ON THE ARITHMETIC TRANSFER CONJECTURE FOR EXOTIC SMOOTH FORMAL MODULI ...

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ON THE ARITHMETIC TRANSFER CONJECTURE FOR EXOTIC
SMOOTH FORMAL MODULI SPACES
M.RAPOPORT,B.SMITHLING,ANDW.ZHANG
Abstract. IntherelativetraceformulaapproachtothearithmeticGan–Gross–Prasadcon-
jecture,weformulatealocalconjecture(arithmetictransfer)inthecaseofanexoticsmooth
formalmodulispaceofp-divisiblegroups,associatedtoaunitarygrouprelativetoaramified
quadraticextensionofap-adicfield. Weproveourconjectureinthecaseofaunitarygroup
inthreevariables.
Contents
1. Introduction 1
Part 1. The conjectures 7
2. Group-theoretic setup and orbit matching 7
3. The moduli space 11
4. Intersection numbers 21
5. Conjectures and main results 24
Part 2. Geometric side 33
6. The Serre map 33
7. Special divisors as sums of quasi-canonical divisors 36
8. Reduction to the Lie algebra 37
9. Explicit calculations for the Lie algebra 43
Part 3. Analytic side 47
10. Inputs from local harmonic analysis 47
11. Reduction to the Lie algebra 53
12. Reduction to the reduced set 59
13. Application of the germ expansion principle 61
14. The germ expansion of a function with special transfer 62
15. Comparison 71
Part 4. Germ expansion 74
16. Statement of the germ expansion 74
17. Germ expansion around x =0 75
0
18. Germ expansion around x (cid:54)=0 79
0
19. Germ expansion for u , and matching 85
red
20. Proofs of Theorems 16.2 and 16.5 89
References 90
1. Introduction
The theorem of Gross and Zagier [7] relates the Neron–Tate heights of Heegner points on
modular curves to special values of derivatives of certain L-functions. This has been generalized
in various ways to higher-dimensional Shimura varieties. One such generalization, which is still
conjectural, has been proposed by Gan–Gross–Prasad [5] and the third-named author [34, 35].
1
2 M.RAPOPORT,B.SMITHLING,ANDW.ZHANG
This arithmetic Gan–Gross–Prasad conjecture is inspired by the (usual) Gan–Gross–Prasad
conjecture relating period integrals on classical groups to special values of certain L-functions.
In [9] Jacquet and Rallis proposed a relative trace formula approach to this last conjecture in
the case of unitary groups, which led them to formulate two local conjectures in this context: a
fundamentallemma(FL)conjecture, andasmoothtransfer(ST)conjecture. Bothoftheirlocal
conjectures are now proved to a large extent, the first for p(cid:29)0 thanks to the work of Yun [32]
(andGordon[6]),andthesecondforarbitraryp-adicnon-archimedeanfieldsbythethird-named
author [37].
In [35] the third-named author proposed a relative trace formula approach to the arithmetic
Gan–Gross–Prasadconjecture. Inthiscontext,heformulatedthearithmeticfundamentallemma
(AFL) conjecture, cf. [35, 23]. The AFL conjecturally relates the special value of the derivative
of an orbital integral to an arithmetic intersection number on a Rapoport–Zink formal moduli
space of p-divisible groups attached to a unitary group. The AFL is proved for low ranks of the
unitary group (n = 2 and 3) in [35], and for arbitrary rank n and minuscule group elements
in [23]. A simplified proof for n = 3 appears in [13]. At present, the general case of the AFL
seems out of reach, even though Yun has obtained interesting results concerning the function
field analog [33].
In the present paper, we address an arithmetic transfer (AT) analog of the ST conjecture in
the arithmetic context, in a very specific case; we refer to [20] for a more general context in
which we expect such arithmetic analogs of ST. The special feature of the case at hand is that,
despite the fact that we take the unitary group to be ramified, the corresponding RZ space is
smooth, cf. [16]. For this reason we speak of exotic smoothness.
Now that we have explained the title of the paper, let us describe its contents in more detail.
Let p be an odd prime number, and let F be a finite extension of Q . Let F/F be a
0 p 0
quadratic field extension. We denote by a (cid:55)→ a the non-trivial automorphism of F/F , and by
0
η = η the corresponding quadratic character on F×. Let e := (0,...,0,1) ∈ Fn, and let
F/F0 0 0
GL (cid:44)→GL bethenaturalembeddingthatidentifiesGL withthesubgroupfixingeunder
n−1 n n−1
left multiplication, and fixing the transposed vector te under right multiplication. Let
S :={s∈Res GL |ss=1},
n F/F0 n
with its action by conjugation of GL . On the other hand, let W and W be the respective
n−1 0 1
split and non-split non-degenerate F/F -hermitian spaces of dimension n. For i = 0 and 1, fix
0
a vector u ∈W of length 1, and denote by W(cid:91) the orthogonal complement of the line spanned
i i i
by u . The unitary group U(W(cid:91)) acts by conjugation on U(W ).
i i i
We now explain the matching relation between regular semi-simple elements of S (F ) and
n 0
of U(W )(F ) and U(W )(F ). Here an element of S (F ), resp. of U(W )(F ), is called regular
0 0 1 0 n 0 i 0
semi-simple (rs) if its orbit under GL , resp. U(W(cid:91)), is Zariski-closed of maximal dimension.
