Topological flatness of local models for ramified unitary groups. I. The odd dimensional case

Abstract

Local models are certain schemes, defined in terms of linear-algebraic moduli problems, which give étale-local neighborhoods of integral models of certain p-adic PEL Shimura varieties defined by Rapoport and Zink. When the group defining the Shimura variety ramifies at p, the local models (and hence the Shimura models) as originally defined can fail to be flat, and it becomes desirable to modify their definition so as to obtain a flat scheme. In the case of unitary similitude groups whose localizations at Qp are ramified, quasi-split GUn, Pappas and Rapoport have added new conditions, the so-called wedge and spin conditions, to the moduli problem defining the original local models and conjectured that their new local models are flat. We prove a preliminary form of their conjecture, namely that their new models are topologically flat, in the case n is odd.

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arXiv:1002.3520v2 [math.AG] 27 Sep 2010
TOPOLOGICAL FLATNESS OF LOCAL
MODELS FOR RAMIFIED UNITARY GROUPS.
I. THE ODD DIMENSIONAL CASE
BRIAN D. SMITHLING
Abstract. Local models are certain schemes, defined in terms of linear-alge-
braic moduli problems, which give ´ etale-local neighborhoods of integral models
of certain p-adic PEL Shimura varieties defined by Rapoport and Zink. When
the group defining the Shimura variety ramifies at p, the local models (and
hence the Shimura models) as originally defined can fail to be flat, and it
becomes desirable to modify their definition so as to obtain a flat scheme.
In the case of unitary similitude groups whose localizations at Qp are rami-
fied, quasi-split GUn, Pappas and Rapoport have added new conditions, the
so-called wedge and spin conditions, to the moduli problem defining the orig-
inal local models and conjectured that their new local models are flat. We
prove a preliminary form of their conjecture, namely that their new models
are topologically flat, in the case n is odd.
1. Introduction
A basic problem for a given Shimura variety is to define a reasonable model
for the variety over the ring of integers of its reflex field. In [RZ], Rapoport and
Zink define natural models for certain PEL Shimura varieties with parahoric level
structure at p over the ring of integers in the completion of the reflex field at any
place dividing p. In addition, they reduce many aspects of the study of these models
to the associated local models; these give ´ etale-local neighborhoods of the Rapoport-
Zink models which are defined in terms of purely linear-algebraic moduli problems,
and thus — at least in principle — are more amenable to direct investigation.
There has been much study of local models in various cases since the appearance
of Rapoport and Zink’s book; see, for example, work of Pappas [P], G¨ ortz [G1,
G2,G3,G4], Haines and Ngˆ o [HN1], Pappas and Rapoport [PR1,PR2,PR3,PR4],
Kr¨ amer [K], Arzdorf [A], and the author [Sm1]. Beginning with Pappas’s paper [P],
it has come to be understood that when the group defining the Shimura variety
is ramified at p — that is, the base change of the group to Qp splits only after
passing to a ramified extension of Qp— then the associated model and local model
need not satisfy one of the most basic criteria for reasonableness, namely they need
not be flat. In such cases, the models and local models defined by Rapoport and
Zink have come to be renamed naive models and local models, respectively, with
the true models and local models defined as the scheme-theoretic closures of the
generic fibers in the respective naive models and local models.
2010 Mathematics Subject Classification. Primary 14G35; Secondary 05E15; 11G18; 17B22.
Key words and phrases. Local model; Shimura variety; unitary group; Iwahori-Weyl group,
admissible set.
1

