Supersingular abelian varieties - Australian Mathematical Society

20 Pages·2005·0.16 MB·English

J.Aust. Math. Soc. 78(2005),373–392
ON THE MASS FORMULA OF SUPERSINGULAR ABELIAN
VARIETIES WITH REAL MULTIPLICATIONS
CHIA-FUYU
(Received1July2003;revised12November2003)
CommunicatedbyK.F.Lai
Abstract
Ageometricmassconcerningsupersingularabelianvarietieswithrealmultiplicationsisformulatedand
relatedtoanarithmeticmass. Wedeterminetheexactgeometricmassformulaforsuperspecialabelian
varieties of Hilbert-Blumenthal type. As an application, we compute the number of the irreducible
componentsofthesupersingularlocusofsomeHilbert-Blumenthalvarietiesintermsofspecialvaluesof
thezetafunction.
2000Mathematicssubjectclassification:primary11F41;secondary: 11E41.
Keywords and phrases: mass formula, supersingular locus, superspecial abelian varieties, Hilbert-
Blumenthalvarieties.
1. Introduction
For a reductive group G over Q which is R-anisotropic and an open compact sub-
group K of G.A /, one can define the mass of .G;K/. The attached mass is a
f
weighted class number. It measures the size of K as well as the class number; but
more importantly it behaves well when the group G or the level K varies. The im-
portance of the mass formula is its uniform way to organize arithmetic objects and
the measurement of these arithmetic objects. The computation of the mass formula
hasalonghistory,datingfromthepioneersSmith,Minkowski,andSiegel. Thereare
many important contributions to the Tamagawa numbers due to the works of Weil,
Ono,Langlands,Harder,LaiandKottwitz, andtotheexactformulas duetothoseof
Shimura,Hashimoto-Ibukiyama,Prasad,Gross,Gan-Yuandmanyothers.
ForasmallclassofgroupsG,themasscanalsoarisefromspecialabelianvarieties
(cid:13)c 2005AustralianMathematicalSociety1446-8107/05$A2:00C0:00
373
374 Chia-FuYu [2]
incharacteristic p. ThecelebratedDeuringmassformulasays
X 1 p−1
D ;
#AutE 24
E
where E runsovertheisomorphismclassesofsupersingularellipticcurves. Thegoal
of this paper is to establish the connection between the geometrically defined mass
and the arithmetically defined mass. Then one can either access the existing mass
formulas,orverifya(arithmeticorgeometric)massformulabyadifferent(geometric
orarithmetic)method.
In this paper weshow that the mass for certain special polarized abelian varieties
withrealmultiplicationscanbeanarithmeticmassforsome.G;K/;seeCorollary2.5.
Thefeatures are: these aresupersingular points and neednotto besuperspecial; the
polarizations can be inseparable; and the formulation does not require the existence
ordefinitionofthemodulispace. Thelatterindicatesthatthereisnonewdifficultyto
establish suchconnection evenwhenthemoduli spaceeither hasbadreduction oris
notwell-behaved.
We use Shimura’s exact mass formula [9] to express the geometric mass in term
of special values of the zeta function up to precise local terms; see equation (3) and
Theorem2.7. Thespecialcasewhenthetotallyrealnumberfield F isQ,theabelian
varieties are superspecial, and the polarizations are principal, the geometric mass
formulareducestoaresultofEkedahlandvanderGeer[3]obtainedbythegeometric
method. OurformulationcanbegeneralizedtobasicpointsinarbitraryPEL-Shimura
variety modulo p (with modification due to the Hasse principle). Doing that will
requiremuchmoreworkandweplantocarryitoutinasubsequentpaper.