n−1 i
For each i, choose a basis of W by first choosing a basis of W(cid:91) and then appending u to it.
i i i
This identifies U(W(cid:91))(F ) with a subgroup of GL (F) and U(W )(F ) with a subgroup of
i 0 n−1 i 0
GL (F). Anelementγ ∈S (F ) issaidtomatch anelementg ∈U(W )(F ) ifbothelements
n n 0 rs i 0 rs
areconjugateunderGL (F)whenconsideredaselementsinGL (F). Thismatchingrelation
n−1 n
induces a bijection
(cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3)
U(W )(F ) (cid:113) U(W )(F ) (cid:39) S (F ) ,
0 0 rs 1 0 rs n 0 rs
cf. [35, §2], where the brackets indicate the sets of orbits under U(W(cid:91))(F ), resp. GL (F ).
i 0 n−1 0
Dual to the matching of elements is the transfer of functions, which is defined through
weighted, resp. ordinary, orbital integrals. For a function f(cid:48) ∈ C∞(S (F )), an element γ ∈
c n 0
S (F ) , and a complex parameter s∈C, we define the weighted orbital integral
n 0 rs
(cid:90)
Orb(γ,f(cid:48),s):= f(cid:48)(h−1γh)|deth|sη(deth)dh,
GLn−1(F0)
as well as its special value
Orb(γ,f(cid:48)):=Orb(γ,f(cid:48),0).
ON THE ARITHMETIC TRANSFER CONJECTURE FOR EXOTIC SMOOTH MODULI SPACES 3
HeretheHaarmeasureonGL (F )isnormalizedsothatvol(GL (O ))=1. Forafunction
n−1 0 n−1 F0
f ∈C∞(U(W )(F )) and an element g ∈U(W )(F ) , we define the orbital integral
i c i 0 i 0 rs
(cid:90)
Orb(g,f ):= f (h−1gh)dh.
i i
U(Wi(cid:91))(F0)
Then the function f(cid:48) ∈ C∞(S (F )) is said to transfer to the pair of functions (f ,f ) in
c n 0 0 1
C∞(U(W )(F ))×C∞(U(W )(F )) if
c 0 0 c 1 0
ω(γ)Orb(γ,f(cid:48))=Orb(g,f )
i
whenever γ ∈S (F ) matches the element g ∈U(W )(F ) . Here
n 0 rs i 0 rs
ω: S (F ) −→C×
n 0 rs
is a fixed transfer factor [37, p. 988], and the Haar measures on U(W(cid:91))(F ) are fixed. The ST
i 0
conjecture asserts that for any f(cid:48), a transfer (f ,f ) exists (non-uniquely), and that any pair
0 1
(f ,f ) arises as a transfer from some (non-unique) f(cid:48). The FL conjecture asserts a specific
0 1
transfer relation in a completely unramified situation.
WhenF/F isunramified,ifonetakesforω: S (F ) →{±1}thenatural transferfactor(see
0 n 0 rs
[23, (1.5)]), and normalizes the Haar measure on U(W(cid:91))(F ) by giving a hyperspecial maximal
0 0
compactsubgroupvolumeone,thentheFLconjectureassertsthat1 transfersto(1 ,0),
Sn(OF0) K0
where K ⊂U(W )(F ) denotes a hyperspecial maximal open subgroup.
0 0 0
By contrast, when F/F is ramified, there is no natural choice of a conjugacy class of open
0
compact subgroups K , no natural choice of a transfer factor, and no natural candidate for f(cid:48)
0
transferring to (1 ,0).
K0
We next pass to the AFL conjecture, which requires, just as in the FL conjecture, that F/F
0
is unramified. We take the same transfer factor as in the FL conjecture, and the same fixed
Haar measure on U(W(cid:91))(F ). For f(cid:48) ∈C∞(S (F )) and γ ∈S (F ) , set
0 0 c n 0 n 0 rs
d (cid:12)
∂Orb(γ,f(cid:48)):= (cid:12) Orb(γ,f(cid:48),s).
ds(cid:12)s=0
Then the AFL conjecture asserts that
(cid:0) (cid:1)
ω(γ)∂Orb γ,1 =−Int(g)·logq, (1.1)
Sn(OF0)
whenever γ ∈ S (F ) matches g ∈ U(W )(F ) (note that the FL conjecture asserts that
n 0 rs 1 0 rs
Orb(γ,1 )=0 for such γ). Here q denotes the number of elements in the residue field of
Sn(OF0)
F .
0
The term Int(g) requires explanation. Let N = N denote the formal scheme over
n F/F0,n
SpfO whichrepresentsthefollowingfunctoronthecategoryofO -schemesS suchthatp·O is
F˘ F˘ S
alocallynilpotentidealsheaf. ThefunctorassociatestoSthesetofisomorphismclassesoftuples
(X,ι,λ,ρ) where X is a formal p-divisible O -module of relative height 2n and dimension n,
F0
whereι: O →End(X)isanactionofO satisfyingtheKottwitzconditionofsignature(1,n−1)
F F
on Lie(X) (cf. [10, §2]), where λ is a principal polarization whose Rosati involution induces the
automorphisma(cid:55)→aonι(O ),andwhereρ: X× S →X × Sisaframingoftherestriction
F S n Speck
ofX tothespecialfiberS ofS,compatiblewithιandλinacertainsense,cf.[23,§2]. ThenN
n
is formally smooth of relative formal dimension n−1 over SpfO . The automorphism group
F˘
(inacertainsense)oftheframingobjectX canbeidentifiedwithU(W )(F ); itactsonN by
n 1 0 n
changing the framing. Let E be the canonical lifting of the formal O -module of relative height
F
1 and dimension 1 over SpfO , with its canonical O -action ι and its natural polarization λ .
