A refinement of the Hodge stratification for connected reductive groups

TL;DR

This paper investigates the structure of certain stratifications in connected reductive groups over p-adic fields, establishing conditions under which Newton points are constant within these stratifications and relating them to moduli schemes of abelian varieties.

Contribution

It refines the understanding of Hodge stratifications for reductive groups, providing explicit bounds for N where Newton points stabilize, and connects these to existing moduli space stratifications.

Findings

Newton point is constant on H_{ulu} N(G) for large N

Explicit bounds for N are computed for specific classes of groups

Connections established between group stratifications and moduli schemes of abelian varieties

Abstract

For connected reductive groups G over a finite extension F of Q_p and L the maximal unramified extension of F we study the sets H_{\mu, N}(G) of elements b in G(L) with given Hodge points of (b\sigma), (b\sigma)^2, ..., (b\sigma)^N. We explain the relationship to stratifications of some moduli scheme of abelian varieties defined by Goren and Oort respectively Andreatta and Goren. We show that for sufficiently large N the Newton point is constant on the sets H_{\mu, N}(G) and compute such N for certain classes of groups.