In a part of this paper we determine the remaining local terms in the case of
superspecialabelianvarietiesofHilbert-Blumenthaltype;seeTheorem3.7. Usingthis
geometricmassformula,wecandeterminethenumberoftheirreduciblecomponents
ofthesupersingular locus, followingthe methodsin [6]and[11]. Inthelast section
we determine this number in some restricted cases, particularly when p is totally
ramified in F; see Theorem 4.4. We note that our approach gives an explanation
of the interesting result of Bachmat and Goren [1] on the supersingular locus of
Hilbert modular surfaces. Finally we note that there is an interesting connection of
supersingularpointswiththetheoryofmodularformsmodulo p;see[4]and[8].
2. Supersingularpointsandthemassformula
2.1. Shimura’sexactmassformula Let B beaquaterniondivisionalgebraovera
totallyrealnumberfield F,andlet(cid:27) denotethestandardinvolutionof B. LetV bea
[3] Supersingularabelianvarieties 375
left B-moduleofrankm,andlet’ beaquaternionHermitianformonV,thatis,
’ V V (cid:2)V ! B
such that ’.x;y/ D ’.y;x/(cid:27) and ’.ax;by/ D a’.x;y/b(cid:27) for all x;y 2 V and
a;b 2 B. LetG’ denotetheunitarygroupattachedto’. Itisareductivegroupover
F andwewillregarditasareductivealgebraicgroupoverQviatheWeilrestriction
of scalars from F to Q. Assume that G’.R/ is compact. For any open compact
subgroup K of G’.A /,the mass of K is definedas follows. Letfc ;c ;:::;c gbe
f 1 2 h
a(complete)setofrepresentativesofthedoublecosetspaceG’.Q/nG’.A /=K,and
f
let 0 VD G’.Q/\c Kc−1. Notethat each 0 is finite bythe assumption of G’.R/.
i i i i
Themassof K isdefinedtobe
Xh 1
mass.K/VD :
#0
iD1 i
Forthegeneraldefinitionofthemassof K (withoutthecompactnessassumption),
we refer to [9, Introduction, page 67]. It follows easily from the definition that for
twoopencompactsubgroups K (cid:26) K ,onehasmass.K /DTK V K Umass.K /.
1 2 1 2 1 2
Chooseamaximalorder O of B. Let L bean O -lattice in V whichismaximal
B B
alongthelattices onwhich ’ takesits valuein O . Let K bethemaximal compact
B 0
subgroup of G’.A / which stabilizes the adelic lattice L ⊗ ZO. In [9, Introduction,
f Z
page68]Shimuragivesanexplicitformula
Ym n (cid:2) (cid:3) o
(1) mass.K /DjD jm2 D1=2 .2k−1/W.2(cid:25)/−2k d(cid:16) .2k/
0 F F F
kD1
Ys Ym (cid:8) (cid:9)
(cid:2) .N. /k C.−1/k ;
i
p
iD1 kD1
where D isthediscriminantof F overQ,d isthedegreeof F,andf g arethe
F i 1(cid:20)i(cid:20)s
p
primesof O atwhich B isramified.
F
Notethatthemaximalityof L isalocalproperty. Thatis, L ismaximalifandonly
ifeach L ismaximalforallprimes of F.
p p
2.2. Onecanexpresstheexactformulaintermsofspecialvaluesofzetafunctions
atnegativeintegers. Let F beatotallyrealfieldofdegreed. Set
3 .s/ VD As0.s=2/d(cid:16) .s/; A VD D1=2(cid:25)−d=2:
F F F
Thenonehasthefunctionalequationfor(cid:16) .s/
F
3 .1−s/D3 .s/:
F F
376 Chia-FuYu [4]
Itgives
0..1−s/=2/d
(cid:16) .s/D D1=2D−s(cid:25)−d=2Cds (cid:16) .1−s/:
F F F 0.s=2/d F
Lets D2k,onegets
0..1−2k/=2/d
(cid:16) .2k/D D1=2D−2k(cid:25)−d=2C2dk (cid:16) .1−2k/:
F F F 0.k/d F
Ontheotherhand,onehas
p
0..1−2k/=2/ .−1/k22k−1.k−1/W0.1=2/ .−1/k22k−1 (cid:25)
D .2k−1/W D :
0.k/ .k−1/W .2k−1/W
Thefactorinthemiddleof(1)canberewrittenasfollows.
