$ \ Mathbb {p}^1 $上的二次合理地图的空间，带有$ n $ n $ - periodic关键点。我们表明，如果$ g_n $在$ \ mathbb {q} $上不可记述，则$ \ mathrm {per} _n（0）$在$ \ mathbb {c} $上是不可约的。为此，我们展示了$ \ mathbb {q} $ - 使用可允许的封面完成Hurwitz Space的完成，在投影完成$ \ mathrm {per} _n（0）$的揭示完成中。相比之下，算术动力学中的统一界定猜想暗示，对于足够大的$ n $，$ \ mathrm {per} _n（0）$本身没有$ \ mathbb {q} $ - 理性点。
We establish an implication between two long-standing open problems in complex dynamics. The roots of the $n$-th Gleason polynomial $G_n\in\mathbb{Q}[c]$ comprise the $0$-dimensional moduli space of quadratic polynomials with an $n$-periodic critical point. $\mathrm{Per}_n(0)$ is the $1$-dimensional moduli space of quadratic rational maps on $\mathbb{P}^1$ with an $n$-periodic critical point. We show that if $G_n$ is irreducible over $\mathbb{Q}$, then $\mathrm{Per}_n(0)$ is irreducible over $\mathbb{C}$. To do this, we exhibit a $\mathbb{Q}$-rational smooth point on a projective completion of $\mathrm{Per}_n(0)$, using the admissible covers completion of a Hurwitz space. In contrast, the Uniform Boundedness Conjecture in arithmetic dynamics would imply that for sufficiently large $n$, $\mathrm{Per}_n(0)$ itself has no $\mathbb{Q}$-rational points.
