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MOD P-POINTS on SHIMURA VARIETIES of PARAHORIC LEVEL

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MOD P-POINTS on SHIMURA VARIETIES of PARAHORIC LEVEL MOD p-POINTS ON SHIMURA VARIETIES OF PARAHORIC LEVEL POL VAN HOFTEN, WITH AN APPENDIX BY RONG ZHOU Abstract. We study the Fp-points of the Kisin-Pappas integral models of abelian type Shimura varieties with parahoric level structure. We show that if the group is quasi-split and unramified, then the mod p isogeny classes are of the form predicted by the Langlands-Rapoport conjecture (c.f. Conjecture 9.2 of [56]). We prove the same results for quasi-split and tamely ramified groups when their Shimura varieties are proper. The main innovation in this work is a global argument that allows us to reduce the conjecture to the case of a very special parahoric, which is handled in the appendix. This way we avoid the complicated local problem of understanding connected components of affine Deligne-Lusztig varieties for general parahoric subgroups. Along the way, we give a simple irreducibility criterion for Ekedahl-Oort and Kottwitz-Rapoport strata. 1. Introduction and statement of results 1.1. Introduction. In [41], Langlands outlines a three-part approach to prove that the Hasse-Weil zeta functions of Shimura varieties are related to L-functions of automorphic forms. The first and third part are ‘a matter of harmonic analysis’, we refer the reader to [70] for an introduction. The second part is about describing the mod p points of suitable integral models of Shimura varieties, which is the central topic of this article. A conjectural description of the mod p points of (conjectural) integral models of Shimura varieties was first given by Langlands in [40] and was later refined by Langlands- Rapoport and Rapoport [42,56,57]. Together with the test function conjecture of Haines-Kottwitz [20], which was recently proven by Haines-Richarz [21], this conjecture is the main geometrical input to the Langlands-Kottwitz method for Shimura varieties of parahoric level. To explain these conjectures, we first need to introduce some notation. Let (G; X) be a Shimura datum, let p be a prime number, let Up ⊂ G(Qp) be a parahoric subgroup p p p and let U ⊂ G(Af ) be a compact open subgroup. For sufficiently small U there is a Shimura variety p ShU Up , which is a smooth quasi-projective variety defined over the reflex field E. Let v j p be a prime p of E, then conjecturally there should be a canonical integral model SU Up over OE;(v). When Up is hyperspecial, canonical models should be smooth and are unique if they satisfy a certain extension property (c.f. [47]). Recent work [50] of Pappas defines a notion of canonical integral models when Up is an arbitrary parahoric and proves that they are unique if they exist. Then conjecturally there should be a bijection (see Conjecture 4.2.2, Section 5 of [42] and Conjecture 9.2 of [56]) a (1.1.1) lim p ( ) ' S(φ); − SU Up Fp U p φ 2010 Mathematics Subject Classification. Primary 11G18; Secondary 14G35. Key words and phrases. Shimura varieties, affine Deligne-Lusztig varieties, shtukas. This work was supported by the Engineering and Physical Sciences Research Council [EP/L015234/1], The EPSRC Centre for Doctoral Training in Geometry and Number Theory (The London School of Geometry and Number Theory), University College London and King’s College London. 1 2 POL VAN HOFTEN where S(φ) = lim I ( )nX (φ) × Xp(φ)=U p: − φ Q p U p Let us elaborate. The sets S(φ) are supposed to correspond to points in a single isogeny class, with p Xp(φ) parametrising p-power isogenies, X (φ) parametrising prime to p isogenies and Iφ(Q) the group p p ur ur of self quasi-isogenies. The set X (φ) is a G(Af )-torsor and Xp(φ) is a subset of G(Qp )=G(Zp ), where G=Zp is the parahoric group scheme with G(Zp) = Up. In the unramified PEL case, (1.1.1) corresponds to Rapoport-Zink uniformisation of isogeny classes (see Section 6 of [55]), with Xp(φ) corresponding to the set of Fp-points of a Rapoport-Zink space. This is why we will often refer to (1.1.1) as uniformisation of isogeny classes. Uniformisation of isogeny classes for Hodge type Shimura varieties is often assumed in recent work in the area, c.f. [22,30]. The indexing set of (1.1.1) consists of admissible morphisms φ : Q ! G, where Q is a certain explicit groupoid. The category of representations of Q contains the isogeny category of abelian varieties over Fp as a full