(cid:2) (cid:3)
(2) D1=2 .2k−1/W.2(cid:25)/−2k d(cid:16) .2k/
F F
(cid:20) p (cid:21)
(cid:2) (cid:3) .−1/k22k−1 (cid:25) d
D DF1−2k .2k−1/W.2(cid:25)/−2k d .2k−1/W (cid:25)−d2C2dk(cid:16)F.1−2k/
(cid:20) (cid:21)
.−1/k d
D D1−2k (cid:16) .1−2k/:
F 2 F
Henceweget
( )
(cid:20) (cid:21)
Ym .−1/k d Ys Ym (cid:8) (cid:9)
(3) mass.K /D (cid:16) .1−2k/ .N. /k C.−1/k :
0 2 F pi
kD1 iD1 kD1
2.3. If F D Q and B isthe quaternion algebra whichis ramified atf1; pg,then
wehave
( )
.−1/m.mC1/=2 Ym Ym (cid:8) (cid:9)
(4) mass.K /D (cid:16).1−2k/ .pk C.−1/k :
0 2m
kD1 kD1
If m D 1and B isramified onlyattheprimes over p andinfinite places, then we
have
(cid:20) (cid:21)
−1 d Ys
(5) mass.K /D (cid:16) .−1/ f.N. /C.−1/g:
0 2 F i
p
iD1
2.4. Inthefollowing,thegroundfieldk isalgebraicallyclosedofcharacteristic p.
Letx D .A ;(cid:21) ;(cid:19) /beasupersingularpolarizedabelian O -varietyoverk ofdimen-
0 0 0 F
siong Dmd. LetG denotethegroupschemeoverSpecZwhosegroupof R-points,
x
foreachcommutativering R,isG .R/D f(cid:30) 2 .End .A /⊗ R/(cid:2);(cid:30)0(cid:30) D1g,where
x OF 0
themap(cid:30) 7!(cid:30)0 istheRosatiinvolutioninducedby(cid:21) .
0
Let3 denotethesetoftheisomorphismclassesofpolarizedabelian O -varieties
x F
.A;(cid:21);(cid:19)/ofdimension goverk suchthat
[5] Supersingularabelianvarieties 377
(i) the Dieudonne´ module M.A/ is isomorphic to M.A / as quasi-polarized
0
Dieudonne´ O ⊗Z -modules,and
F p
(ii) the Tate module T.A/ is isomorphic to T.A / asnon-degenerate alternating
l l 0
O ⊗Z-modulesforalll 6D p.
F l
Thecondition (i) implies that A issupersingular. Let30 bethesubset of3 that
x x
consistsofobjects.A;(cid:21);(cid:19)/suchthat
(iii) thereexistsanelement(cid:30) 2Hom .A ;A/⊗Qsuchthat(cid:30)(cid:3)(cid:21)D(cid:21) .
OF 0 0
THEOREM 2.1 ([11, Theorem 10.5]). There is a natural bijection of pointed sets
between30 andG .Q/nG .A /=G .ZO/.
x x x f x
PROPOSITION2.2. Let.Ai;(cid:21)i;(cid:19)i/,i D1;2betwosupersingularpolarizedabelian
O -varietiesoverk. Thenthereexists’ 2Hom .A ;A /⊗Qsuchthat’(cid:3)(cid:21) D(cid:21) .
F OF 1 2 2 1
Thatis,thecondition(iii)ofSubsection2.4isautomatic.
PROOF. Choose an isogeny ’ V A1 ! A2. By Noether-Skolem’s theorem, ’ can
be chosen to be O -linear. Let (cid:21)0 VD ’(cid:3)(cid:21) and (cid:3) and (cid:3)0 be the Rosati involutions
F 1 2 1 1
inducedby(cid:21) and (cid:21)0 on E VD End.A /⊗Q, respectively. By[12, Satz1.1], there
1 1 1 1
existsapositiveelementc 2 E1(cid:2) withc D c(cid:3)1 suchthat x(cid:3)01 D c−1x(cid:3)1cforallx 2 E1.