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2BRIAN D. SMITHLING
Sparked by Pappas’s observation, it has become an important part of the subject
to better understand local models in the presence of ramification; see [G4,P,PR1,
PR2,PR3, PR4]. In particular, it is an interesting problem to obtain a moduli-
theoretic description of the local model in such cases, ideally by refining the moduli
problem that describes the naive local model. This is the problem of concern in this
paper and its sequel [Sm3], in one of the basic cases in which ramification arises: a
unitary similitude group attached to an imaginary quadratic number field ramified
at p ?= 2.
More precisely, let F/F0be a ramified quadratic extension of discretely valued,
non-Archimedean fields of residual characteristic not 2. Endow Fn, n ≥ 3, with the
F/F0-Hermitian form φ specified by the values φ(ei,ej) = δi,n+1−jon the standard
basis vectors e1,..., en, and consider the reductive group GUn:= GU(φ) over F0.
In this paper we consider exclusively the case that n is odd; the case of even n
will be treated in [Sm3]. In the odd case, every parahoric subgroup of GUn(F0)
can be described as the stabilizer of a self-dual periodic lattice chain in Fn, and
the conjugacy classes of parahoric subgroups can be naturally parametrized by
the nonempty subsets of {0,...,m}, where n = 2m + 1; see (3.7.1) or, for more
details, [PR4, §1.2.3(a)]. Identifying G⊗F0F ≃ GLn,F×Gm,F, let µr,sdenote the
cocharacter?1(s),0(r),1?of D×Gm,F, where D denotes the standard maximal torus
of diagonal matrices in GLn,F and r and s are nonnegative integers with r+s = n.
In the special case that F0= Qpand (Fn,φ) is isomorphic to the Qp-localization of
a Hermitian space (Kn,ψ) with K an imaginary quadratic number field, the pair
(r,s) denotes the signature of (Kn,ψ) and µr,s is a cocharacter obtained in the
usual way from the Shimura datum attached to the associated unitary similitude
group, as in [PR4, §1.1].
For nonempty I ⊂ {0,...,m}, we may then consider the naive local model Mnaive
in the sense of [RZ]; this is a projective OF-scheme whose explicit definition we recall
in §2.3. As [P] observes, the naive local model fails to be flat over OF in general.
In response, the papers [P,PR4] attempt to correct for non-flatness by adding new
conditions to the moduli problem defining Mnaive
condition to define a closed subscheme M∧
§2.4); and then Pappas and Rapoport add a further condition, the spin condition,
to define a third closed subscheme Mspin
I
The schemes Mnaive
I
, M∧
I
, and the honest local model Mloc
generic fiber, and Pappas and Rapoport conjecture the following.
I
I
: first Pappas adds the wedge
I⊂ Mnaive
I
, the wedge local model (see
⊂ M∧
I, the spin local model (see §2.5).
I, Mspin
I
all have common
Conjecture (Pappas-Rapoport [PR4, 7.3]). Mspin
Mloc
I
inside Mnaive
I
; or in other words, Mspin
I
coincides with the local model
is flat over OF.
I
Although the conjecture remains open in general, Pappas and Rapoport have
obtained a good deal of computer evidence in support of it. The main result of this
paper is essentially a preliminary form of the conjecture, which we state precisely
as follows.
Main Theorem. The schemes Mspin
in other words, the underlying topological spaces of Mspin
I
and M∧
Iare topologically flat over OF; or
, M∧
I
I, and Mloc
I
coincide.
See (5.5.5) and (5.6.3); recall that a scheme over a regular, integral, 1-dimensional
base scheme is topologically flat if its generic fiber is dense. The theorem notwith-
standing, the scheme structures on M∧
Iand Mspin
I
really can differ, and it is only

