Some unlikely intersections between the Torelli locus and Newton strata in ${\mathcal{A}}_g$

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 $\mathbb{Z}\mathrm{/}p\mathbb{Z}$-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 $\frac{2g}{3}-\frac{2p(p-1)}{3}\geq 2k(p-1)$,thereexistsacurveofgenus$g$whoseNewtonpolygonhasslopes${0,1}^{g-k(p-1)} \sqcup {\frac{1}{2}}^{2k(p-1)}$. 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 $\{C_g{\}}_{g\ge 1}$, 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 $\mathbb{Z}\mathrm{/}p\mathbb{Z}$-covers of curves due to the author, in addition to recent work of Booher-Pries on the realization of this Hodge bound.