Asx 2 F,cliesinC VDEnd .A /⊗Q.
OF 1
Let P be the the algebraic variety over Q defined by fX 2 C;X(cid:3)1X D cg. It is
atorsor under the algebraic group G1 VD fg 2 C(cid:2);g(cid:3)1g D 1g overQ byleft action.
Henceitformsanelement(cid:24) inH1.Q;G /. By[5,Lemma2.11],P.R/6D;. SinceG
1 1
issemi-simpleandsimply-connected, H1.Q ;G / D0. Therefore(cid:24) islocallytrivial
p 1
everywhere. BytheHasseprinciplefor G ,(cid:24) istrivial. Thenthereexistsanelement
1
g 2 C.Q/ such that g(cid:3)1g D c. Replacing ’ by ’g, we have (cid:3)1 D (cid:3)01, hence that
(cid:21)0 D q(cid:21) for some q 2 Q(cid:2). Note that q is positive. This follows from the fact
1 1
thatthepolarizationslieinthepositiveconeoftheNe´ron-SeverigroupNS.A /⊗R.
1
Therefore q is a norm of C and we can find a ’ 2 Hom .A ;A /⊗Q such that
OF 1 2
’(cid:3)(cid:21) D(cid:21) . Thiscompletestheproof.
2 1
COROLLARY2.3. 30x D3x.
LEMMA2.4. Let.A;(cid:21);(cid:19)/ 2 3x andTcU bethe corresponding double coset. Then
Aut.A;(cid:21);(cid:19)/ ’0 ,where0 VD G .Q/\cG .ZO/c−1.
c c x x
PROOF. LetG0 bethegroupschemeoverSpecZattachedto.A;(cid:21);(cid:19)/definedasin
Subsection2.4. Thatis,foranycommutativering R,
G0.R/Df(cid:11) 2.End .A/⊗ R/;(cid:11)0(cid:11) D1g;
OF
378 Chia-FuYu [6]
wherethemap(cid:11) 7!(cid:11)0 istheRosatiinvolutioninducedby(cid:21).
Choose a map (cid:30) 2 Hom .A ;A/⊗Q such that (cid:30)(cid:3)(cid:21) D (cid:21) . Then the element
OF 0 0
c 2 G .A /hastheproperty
x f
((cid:3)) (cid:30)c 2Isom..A .l/;(cid:21) ;(cid:19) /;.A.l/;(cid:21);(cid:19)//; 8l:
l 0 0 0
Notethat(cid:11) 2Aut.A;(cid:21);(cid:19)/ifandonlyif(cid:11) 2 G0.Q/and(cid:11) 2Aut.A.l/;(cid:21);(cid:19)/foralll.
l
Themap(cid:30)givesanisomorphismG .Q/! G0.Q/whichsends(cid:12)to(cid:30)(cid:12)(cid:30)−1 DV(cid:11).
x
From((cid:3)),wehave(cid:11) 2 G0.ZO/ifandonlyif.(cid:30)c/−1(cid:11).(cid:30)c/2 G .ZO/. Hence,(cid:11) 2 G0.ZO/
x
ifandonlyifc−1(cid:12)c 2 G .ZO/. Thiscompletestheproof.
x
X
COROLLARY2.5. mass.3x/ VD #Aut.1A;(cid:21);(cid:19)/ Dmass.Gx.ZO//.