F˘ F E E
There is a natural closed embedding of N into N ,
n−1 n
(cid:47)(cid:47)
δ : N N
N n−1 n
(cid:31) (cid:47)(cid:47)
Y Y ×E.
Here all auxiliary structure (O -action, polarization, framing) has been suppressed from the
F
notation, and E denotes E, with ι replaced by its conjugate. Let
E
∆⊂N × N
n−1 SpfOF˘ n
4 M.RAPOPORT,B.SMITHLING,ANDW.ZHANG
denotethegraphofδ . ThenInt(g)isdefinedastheintersectionnumberof∆withitstranslate
N
under the automorphism 1×g of N × N ,
n−1 SpfOF˘ n
(cid:0) L (cid:1)
Int(g)=χ O ⊗ O .
∆ (1×g)∆
This concludes the statement of the AFL conjecture.
It should be true in the situation of the AFL that for any f(cid:48) ∈ C∞(S (F )) with transfer
c n 0
(1 ,0), there exists a function f(cid:48) ∈C∞(S (F )) such that
K0 corr c n 0
ω(γ)∂Orb(γ,f(cid:48))=−Int(g)·logq+ω(γ)Orb(γ,f(cid:48) )
corr
whenever γ ∈ S (F ) matches g ∈ U(W )(F ) . This would follow along the lines of Lemma
n 0 rs 1 0 rs
5.18 below from a conjectural density principle on weighted orbital integrals. See Conjecture
5.16 for the statement of the density principle in the setting of this paper.
Now we come to the formulation of our AT conjecture. We assume for this that F/F is
0
ramified. We then modify the definition of the formal moduli space N = N by slightly
F/F0,n n
changing the conditions on the tuples(X,ι,λ,ρ). Namely, inaddition to theKottwitz condition
of signature (1,n−1), we impose on ι the Pappas wedge condition of signature (1,n−1), and
the spin condition. The latter condition states that for a uniformizer π of F, the endomorphism
ι(π)|Lie(X) is nowhere zero on S. Furthermore, we change the condition that λ is principal to
the condition
Ker(λ)⊂X[ι(π)] with |Ker(λ)|=q2(cid:98)n/2(cid:99).
It turns out that N is again formally smooth of relative formal dimension n−1 over SpfO ,
n F˘
and is essentially proper when n is even. We stress that this result is quite surprising in the
presence of ramification.
The morphism δ : N → N can be defined exactly as before when discussing the AFL
N n−1 n
setup, provided that n is odd, since then 2(cid:98)n−1(cid:99)=2(cid:98)n(cid:99). We then define Int(g) as before. Our
2 2
AT conjecture is as follows.
Conjecture 1.1. Let F/F be ramified, and let n≥3 be odd.
0
(a) There exists a function f(cid:48) ∈C∞(S(F )) with transfer (1 ,0) such that
c 0 K0
2ω(γ)∂Orb(γ,f(cid:48))=−Int(g)·logq (1.2)
for any γ ∈S(F ) matching an element g ∈U(W )(F ) .
0 rs 1 0 rs
(b) For any f(cid:48) ∈ C∞(S(F )) with transfer (1 ,0), there exists a function f(cid:48) ∈ C∞(S(F ))
c 0 K0 corr c 0
such that
2ω(γ)∂Orb(γ,f(cid:48))=−Int(g)·logq+ω(γ)Orb(γ,f(cid:48) )
corr
for any γ ∈S(F ) matching an element g ∈U(W )(F ) .
0 rs 1 0 rs
Here K denotes the maximal compact subgroup stabilizing a nearly π-modular lattice Λ in
0 0
W (see (5.3) below), and ω is the transfer factor defined in (5.5) below. The Haar measure on
0
U(W(cid:91))(F ) is defined by volK(cid:91) =1 for a special maximal compact subgroup K(cid:91) of U(W(cid:91))(F ).
0 0 0 0 0 0
We also formulate a “homogeneous” variant of the AT conjecture (Conjecture 5.3), which we
show is equivalent to the above conjecture in §5.2. Note that between the statements of the
AFL conjecture (1.1) and the AT conjecture (1.2), there is a discrepancy of a factor of 2. This
is a genuine difference between the unramified and ramified cases, and we refer to [21] for an
explanation by way of a global comparison between the height pairing and the derivative of a
relative trace formula.
Ourmainresultconcernsthefirstnon-trivialcasen=3oftheATconjecture. Moreprecisely,
we prove the following.
Theorem 1.2. Let F = Q , and let n = 3. Then Conjecture 1.1 holds true. In addition, for
0 p
any g ∈ U(W )(F ) , the intersection of ∆ and (1×g)∆, if non-empty, is an artinian scheme
1 0 rs
with two points, and Int(g)=length(∆∩(1×g)∆).