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TOPOLOGICAL FLATNESS OF RAMIFIED UNITARY LOCAL MODELS. I.3
Mspin
I
[Sm3] that the spin local models Mspin
are also topologically flat; here the schemes M∧
agree set-theoretically.
Following G¨ ortz [G1] (see also [G2,G4,PR1,PR2,PR3,PR4,Sm1]), the key tech-
nique in the proof of the Main Theorem is to embed the special fiber of Mnaive
an appropriate affine flag variety, where it and the special fibers of M∧
Mloc
I
become stratified into finitely many Schubert cells. Pappas and Rapoport
show that the Main Theorem follows from showing that the Schubert cells in the
special fibers of M∧
I
are indexed by the µr,s-admissible set; see §5.6.
We solve this problem by translating it into an equivalent one for GSp2m, which
in turn boils down to obtaining a concrete description of the admissible set for the
cocharacter
of GSp2m. Here we make key use of a result of
Haines and Ngˆ o [HN2] that describes admissible sets for GSp2m in terms of per-
missible sets for GL2m. As a byproduct of our considerations, we show that the
notions of?2(s),1(2m−2s),0(s)?-admissibility and?2(s),1(2m−2s),0(s)?-permissibility
for GSp2mare equivalent.
Finally, we remark that failure of flatness of the models in [RZ] is not a phe-
nomenon related solely to ramification: as observed by Genestier, the local models
in [RZ] attached to split even orthogonal groups also fail to be flat in general. See
[PR4,Sm1].
We now outline the contents of the paper. Sections 2–4 consist almost entirely
of review from [PR4]: in §2 we review the definitions of the various local models,
in §3 we review some group-theoretic aspects of ramified GUn, and in §4 we review
the embedding of the special fiber of the local model into an appropriate affine
flag variety attached to GUn. In §5 we obtain combinatorial descriptions of the
Schubert cells contained in the special fibers of M∧
variety, and we reduce the Main Theorem to showing that these cells are indexed
by the µr,s-admissible set. In §6 we solve this last problem, as described above.
that is conjectured to be flat in general; see [PR4, 7.4(iv)]. We shall show in
attached to even ramified unitary groups
I
Iand Mspin
I
usually do not even
I
in
I, Mspin
I
, and
Iand Mspin
?2(s),1(2m−2s),0(s)?
Iand Mspin
I
inside the affine flag
Acknowledgments. I thank Michael Rapoport for his encouragement to work on
this problem and for introducing me to the subject. I also thank him and Robert
Kottwitz for reading and offering comments on a preliminary version of this article;
and the referee for carefully reading the article and suggesting some corrections and
improvements. Part of the writing of this article was undertaken at the Hausdorff
Research Institute for Mathematics in Bonn, which I thank for its hospitality and
excellent working conditions.
Notation. We fix once and for all an odd integer n = 2m + 1 ≥ 3 and a partition
n = s + r with 0 ≤ s ≤ m, so that s < r. Although almost everything we shall do
will depend on n and this partition, we shall usually not embed these choices into
the notation.
We let F/F0 denote a ramified quadratic extension of discretely valued, non-
Archimedean fields with respective rings of integers OF and OF0, respective uni-
formizers π and π0satisfying π2= π0, and common residue field k of characteristic
not 2. We also employ an auxiliary ramified quadratic extension K/K0of discretely
valued, non-Archimedean Henselian fields with respective rings of integers OKand
OK0, respective uniformizers u and t satisfying u2= t, and the same residue field

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4BRIAN D. SMITHLING
k; eventually K and K0will be the fields of Laurent series k((u)) and k((t)), re-
spectively. We put Γ := Gal(K/K0), and we write x ?→ x for the action of the
nontrivial element of Γ on K; then u = −u. Abusing notation, we continue to write
x ?→ x for the R-algebra automorphism of K ⊗K0R induced by any base change
K0→ R.
We relate objects by writing ≃ for isomorphic,∼= for canonically isomorphic, and
= for equal.
Given a vector v ∈ Rl, we write v(j) for its jth entry, Σv for the sum of its entries,
and v∗for the vector in Rldefined by v∗(j) = v(l+1−j). Given another vector w, we
write v ≥ w if v(j) ≥ w(j) for all j. We write d for the vector (d,d,...,d), leaving
it to context to make clear the number of entries. The expression
denotes the vector with d repeated i times, followed by e repeated j times, and so
on.
We write Slfor the symmetric group on 1,..., l, and S∗
S∗
?d(i),e(j),...?
lfor its subgroup
l:=?σ ∈ Sl
??σ(l + 1 − j) = l + 1 − σ(j) for all j ∈ {1,...,l}?.
2. Unitary local models
We begin by recalling the definition and some of the discussion of local models
for odd ramified unitary groups from [PR4]. Let I ⊂ {0,...,m} be a nonempty
subset.
2.1. Pairings. Let V := Fn. In this subsection we introduce some pairings on V
and notation related to them which we’ll use throughout the article.
Let e1, e2,..., endenote the standard ordered F-basis in V , and let
φ: V × V −→ F
be the F/F0-Hermitian form on V whose matrix with respect to the standard basis
is


1
(2.1.1)

1
...