.A;(cid:21);(cid:19)/23x
2.5. The semi-simple algebra C VD End .A /⊗Q is the centralizer of F in
x OF 0
thesimplealgebra M .End0.E//,where E isasupersingularellipticcurve. Wehave
g
C ’ M .B/,where B isaquaternionalgebraoverthetotallyrealfield F ramifiedat
x m
allrealplacesandunramifiedatallprimesnotdividing p. Sincethelocalinvariantsof
thecommutantC coincideswiththoseofF⊗ End0.E/,wehaveB ’ F⊗ End0.E/.
x Q Q
Therefore,
C ⊗R’ M .H/; C ⊗Q ’ M .F ⊗Q/; l 6D p
x m Y x l 2mY l
C ⊗Q ’ M .B /(cid:2) M .F /;
x p m 2m
jp;g Vodd p jp;g Veven p
p p p p
where g VD TF V Q U. The Rosati involution induces the standard involution on
p
M .H/ pand M p.B /, which we will denote by (cid:3). It also induces the symplectic
m m
involutionon M .pF ⊗Q/and M .F /. Therefore,wehave
2m l 2m
p
G .R/’fg 2 M .H/;g(cid:3)g D1g; G .Q/’Sp .F ⊗Q/; l 6D p
x m Y x Yl 2m l
G .Q /’ G (cid:2) Sp .F /;
x p 2m
jp;g Vodd p jp;g Veven p
p p p p
whereG Dfg 2 M .B /;g(cid:3)g D1g. Wewillchoosesuchisomorphismsandreplace
m
’byD.p p
LEMMA2.6. Thereexistaleft B-module V,aquaternionHermitianfrom’ on V,
and a maximal lattice L in the sense of Shimura (see Subsection 2.1) such that
G’ ’ G overQand K definedinSubsection2.1is
x Y 0 Y Y
K D Sp .O ⊗Q/(cid:2) K (cid:2) Sp .O /;
0 2m F l 0; 2m
l6Dp jp;g Vodd p jp;g Veven p
p p p p
where O istheringofintegersin F and K Dfg 2 M .O /;g(cid:3)g D1g.
0; m B
p p P p p
PROOF. LetV D B(cid:8)g with’.x;y/ D xiyNi(cid:27) andlet L D OB(cid:8)g.
[7] Supersingularabelianvarieties 379
2.6. The computation of mass.G .ZO// is to relate with the mass of the standard
x
one, K ,andthentoapplyShimura’smassformula.
0
ForanytwoopencompactsubgroupsK ;K ofG .A /,ofG .Q /,orofG .Q/,
1 2 x f x p x l
wedefinearationalnumber(cid:22) .K =K /with(cid:15)D f; p orl inthreerespectivecases,
(cid:15) 2 1
by
(cid:22) .K =K /VD TK V K \K U−1TK V K \K U:
(cid:15) 2 1 1 1 2 2 1 2
Wehavethefollowingproperties
(
TK V K U if K (cid:26) K ;
(i) (cid:22) .K =K /D 2 1 1 2
(cid:15) 2 1 TK V K U−1 if K (cid:27) K :
Q 1 2 Q 1 2
(ii) If K D K and K D K , where l runs over all primes of Q, then
1 Q l 1;l 2 l 2;l
(cid:22) .K =K /D (cid:22).K =K /.
f 2 1 l l 2;l 1;l
(iii) mass.K /D(cid:22) .K =K /mass.K /.
1 f 2 1 2
FromCorollary2.5,wehave
(6) mass.3 / D(cid:22) .K =G .ZO//mass.K /:
x f 0 x 0
Iflisprimetothedegreeofthepolarization(cid:21) ,thenG .Z/isisomorphictoaproduct
0 x l
ofSp .O /and(cid:22).K =G .Z//D1. Therefore,wehave
2m v l 0;l x l
( )
Y
(7) mass.3 / D (cid:22).K =G .Z// mass.K /:
x l 0;l x l 0
lDp;orljdeg(cid:21)0
THEOREM2.7. Letnotationbeasabove. Ifm D1ordeg(cid:21)0isapowerof p,thenwe
have the simplified mass formula mass.3 / D (cid:22) .K =K /mass.K /, where K
x p 0;p M0 0 M0
isthegroupofautomorphismsofthequasi-polarizedDieudonne´ O ⊗Z -moduleM
F p 0
attachedto x.