We also prove a Lie algebra version of the above theorem; see Theorem 5.23.
Let us comment on the proof of Theorem 1.2. In those cases in which the AFL conjecture
has been established, the proof proceeds by calculating explicitly both sides of the conjectured
identity and comparing the results. This approach fails for the AT conjecture because the
ON THE ARITHMETIC TRANSFER CONJECTURE FOR EXOTIC SMOOTH MODULI SPACES 5
left-hand side of the identity is not well-determined by the pair of transfer functions (1 ,0).
K0
UnliketheAFLsituation,thereisnocanonicalchoiceforthefunctionf(cid:48);infact,f(cid:48) cannotcome
fromtheIwahoriHeckealgebra,cf.Remark5.4(iii). Inparticular,wenotethatthecharacteristic
function1 hasvanishingorbitalintegralsatall regularsemi-simpleelementsintheramified
setting,i.eS.(iOtFt0r)ansfersto(0,0). Insteadweprovepart(b)ofConjecture1.1(inthecaseF =Q
0 p
and n=3) by showing that, for any f(cid:48) as in the statement, the sum
2ω(γ)∂Orb(γ,f(cid:48))+Int(g)·logq (1.3)
is an orbital integral function, i.e. of the form ω(γ)Orb(γ,f(cid:48) ) for a suitable function f(cid:48) .
corr corr
Then part (a) of Conjecture 1.1 follows easily.
To prove that (1.3) is an orbital integral function, we first remark that (1.3) may be viewed
as a function on an open subset of the categorical quotient of S by GL , and then use the
n n−1
fact [37] that the desired property may be checked locally on the base. To achieve this goal, we
proceedintwosteps. First,wedetermineexplicitlyInt(g). Second,wedevelopagermexpansion
around each point of the categorical quotient, which is sufficiently explicit that it determines
ω(γ)∂Orb(γ,f(cid:48)) up to a local orbital integral function. Putting these two steps together, we
check that (1.3) is an orbital integral function. The description just given is inaccurate, insofar
as we first perform a reduction to a Lie algebra analog. Here the Cayley transform from [37]
plays a key role. For the first step we use, similarly to [10], the results of Gross and Keating [1]
onquasi-canonical liftings (thereductionto[1]in[10,§8]istransposedheretotheramifiedcase;
in fact, we found a drastic simplification of the proof (due to Zink) in loc. cit., which applies
to both the unramified and the ramified case). For the second step, we base ourselves on the
results on local harmonic analysis in [36], which we complete and make more explicit in various
ways.
Additionally, letuspointouttwogroup-theoreticfeaturesinthecasen=3whichseemtobe
important. ThefirstistheexceptionalisomorphismSL (cid:39)SU . Onegeometricmanifestationof
2 2
thisisthatthereisanaturalisomorphismbetweenMandaconnectedcomponentofN ,where
2
M is the Lubin–Tate deformation space over SpfO of the formal O -module of dimension 1
F˘ F0
and height 2; see Proposition 6.3. To state the second feature, we note that the aforementioned
maximal compact subgroups K and K(cid:91) have symplectic reductions; see Remark 5.2. When
0 0
n=3, associated to the reduction of K(cid:91) is a second exceptional isomorphism Sp (cid:39)SL , which
0 2 2
plays a role in reducing the conjecture to a Lie algebra version; see the proof of Lemma 11.1.
Let us also remark on the restriction to F =Q in Theorem 1.2. While we certainly expect
0 p
that the overall framework of this paper should be valid for any p-adic field F , there are a few
0
instances, all of which occur when working with N or related formal schemes, where we need
n
toappealtoresultsintheliteraturewhichareonlyestablishedatthelevelofgeneralityofq =p
or F = Q . Indeed, strictly speaking, this is already the case for the representability result in
0 p
[25] which is needed to know that N is a formal scheme in the first place! In accordance with
n
our expectations, we will use the general notation q and F throughout the paper, but when
0
working in a context where formal schemes are present, we will always tacitly take q = p and
F =Q . By contrast, the parts of this paper lying in the realm of harmonic analysis are valid
0 p
without any restriction on F .
0
Now let us comment on the possibility of extending our main result to odd integers n > 3.
The difficulties seem formidable. First, one would have to deal with degenerate intersections.
Related to this is the fact that the reduction procedure to a Lie algebra analog breaks down. In
fact,weareunabletoevenformulateaconjecturalLiealgebraversionofConjecture1.1,sincewe
are lacking a reasonable definition of an intersection multiplicity in this context; see Conjecture
5.10 below, in which we have to assume that the intersection is artinian. The second difficulty
is that our knowledge of local harmonic analysis when n > 3 is not advanced enough; even a
germ expansion principle is missing beyond the case of n = 3 [36]. One possibility for making
further progress would be to consider Conjecture 1.1 only for elements γ and g that satisfy
certain simplifying restrictions, in the spirit of [23] (which considers only minuscule elements).
Onthepositiveside,thereareotherinstancesofATconjectures. Indeed,in[20],weformulate
AT conjectures for F/F ramified (as in the present paper) and n even, and also for F/F
0 0
6 M.RAPOPORT,B.SMITHLING,ANDW.ZHANG
unramified and any n. The methods developed in the present paper can be applied to some
low-dimensional cases of them, cf. [20].