.
We attach to φ the alternating and symmetric F0-bilinear forms V ×V → F0given
respectively by
?π−1φ(x,y)?
For any OF-lattice Λ ⊂ V , we denote by?Λ the φ-dual of Λ,
(2.1.3)
(2.1.2)?x,y? :=1
2TrF/F0
and(x,y) :=1
2TrF/F0
?φ(x,y)?.
?Λ :=?x ∈ V
?Λ =?x ∈ V
??φ(Λ,x) ⊂ OF
???Λ,x? ⊂ OF0
?.
Then?Λ is also the ? , ?-dual of Λ,
?;
and?Λ is related to the ( , )-dual?Λs:= {x ∈ V | (Λ,x) ⊂ OF0} by the formula
induce perfect OF0-bilinear pairings
?Λs= π−1?Λ. Both?Λ and?Λsare OF-lattices in V , and the forms ? , ? and ( , )
(2.1.4)Λ ×?Λ
for all Λ.
? , ?
− − → OF0
andΛ ×?Λs( , )
− − → OF0

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TOPOLOGICAL FLATNESS OF RAMIFIED UNITARY LOCAL MODELS. I.5
2.2. Standard lattices. For i = nb + c with 0 ≤ c < n, we define the standard
OF-lattice
c
?
j=1
(2.2.1)Λi:=
π−b−1OFej+
n
?
j=c+1
π−bOFej⊂ V.
Then?Λi= Λ−ifor all i, and the Λi’s form a complete, periodic, self-dual lattice
··· ⊂ Λ−2⊂ Λ−1⊂ Λ0⊂ Λ1⊂ Λ2⊂ ··· ,
which we call the standard lattice chain. More generally, for any nonempty subset
I ⊂ {0,...,m}, we denote by ΛI the periodic, self-dual subchain of the standard
chain consisting of all lattices of the form Λifor i ∈ nZ±I. Of course, in this way
Λ{0,...,m}denotes the standard chain itself.
The standard lattice chain admits the following obvious trivialization. Let ǫ1,...,
ǫndenote the standard basis of On
all other standard basis elements to themselves. Then there is a unique isomorphism
of chains of OF-modules
chain
F, and let βi: On
F→ On
Fmultiply ǫiby π and send
(2.2.2)
···??
??Λ0??
??Λ1??
??···??
??Λn??
??···
···
βn??On
F
∼
??
β1??On
F
∼
??
β2??···
βn??On
F
∼
??
β1??···
such that the leftmost displayed vertical arrow identifies the ordered OF-basis
ǫ1,...,ǫnof On
in the top and bottom rows in (2.2.2), we get an analogous trivialization of ΛI for
any I.
Fwith the ordered basis e1,...,enof Λ0. Restricting to subchains
2.3. Naive local models. We now review the definition of the naive local models
from [PR4, §1.5].
Recall our fixed partition n = s + r with s < r. The naive local model Mnaive
the following contravariant functor on the category of OF-algebras. Given an OF-
algebra R, an R-point in Mnaive
I
consists of, up to an obvious notion of isomorphism,
• a functor
ΛI
??(OF⊗OF0R-modules)
Λi?
where ΛI is regarded as a category by taking the morphisms to be the
inclusions of lattices in V ; together with
• an inclusion of OF⊗OF0R-modules Fi֒→ Λi⊗OF0R for each i ∈ nZ ± I,
functorial in Λi;
satisfying the following conditions for all i ∈ nZ ± I.
(LM1) Zariski-locally on SpecR, Fi embeds in Λi⊗OF0R as a direct R-module
summand of rank n.
(LM2) The isomorphism Λi⊗OF0R− → Λi−n⊗OF0R obtained by tensoring Λi
πΛi= Λi−nidentifies Fiwith Fi−n.
(LM3) The perfect R-bilinear pairing
I
is
??Fi,
∼π
− →
∼
(Λi⊗OF0R) × (Λ−i⊗OF0R)
induced by (2.1.4) identifies F⊥
? , ?⊗R
− − − − − → R
i ⊂ Λ−i⊗OF0R with F−i.