Asaconsequence,thefollowingcorollaryisageneralization oftheDeuringmass
formula. Recently, Ekedahl and van der Geer [3] compute the intersections of cy-
cle classes on the moduli space . They obtained this as a consequence of the
g
A
Hirzebruch-Mumfordproportionalityprinciple.
COROLLARY2.8. Let3bethesetoftheisomorphismclassesofprincipallypolar-
izedsuperspecialabelianvarietiesoverk ofdimensiong. Then
( )
X 1 .−1/g.gC1/=2 Yg Yg (cid:8) (cid:9)
D (cid:16).1−2k/ .pk C.−1/k :
#Aut.A;(cid:21)/ 2g
.A;(cid:21)/23 kD1 kD1
380 Chia-FuYu [8]
PROOF. Takex D.A0;(cid:21)0/tobeaprincipallypolarizedsuperspecialabelianvariety
overk ofdimension g. NotethattheDieudonne´ module M ofx isisomorphictothe
0
product of g copies of separably quasi-polarized rank two supersingular Dieudonne´
modules. It follows that 3 D 3 and the group K of the automorphisms of M
x M0 0
is K D f(cid:11) 2 M .O /;(cid:11)(cid:3)(cid:11) D 1g, where B is the quaternion division algebra
M0 g Bp p
overQ . Therefore,weobtainmass.3/Dmass.K /andfinishtheproofby(4).
p 0
3. SuperspecialpointsofHBtypeandthemassformula
We keep the notations in the previous section, except that we denote the totally
realfield by F andreserve thesymbol F for the Frobenius operator onaDieudonne´
module. Let beaprimeofO overp,andlete and f denotetheramificationdegree
F
andresidue dpegreeof ,respectively. Intherepst ofthpis paper, wewillonly treat the
Hilbert-Blumenthalcapses,namelym D1andd D g. Let VD O ⊗Z D(cid:8) .
F p jp
O p Op
3.1. We first recall the classification of superspecial quasi-polarized Dieudonne´
-modules[10]. Let x D .A ;(cid:21) ;(cid:19) /beasuperspecial polarizedabelian O -variety
0 0 0 F
Oover k and let M be its covariant Dieudonne´ module. We have M D (cid:8) M and
0 0 jp
willdescribe M foreach . p p
LetebetheLpietypeofpM anda bethea-typeof M . Werecalltheirdefinitions
in[10,Section1]below: p p
M
e.M / D.fe1i;e2ig/i2Z=f Z () Lie.M /’ kT(cid:25)U=.(cid:25)e1i/(cid:8)kT(cid:25)U=.(cid:25)e2i/
p p p i2Z=f Z M
p
a.M / D.fa1i;a2ig/i2Z=f Z () M =.F;V/M ’ kT(cid:25)U=.(cid:25)a1i/(cid:8)kT(cid:25)U=.(cid:25)a2i/;
p p p p i2Z=f Z
p
where(cid:25) isauniformizerof O andtheright-handsidesareisomorphismsof O ⊗
Z
p
k ’(cid:8) kT(cid:25)U=.(cid:25)e /-moduples. p
i2Z=f Z p
As M ipssuperspecial,eisequaltoa andithastheform
p
.fe ;e g;fe −e ;e −e g;fe ;e g;:::/
1 2 1 2 1 2
p p
forsomeintegerse ,e with0 (cid:20) e (cid:20) e (cid:20) e ;see[10,Section2]. When f isodd,
1 2 1 2
it satisfies an additional condition e Ce Dpe . We say M is of type .ep;e / for
1 2 1 2
convenience. p p
Let −1 D .(cid:25)−d/ be the inverse of the different of over Z . There is a
p
uniqueDWp⊗ -bilinear pairing .(cid:1);(cid:1)/ V M (cid:2) M ! WO⊗p such that hx;yi D
Tr .(cid:25)−Od.px;y//. Wehave O ⊗W Dp(cid:8) p Wi bytheOapctionof ur andthis
W⊗ =W i2Z=f Z
alsogOipvesthedecomposition M Dp(cid:8) Mi. p Op
i2Z=f Z
p p p
[9] Supersingularabelianvarieties 381
LEMMA3.1. (1) If f is even, then there is a Wi-basis Xi;Yi for Mi for each
i 2Z=f Zsuchthat p p
(
p
(cid:25)n ifi isodd;
(i) .X ;Y/D
i i (cid:25)nCe −e1−e2 ifi iseven;
p
foralli 2Z=f (Zandsomen 2Z. (
(ii) FX Dp −(cid:25)e1YiC1 ifi isodd; FY D v(cid:25)e2XiC1 ifi isodd;
i −(cid:25)ep−e2YiC1 ifi iseven; i v(cid:25)ep−e1XiC1 ifi iseven;
wherev(cid:25)e D p.