Letusfinallycommentontheglobalmotivationbehindalloftheseconjectures. TheAFLwas
introducedbythethird-namedauthorin[35]inthecontextofarelativetraceformulaapproach
tothehermitiancaseofthearithmeticGan–Gross–Prasadconjecture,tofacilitateacomparison
between geometric and analytic sides at inert primes of the unitary group. In forthcoming work
[21] we will explain how our AT conjectures facilitate analogous comparisons in this framework,
which in particular encompass the ramified primes of the unitary group. We refer to loc. cit. for
details.
We now give an overview of the contents of this paper. The paper consists of four parts.
In Part 1, we give the group-theoretic setup (in its homogeneous, its inhomogeneous, and its
Lie algebra versions); we define the formal moduli spaces of p-divisible groups, and establish
some structural properties for them; and we define the arithmetic intersection numbers that
enter into the formulation of our conjectures and results. In §5, we formulate our main results.
In Part 2, we explicitly calculate the arithmetic intersection numbers in the case n = 3, by
reduction to the Lie algebra and by relating this case to the Gross–Keating formulas.
In Part 3, we tackle the left-hand side of the identities to be proved in Theorem 1.2. This is
done by reducing the problem to one on the reduced subset of the Lie algebra. The rest of part
3 is devoted to explicitly evaluating the germ expansion of the orbital integral of a function f(cid:48)
with transfer (1 ,0), and then making the comparison with the result of part 2. At the end of
K0
part 3, the proof of Theorem 1.2 is complete.
In Part 4, we prove the germ expansion of the orbital integral of a general function f(cid:48). This
part of the paper can be read independently of the rest and lies squarely in the domain of local
harmonic analysis for the Jacquet–Rallis relative trace formula approach to the Gan–Gross–
Prasad conjecture.
Acknowledgments. We are grateful to A. Mihatsch for his help; in particular, he alerted us
to the difficulties arising when considering degenerate intersections. We also thank B. Gross,
S. Kudla, P. Scholze, and Z. Yun for helpful discussions. We finally thank the referees for their
remarks on the text.
M.R.issupportedbyagrantfromtheSimonsfoundationandbytheDeutscheForschungsge-
meinschaftthroughthegrantSFB/TR45. B.S.issupportedbyanNSAgrantH98230-16-1-0024.
W.Z. is supported by NSF grants DMS #1301848 and #1601144, and by a Sloan research fel-
lowship.
Notation. We list here some notation that we use throughout the whole paper. We denote by
p an odd prime number.
F/F is a ramified quadratic extension of finite extensions of Q . We denote by a (cid:55)→ a the
0 p
nontrivial automorphism of F/F , and by η = η the corresponding quadratic character on
0 F/F0
F×. When h is a square matrix with entries in F , we sometimes abbreviate η(deth) to η(h).
0 0
Sincep(cid:54)=2,wemayanddochooseuniformizersπofF and(cid:36)ofF suchthatπ2 =(cid:36). Wedenote
0
by k an algebraic closure of the common residue field k of F and F , and we set q := #k. As
0
stated above, when working in an algebro-geometric context where formal schemes are present,
we always understand that q =p and F =Q . We denote the group of squares in F× by F×,2.
0 p 0 0
We denote the group of norm 1 elements in F× by
F1 :={a∈F |aa=1}.
WedenotebyF˘ thecompletionofamaximalunramifiedextensionofF ,andbyF˘ :=F˘ ⊗ F
0 0 0 F0
the analogous object for F.
A polarization on a p-divisible group X is an anti-symmetric isogeny X → X∨, where X∨
denotes the dual. We use a superscript ◦ to denote the operation −⊗ZQ on groups of homo-
morphisms, so that for example
Hom◦(X,Y):=Hom(X,Y)⊗ZQ,
ON THE ARITHMETIC TRANSFER CONJECTURE FOR EXOTIC SMOOTH MODULI SPACES 7
whereY isanotherp-divisiblegroup. Foranyquasi-isogenyρ: X →Y andpolarizationλonY,
we define the pullback polarization
ρ∗(λ):=ρ∨◦λ◦ρ.
We denote by E the unique (up to isomorphism) formal O -module of relative height 2 and
F0
dimension 1 over Speck. We set
O :=End (E) and D :=O ⊗ F .
D OF0 D OF0 0
Thus D is “the” quaternion division algebra over F , and O is its maximal order. Since F/F
0 D 0
is ramified, any F -embedding of F into D makes E into a formal O -module of relative height
0 F
1. We fix such an embedding
ιE: F −→D
onceandforall, andwealwaysunderstandEtobeaformalOF-moduleviaιE. WedenotebyE
thecorrespondingcanonicalliftofEoverSpfO , equippedwithitsO -actionι andO -linear
F˘ F E F
framing isomorphism ρ : E −∼→ E. We denote by E the same object as E, except where the
E k
OF-action ιE is equal to ιE precomposed by the nontrivial automorphism of F/F0; and ditto for
E in relation to E, which is furthermore equipped with the same framing ρ := ρ on the level
E E
of O -modules over Speck. Of course E is a formal O -module of relative height 1 in its own
F0 F
right, but note that E is not its canonical lift. In §3.3 we will specify a principal polarization λE
onE,andwewilldenotebyλE theprincipalpolarizationonE liftingλE. WewriteλE :=λE and
λ :=λ for the same polarizations when we regard them as defined on E and E, respectively.
