Newton polygon stratification of the Torelli locus in PELtype Shimura varieties
Abstract
We study the intersection of the Torelli locus with the Newton polygon stratification of the modulo $p$ reduction of certain PELtype Shimura varieties. We develop a clutching method to show that the intersection of the open Torelli locus with some Newton polygon strata is nonempty. This allows us to give a positive answer, under some compatibility conditions, to a question of Oort about smooth curves in characteristic $p$ whose Newton polygons are an amalgamate sum. As an application, we produce infinitely many new examples of Newton polygons that occur for smooth curves that are cyclic covers of the projective line. Most of these arise in inductive systems which demonstrate unlikely intersections of the open Torelli locus with the Newton polygon stratification in Siegel modular varieties. In addition, for the twenty special PELtype Shimura varieties found in Moonen's work, we prove that all Newton polygon strata intersect the open Torelli locus (if $p>>0$ in the supersingular cases).
 Publication:

arXiv eprints
 Pub Date:
 November 2018
 arXiv:
 arXiv:1811.00604
 Bibcode:
 2018arXiv181100604L
 Keywords:

 Mathematics  Number Theory;
 primary 11G18;
 11G20;
 11M38;
 14G10;
 14G35;
 secondary 11G10;
 14H10;
 14H30;
 14H40;
 14K10