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arXiv · 1811.00604

Newton polygon stratification of the Torelli locus in PEL-type Shimura varieties

Wanlin Li·Elena Mantovan·Rachel Pries·Yunqing Tang
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Abstract

We study the intersection of the Torelli locus with the Newton polygon stratification of the modulo $p$ reduction of certain PEL-type Shimura varieties. We develop a clutching method to show that the intersection of the open Torelli locus with some Newton polygon strata is non-empty. 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 PEL-type 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).

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Wanlin Li, Elena Mantovan, Rachel Pries, Yunqing Tang. 2018-11-01. Newton polygon stratification of the Torelli locus in PEL-type Shimura varieties. https://arxiv.org/abs/1811.00604

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