E E
We denote the main involution on D by c (cid:55)→ c, and the reduced norm by N. We also write
N for the norm map F× → F×; of course, all of this is compatible with any embedding of F
0
into D. We write v for the normalized valuation on D, and we use π as a uniformizer for D,
D
via ιE. We write v for the normalized (i.e. (cid:36)-adic) valuation on F0. For c ∈ D×, we define the
conjugate embedding
cι : F (cid:47)(cid:47)D
E
(cid:31) (cid:47)(cid:47) (1.4)
a cιE(a)c−1.
We denote by cF the image of cι .
E
GivenavarietyV overSpecF ,wedenotebyC∞(V(F ))thesetoflocallyconstant,compactly
0 c 0
supported functions on the space V(F ), endowed with its (cid:36)-adic topology; and we typically
0
abbreviate this set to C∞(V). We normalize the respective Haar measures on F , F, F×, and
c 0 0
F× such that vol(O )=vol(O )=vol(O×)=vol(O×)=1.
We write A for tFh0e affine linFe, M for Fth0e schemeFof n×n matrices, and 1 for the n×n
n n
identity matrix. We use a subscript S to denote base change to S, and when S = SpecA, we
often use a subscript A instead.
Part 1. The conjectures
In this first part of the paper we introduce the objects involved in the statements of our AT
conjectures. In §5 we state the conjectures and our main results.
2. Group-theoretic setup and orbit matching
We begin by explaining the general group-theoretic setup in this paper and the attendant
matching relation for regular semi-simple elements. We consider three cases: the homogeneous
group setting, the inhomogeneous group setting, and the Lie algebra setting. In this section
n≥2 is an integer.
2.1. Homogenous setting. We begin with the algebraic group
G(cid:48) :=Res (GL ×GL )
F/F0 n−1 n
over F . We consider the following two subgroups of G(cid:48). The first subgroup is
0
H(cid:48) :=Res GL ,
1 F/F0 n−1
8 M.RAPOPORT,B.SMITHLING,ANDW.ZHANG
which is embedded diagonally, via the inclusion of GL in GL sending A(cid:55)→diag(A,1). The
n−1 n
second subgroup is
H(cid:48) :=GL ×GL ,
2 n−1 n
with its obvious embedding into G(cid:48). Let
H(cid:48) :=H(cid:48) ×H(cid:48).
1,2 1 2
Then H(cid:48) acts on G(cid:48) by (h ,h ): γ (cid:55)→ h−1γh . We call an element γ ∈ G(cid:48)(F ) regular semi-
1,2 1 2 1 2 0
simple if it is regular semi-simple for the action of H(cid:48) , i.e. its orbit under H(cid:48) is closed, and
1,2 1,2
its stabilizer is of minimal dimension. In the case at hand, it is equivalent that γ have closed
orbit and trivial stabilizer, which follows from [18, Th. 6.1]. We denote by G(cid:48)(F ) the set of
0 rs
regular semi-simple elements in G(cid:48)(F ).
0
We next consider non-degenerate F/F -hermitian spaces of dimension n. Up to isomorphism
0
there are two of them, a split one W and a non-split one W . They are distinguished by the
0 1
rule
η(cid:0)(−1)n(n−1)/2detW (cid:1)=(−1)i, (2.1)
i
wheredetW :=detJ foranyhermitianmatrixJ (relativetothechoiceofabasis)representing
i i i
the hermitian form. Let i∈{0,1}. Write
U :=U(W ), (2.2)
i i
and let
u ∈W
i i
be a non-isotropic vector, which we call the special vector. We will choose u and u such that
0 1
theirnormsarecongruentmodNF×, whichcanalwaysbedonesincen≥2. LetW(cid:91) denotethe
i
orthogonal complement in W of the line spanned by u , and let
i i
H :=U(W(cid:91)).
i i
Then H naturally embeds into U as the stabilizer of u . Since we assume that u and u have
i i i 0 1
congruent norms mod NF×, W(cid:91) and W(cid:91) are non-isomorphic as hermitian spaces.
0 1
To lighten notation, now set W :=W , and define W(cid:91), U, H, and u analogously. Let
i
G :=H ×U,
W
and consider H as a subgroup of G , embedded diagonally. Then H ×H acts on G via the
W W
rule
(h ,h ): g (cid:55)−→h−1gh .
1 2 1 2
An element g ∈ G (F ) is called regular semi-simple if it is regular semi-simple for the action
W 0
of H ×H. We denote by G (F ) the set of regular semi-simple elements in G (F ).
