论文标题
$ \ mathcal {a} _g $中的Torelli基因座和牛顿层之间的一些不太可能的交集
Some unlikely intersections between the Torelli locus and Newton strata in $\mathcal{A}_g$
论文作者
论文摘要
令$ p $是一个奇怪的素数。特征$ p $中曲线的牛顿多边形可能是什么？同等地，哪个牛顿地层与$ \ Mathcal {a} _g $中的Torelli基因座相交？在本说明中，我们使用$ \ mathbb {z}/p \ mathbb {z} $ - 动作研究了某些曲线的牛顿多边形。这些曲线中的许多曲线在$ \ Mathcal {a} _g $中表现出不太可能的交叉点。这是特别感兴趣的一个例子：修复属$ g $。我们证明，对于任何$ k $，带有$ \ frac {2g} {3} - \ frac {2p（p-1）} {3} {3} \ geq 2k（p-1）$，存在牛顿newton Polygon具有Slopes $ $ \ \ {0,1 \}^g-s的newton $ g $的曲线， \ {\ frac {1} {2} \}^{2k（p-1）} $。这为Oort的猜想提供了证据，即两条曲线的牛顿多边形的合并再次是曲线的牛顿多边形。我们还构建了曲线$ \ {c_g \} _ {g \ geq 1} $的家庭，其中$ c_g $是属$ g $的曲线，其牛顿多边形具有有趣的渐近属性。例如，我们构建了一个曲线系列，其牛顿多边形在下面由图$ y = \ frac {x^2} {4G} $渐近地界定。该证明使用$ \ mathbb {z}/p \ mathbb {z} $的牛顿对绞车的结果 - 作者的曲线封面，除了Booher-Pries在实现此霍奇绑定方面的最新工作外。
Let $p$ be an odd prime. What are the possible Newton polygons for a curve in characteristic $p$? Equivalently, which Newton strata intersect the Torelli locus in $\mathcal{A}_g$? In this note, we study the Newton polygons of certain curves with $\mathbb{Z}/p\mathbb{Z}$-actions. Many of these curves exhibit unlikely intersections between the Torelli locus and the Newton stratification in $\mathcal{A}_g$. Here is one example of particular interest: fix a genus $g$. We show that for any $k$ with $\frac{2g}{3}-\frac{2p(p-1)}{3}\geq 2k(p-1)$, there exists a curve of genus $g$ whose Newton polygon has slopes $\{0,1\}^{g-k(p-1)} \sqcup \{\frac{1}{2}\}^{2k(p-1)}$. This provides evidence for Oort's conjecture that the amalgamation of the Newton polygons of two curves is again the Newton polygon of a curve. We also construct families of curves $\{C_g\}_{g \geq 1}$, where $C_g$ is a curve of genus $g$, whose Newton polygons have interesting asymptotic properties. For example, we construct a family of curves whose Newton polygons are asymptotically bounded below by the graph $y=\frac{x^2}{4g}$. The proof uses a Newton-over-Hodge result for $\mathbb{Z}/p\mathbb{Z}$-covers of curves due to the author, in addition to recent work of Booher-Pries on the realization of this Hodge bound.
