Some unlikely intersections between the Torelli locus and Newton strata in A_g

2020/06/08 by Kramer-Miller, Joe
#11G20 #11M38 #14K10 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2006.04927

Abstract

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 A_g? In this note, we study the Newton polygons of certain curves with ℤ/pℤ-actions. Many of these curves exhibit unlikely intersections between the Torelli locus and the Newton stratification in A_g. Here is one example of particular interest: fix a genus g. We show that for any k with (2g)/(3)-(2p(p-1))/(3)≥ 2k(p-1), there exists a curve of genus g whose Newton polygon has slopes \0,1\^g-k(p-1) \sqcup \(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 ≥ 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=(x²)/(4g). The proof uses a Newton-over-Hodge result for ℤ/pℤ-covers of curves due to the author, in addition to recent work of Booher-Pries on the realization of this Hodge bound.

Related

• Current results on Newton polygons of curves
• Newton polygon stratification of the Torelli locus in PEL-type Shimura varieties
• Some cases of Oort's conjecture about Newton polygons
• Newton Polygons of Sums on Curves II: Variation in p-adic Families
• The Torelli locus and Newton polygons