W 0 rs W 0
Wenowrecallthematchingrelationbetweenregularsemi-simpleelements,asin[35]. Choose
an F-basis for W(cid:91) and complete it to a basis for W by adjoining u. This identifies W(cid:91) with
Fn−1 and W with Fn in such a way that u corresponds to the column vector
e:=(0,...,0,1)
in Fn, and hence determines embeddings of groups U (cid:44)→ Res GL and G (cid:44)→ G(cid:48). We
F/F0 n W
call the embeddings obtained in this way special embeddings. An element γ ∈ G(cid:48)(F ) and an
0 rs
element g ∈ G (F ) are said to match if these two elements, when considered as elements in
W 0 rs
G(cid:48)(F ), are conjugate under H(cid:48) (F ). The matching relation is independent of the choice of
0 1,2 0
special embedding and induces a bijection [35, §2]1
(cid:2)G (F ) (cid:3)(cid:113)(cid:2)G (F ) (cid:3)(cid:39)(cid:2)G(cid:48)(F ) (cid:3).
W0 0 rs W1 0 rs 0 rs
Here on the left-hand side, the square brackets denote the sets of orbits under the respective
actions of H (F )×H (F ) and H (F )×H (F ), whereas the brackets on the right-hand side
0 0 0 0 1 0 1 0
denote the set of orbits under H(cid:48) (F ).
1,2 0
1In loc. cit. only the inhomogeneous version, which will be taken up in the next subsection, is considered.
However,theinhomogeneousversioneasilyimpliesthehomogeneousversion.
ON THE ARITHMETIC TRANSFER CONJECTURE FOR EXOTIC SMOOTH MODULI SPACES 9
2.2. Inhomogeneous setting. Now we pass to the inhomogeneous version of the previous
subsection. Consider the following identifications of algebraic varieties over F . First,
0
H(cid:48)\G(cid:48) −∼→Res GL , γ =(γ ,γ )(cid:55)−→γ−1γ .
1 F/F0 n 1 2 1 2
Second, let
S :=S :={g ∈Res GL |gg =1 }
n F/F0 n n
and
H(cid:48) :=GL .
n−1
Then
Res (GL )(cid:14)GL −∼→S, γ (cid:55)−→γγ−1,
F/F0 n n
and the above two identifications induce an identification on F -rational points
0
G(cid:48)(F )(cid:14)H(cid:48) (F )(cid:39)S(F )(cid:14)H(cid:48)(F ). (2.3)
0 1,2 0 0 0
Here the action of H(cid:48) on S is through conjugation. In other words, the map
G(cid:48)(F )−→S(F ), γ =(γ ,γ )(cid:55)−→s(γ):=(γ−1γ )(cid:0)γ−1γ (cid:1)−1,
0 0 1 2 1 2 1 2
induces the bijection (2.3).
Ontheunitarygroupside,leti∈{0,1}andusethenotationW,U,H,etc.asintheprevious
subsection. Then we similarly have an identification of F -points of algebraic varieties over F ,
0 0
G (F )(cid:14)(cid:0)H(F )×H(F )(cid:1)(cid:39)U(F )(cid:14)H(F ), (g ,g )(cid:55)−→g−1g .
W 0 0 0 0 0 1 2 1 2
Here the action of H(F ) on U(F ) is by conjugation.
0 0
The definitions from the homogeneous setting readily transfer to the inhomogeneous setting.
Thus an element γ ∈ S(F ) is called regular semi-simple if it is regular semi-simple for the
0
action of GL on GL . As in the homogeneous setting, it is again equivalent that γ have
n−1 n
closed orbit and trivial stabilizer, cf. [18, Th. 6.1]. We denote by S(F ) or S (F ) the set
0 rs n 0 rs
of regular semi-simple elements in S(F ). An element g ∈ U(F ) is regular semi-simple if it is
0 0
regular semi-simple for the action of H on U; equivalently, when g is considered as an element
in GL (F) upon choosing a special embedding for U as in the previous subsection, it is regular
n
semi-simplefortheactionofGL onGL . WedenotebyU(F ) thesetofregularsemi-simple
n−1 n 0 rs
elements in U(F ).
0
An element γ ∈S(F ) matches an element g ∈U(F ) if these two elements are conjugate
0 rs 0 rs
underGL (F)whenconsideredaselementsofGL (F),uponchoosingaspecialembeddingfor
n−1 n
U. The matching relation is again independent of the choice of special embedding, and induces
a bijection [35, §2]
(cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3)
U (F ) (cid:113) U (F ) (cid:39) S(F ) ,
0 0 rs 1 0 rs 0 rs
where on the left-hand side are the respective sets of orbits under H (F ) and H (F ), and on
0 0 1 0
the right-hand side the set of orbits under GL (F ). In particular, there is a disjoint union
n−1 0
decomposition
S(F ) =S (cid:113)S , (2.4)
0 rs rs,0 rs,1
where S denotes the set of elements in S(F ) that match with elements in U (F ) .
rs,i 0 rs i 0 rs
2.3. Lie algebra setting. We also consider a Lie algebra version of the inhomogeneous setup.
We introduce the Lie algebra version of S ,
n
(cid:8) (cid:12) (cid:9)
s:=sn := y ∈ResF/F0Mn (cid:12)y+y =0 . (2.5)
Then H(cid:48) =GL acts on s, and we call an element of s(F ) regular semi-simple if its H(cid:48)-orbit
n−1 0
is closed and of maximal dimension. It is again equivalent that the element have closed orbit
and trivial stabilizer. We denote by s(F ) = s (F ) the set of regular semi-simple elements
0 rs n 0 rs
in s(F ).
