AI 中文总结
研究算术动力学中亏格度猜想，证明非等平凡单参数有理映射代数族中不同动力曲线亏格度趋于无穷，结合多种理论证明相关结果，还得到高维类似及应用，如迭代原像和预周期点的一致有界性。
AI 中文摘要
我们证明了算术动力学中的亏格度猜想：对于\(\mathbb{P}^1\)上任何非等平凡的单参数有理映射代数族，不同动力曲线的亏格度趋于无穷。更一般地，在灵活的拉泰斯族之外，每一小段水平曲线的亏格度趋于无穷，且其亏格相对于参数曲线的次数超线性增长。我们还在自然分歧和乘子一般性假设下得到了高维类似结果。作为应用，我们证明了数域上迭代原像的一致有界性结果以及函数域上预周期点的几何一致有界性结果。证明结合了算术等分布、编织流和分歧理论；分歧机制促使亏格和亏格度增长。
英文摘要
We prove the Gonality Conjecture in arithmetic dynamics: for any non-isotrivial one-parameter algebraic family of rational maps on $\mathbb{P}^1_{\mathbb{C}}$, the gonality of distinct dynatomic curves tends to infinity. More generally, outside the flexible Lattès family, every small sequence of horizontal curves has gonality tending to infinity, and its genus grows superlinearly with its degree over the parameter curve. We also obtain higher-dimensional analogues under natural bifurcation and multiplier-genericity hypotheses. As applications, we prove uniform boundedness results for iterated preimages over number fields and geometric uniform boundedness results for preperiodic points over function fields. The proof combines arithmetic equidistribution, woven currents, and bifurcation theory; the bifurcation mechanism is what forces the growth of genus and gonality.
