Integral canonical models for automorphic vector bundles of abelian type

Abstract

We define and construct integral canonical models for automorphic vector bundles over Shimura varieties of abelian type.More precisely, we first build on Kisin's work to construct integral canonical models over OE[1/N ] for Shimura varieties of abelian type with hyperspecial level at all primes not dividing N compatible with Kisin's construction. We then define a notion of an integral canonical model for the standard principal bundles lying over Shimura varieties and proceed to construct them in the abelian type case. With these in hand, one immediately also gets integral models for automorphic vector bundles.

Filtered F-crystals on Shimura varieties of abelian type

“…The proof of this theorem is very similar to that of the main theorem of our previous paper [15], except we now are working locally at a single prime so can use Kisin's models directly, and rather than using properties of Milne's de Rham construction we must use properties of that of Liu and Zhu. Apart from making the notation more generally tidy, 1 We should remark that the caveat at p = 2 may likely be removed.…”

“…this has two advantages. First, the assumption of [15] that Z(G) • be split by a CM field may now be removed, since we may use pure p-adic Hodge theory in place of the theory of CM motives, which underpin Milne's work and the constructions of [15]. Second, combining the present paper with [15] and noting that the constructions are the same gives a (perhaps somewhat roundabout) proof of Liu-Zhu's conjecture [14, 4.9 (ii)] that the padic analytification of Milne's construction agrees with theirs in the case of abelian type Shimura varieties (with Z(G) • split by a CM field of course).…”

On the arithmetic Siegel--Weil formula for GSpin Shimura varieties

“…Then by Kisin [Kis10] (p > 2) and Kim-Madapusi-Pera [KMP16] (p = 2), there exists a smooth integral canonical model M K of Sh K over the localization O F,(ν) . We remark that by the work of Lovering [Lov17], these semi-global integral models over O F,(ν) glue to an integral canonical model over the ring of S-integers of F , for any S containing all finite places v where K v is not hyperspecial.…”

$p$-adic families of modular forms for Hodge type Shimura varieties with non-empty ordinary locus

“…Again E(ρ) is naturally equipped with a filtration. If ρ extends to a Q-representation, then, by [Lov17], we have a canonical integral model E(ρ) + of E(ρ).…”

“…Let (G, X) be a Shimura datum of Hodge type, with associated Shimura variety S. By definition this means that there is a closed immersion S ֒→ S, where S is a Siegel variety. We work with adic spaces, but we also need a good theory of integral model of our varieties and sheaves, to use [Kis10] and [Lov17], so we make some technical assumptions about G and the level structure, that we will not recall in this introduction, see Subsection 1.1 for details. The integral (dominant) weights for S parameterize irreducible algebraic representations of the Levi of a fixed parabolic subgroup of G. These weights can be interpolated by the weight space W, that parameterizes p-adic weights.…”