0
For i∈{0,1}, let
u :=LieU .
i i
An element of u (F ) is called regular semi-simple if its orbit under H is closed and of maximal
i 0 i
dimension. We denote by u (F ) the set of regular semi-simple elements in u (F ).
i 0 rs i 0
As in §2.1, the choice of a basis for W(cid:91), extended by u to a basis for W , determines an
i i i
embedding u (cid:44)→ Res M , which we again call a special embedding. An element y ∈ s(F )
i F/F0 n 0 rs
10 M.RAPOPORT,B.SMITHLING,ANDW.ZHANG
matches an element x ∈ u (F ) if these two elements are conjugate under GL (F) when
i 0 rs n−1
consideredaselementsofM (F). Thematchingrelationisindependentofthespecialembedding
n
and induces a bijection [9, §5]
[u (F ) ](cid:113)[u (F ) ](cid:39)[s(F ) ],
0 0 rs 1 0 rs 0 rs
where on the left-hand side are the respective sets of orbits under H (F ) and H (F ), and
0 0 1 0
on the right-hand side the set of orbits under GL (F ). As before, we get a disjoint union
n−1 0
decomposition
s(F ) =s (cid:113)s ,
0 rs rs,0 rs,1
where s denotes the set of elements in s(F ) that match with elements in u (F ) .
rs,i 0 rs i 0 rs
2.4. Linear algebra characterizations. Wenowrecallthelinearalgebracharacterizationsof
regularsemi-simpleelementsandoftheirmatching,intheinhomogeneousgroupsettingandthe
Lie algebra setting. First we introduce the discriminant ∆, which is a morphism of varieties
∆: Res M (cid:47)(cid:47)Res A
F/F0(cid:31) n (cid:47)(cid:47) F/F0 (2.6)
x det(texi+je)
0≤i,j≤n−1
over F . We remark that beginning in §8 we will actually work with a rescaled version of ∆,
0
cf. (8.12).
An element γ ∈ S(F ) is regular semi-simple if and only if, considering γ as an element of
0
GL (F),thesetsofvectors{γie}n−1 and{teγi}n−1 arebothlinearlyindependent[18,Th.6.1].2
n i=0 i=0
Equivalently, ∆(γ)(cid:54)=0. Hence the regular semi-simple locus inside S is the complement of the
locus ∆=0, i.e. the Zariski-open subscheme
(cid:8) (cid:12) (cid:9)
Srs :=Sn,rs := γ ∈S (cid:12)∆(γ)(cid:54)=0 .
Thus we may write S(F ) and S (F ) interchangeably.
0 rs rs 0
To x ∈ M (F) we associate the following numerical invariants: the n coefficients of the
n
characteristicpolynomialchar (T)∈F[T],andthen−1elements texie∈F fori=1,...,n−1.
x
Then two elements of S(F ) are conjugate under GL (F ) if and only if they have the same
0 rs n−1 0
numerical invariants when considered as elements of M (F), cf. [35].
n
For i ∈ {0,1} and U = U , an element g ∈ U(F ) is regular semi-simple if and only if, when
i 0
consideredas anelementofGL (F)via anyspecial embedding, g satisfies theconditionsabove,
n
i.e. the sets of vectors {gie}n−1 and {tegi}n−1 are both linearly independent. Equivalently,
i=0 i=0
∆(g) (cid:54)= 0, so that the regular semi-simple set is the Zariski open complement to the locus
∆(g) = 0, and we write U(F ) and U (F ) interchangeably. Two elements g ∈ U(F ) and
0 rs rs 0 0 rs
γ ∈ S(F ) are matched if and only if their numerical invariants, when both are considered as
0 rs
elements of M (F), coincide.
n
The theory in the Lie algebra setting is entirely analogous. An element y ∈ s(F ) is regular
0
semi-simpleifandonlyif,consideringyasanelementofM (F),thesetsofvectors{yie}n−1 and
n i=0
{teyi}n−1 are both linearly independent [18, Th. 6.1]. Equivalently, ∆(y) (cid:54)= 0, i.e. the regular
i=0
semi-simple set is the Zariski open complement to the locus ∆(y)=0, and we write s(F ) and
0 rs
s (F ) interchangeably. Furthermore, two elements of s(F ) are conjugate under GL (F )
rs 0 0 rs n−1 0
if and only if they have the same numerical invariants when considered as elements of M (F),
n
cf. [18].
For i ∈ {0,1} and u = u , an element x ∈ u(F ) is regular semi-simple if and only if, when
i 0
considered as an element of M (F) via any special embedding, x satisfies the conditions above,
n
i.e. the sets of vectors {xie}n−1 and {texi}n−1 are both linearly independent. Equivalently,
i=0 i=0
∆(x) (cid:54)= 0, and we again write u(F ) and u (F ) interchangeably. Two elements x ∈ u(F )
0 rs rs 0 0 rs
and y ∈s(F ) are matched if and only if their numerical invariants, when both are considered
0 rs
as elements of M (F), coincide [9, 35].
n
From now on in the paper, we make the blanket assumption that
the special vectors u ∈W and u ∈W have norm 1. (2.7)
0 0 1 1
2In[18],theLiealgebraversionisconsidered,butitiseasytodeducethegroupversionfromthis.