subcategory and is conjecturally closely related to the category of motives over Fp (see p Section 15 of [46]). We also expect that (1.1.1) is compatible with the action of G(Af ) on both sides and that the action of Frobenius on the left hand side should correspond to the action of a certain operator Φ on the right hand side. Kisin [34] proves a slightly weaker version of the conjecture for abelian type Shimura varieties under the assumption that GQp is quasi-split and split over an unramified extension, and that Up is hyperspecial. An important idea in his proof is to show that both admissible morphisms and isogeny classes ‘come from special points’. He deduces the former from Satz 5.3 of [42] and the latter is deduced, after a lengthy dévissage from the abelian type to the Hodge type case, from uniformisation of isogeny classes. In the parahoric case, uniformisation of isogeny classes was proven by Zhou in [67], under the assumption that GQp is residually split. We remind the reader that split implies residually split implies quasi-split and that residually split + unramified implies split. 1.2. Main results. Let (G; X) be a Shimura datum of abelian type and let p > 2 be a prime such that GQp is quasi-split and splits over an unramified extension. Let Up ⊂ G(Qp) be a parahoric p p subgroup and consider the tower of Shimura varieties fShG;U Up gU over the reflex field E with its p p p action of G(Af ), where U varies over compact open subgroups of G(Af ). Then by Theorem 0.1 of [33], p this tower of Shimura varieties has a G(Af )-equivariant extension to a tower of flat normal schemes p p fSG;U Up gU over OE(v) , where v j p is a prime of the reflex field E. Theorem 1. Let (G; X) be as above and suppose that either (Gad;Xad) has no factors of type DH or 1 p that Up is contained in a hyperspecial subgroup. Then there is an G(Af )-equivariant bijection a p p lim p ( ) ' lim I ( )nX (φ) × X (φ)=U − SG;U Up Fp − φ Q p U p φ U p p respecting the action of Frobenius, where the action of Iφ(Q) on Xp(φ) × X (φ) is the natural action ad conjugated by some τ(φ) 2 Iφ (Af ). Here Xp(φ) is the affine Deligne-Lustzig variety of level Up associated to φ, see Section 2.1.4. The indexing set runs over conjugacy classes of admissible morphisms Q ! G, see Section 4.2. As a byproduct of our arguments, we obtain the following result: 1See Appendix B of [46] for a classification of abelian type Shimura varieties into types A; B; C; DR and DH MOD p-POINTS ON SHIMURA VARIETIES OF PARAHORIC LEVEL 3 ad Theorem 2. Let (G; X) be as above and let Up denote a hyperspecial parahoric. Assume that G is -simple and let S fwg be an Ekedahl-Oort stratum that is not contained in the basic locus (the Q U;Fp smallest Newton stratum). Then S fwg ! S U;Fp U;Fp induces a bijection on connected components. Our methods will also prove versions of Theorems 1 and 2 without the assumption that GQp splits over an unramified extension, but always under the assumption that GQp is quasi-split. Moreover we prove irreducibility of Kottwitz-Rapoport strata at Iwahori level. The generalisations of Theorems 1 and 2 are Theorems 5.4.1 and 5.4.3, respectively, which assume that the Shimura varieties in question are proper and not of type A. Our proof of Theorem 5.4.1 proceeds by reduction to the case of an very special parahoric. This case is handled by Rong Zhou in Appendix A, by studying connected components of affine Deligne-Lusztig varieties of very special level and applying the main results of his earlier paper [67]. Ekedahl-Oort strata contained in the basic locus are highly reducible, for example the number of points in the supersingular locus of the modular curve goes to infinity with p. Similarly the basic locus itself is highly reducible. This means that the theorem is false for products of Shimura varieties with b basic ad in one factor and non-basic in the other; this is where the assumption that G is Q-simple comes from. It can be replaced with the assumption that b is Q-non basic, which means that the image of b ad in B(Gi;Qp ) is basic for any Q-factor Gi of G (this terminology was introduced in [38]). It follows from Theorem D of [59] that each Newton stratum contains a minimal EO stratum, that is, an EO stratum that is a central leaf. Central leaves that are not contained in the basic locus are expected to be irreducible, this is often referred to as the ‘discrete part’ of the Hecke-orbit conjecture (c.f.
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