p
(2) If f is odd, then there is a Wi-basis X ;Y for Mi for each i 2 Z=f Z such
i i
that p p p
(i) .X ;Y/D(cid:25)n forsomen 2Z.
i i
(ii) FXi D−(cid:25)e1YiC1;FYi Dv(cid:25)e2YiC1 fori 2Z=f Z,wherev(cid:25)ep D p.
(3) Inparticular, if the pairing .(cid:1);(cid:1)/ is perfect and pM satisfies the Rapoport con-
dition(inthiscase, M isoftype.0;e /),thenthereexisptsa Wi-basisfX ;Ygof Mi
i i
foreachi 2Z=f Zsucphthat p p
p
(i) Y 2.VM/i and.X ;Y/ D1,
i i i
(ii) FX D−Y ; FY D pX ,
i iC1 i iC1
foralli 2Z=f Z.
p
PROOF. See[10,Lemmas4.5–4.6].
LetF0 betheuniquequadraticunramified extensionof F and(cid:28) bethegenerator
ofGal.Fp0=F /. p
p p
LEMMA3.2. Let KM bethegroupofautomorphismsofthequasi-polarizedDieu-
donne´ -module M .p
Op p
(1) Iftheresiduedegree f iseven,then
(cid:26)(cid:18) p (cid:19) (cid:27)
a b
K ’ 2SL .O /;c (cid:17)0 mod (cid:25)e2−e1 :
M c d 2 F
p p
(2) Iftheresiduedegree f isodd,then
p (cid:26)(cid:18) (cid:19) (cid:27)
a b
KM ’ (cid:25)e2−e1b(cid:28) a(cid:28) 2SL2.OF0 / :
p p
PROOF. Let(cid:30) 2 KM . ChooseaWi-basisfXi;Yigfor Mi asinLemma3.1. Write
(cid:18) p(cid:19) (cid:18) (cid:19) (cid:18) (cid:19) p
(cid:30).X / X a b
i D A i ; A D i i 2GL .Wi/:
(cid:30).Y/ i Y i c d 2
i i i i
382 Chia-FuYu [10]
It follows from .(cid:30).X /;(cid:30).Y// D .X ;Y/ that A 2 SL .Wi/. Note that we have
i i i i i 2
F2.X / D −pX and F2.Y / D −pY foralli. Thecondition (cid:30)F2 D F2(cid:30) gives
i iC2 i iC2
A D A.2/ foralli 2Z=f Z,wherewewrite A.n/ for A(cid:27)n.
iC2 i
(1)If f iseven,thenfApgaredeterminedby A and A ,and A ;A 2 SL .O /.
i 0 1 0 1 2 F
Wehave p p
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
F.X0/ D J X1 ; J D 0 −(cid:25)ep−e2 ;
(cid:18)F.Y0/(cid:19) 1(cid:18)Y1(cid:19) 1 (cid:18)v(cid:25)ep−e1 (cid:19)0
F.X1/ D J X2 ; J D 0 −(cid:25)e1 :
F.Y1/ 0 Y2 0 v(cid:25)e2 0
The condition F(cid:30) D (cid:30)F gives J A D A.1/J and J A.1/ D A J . These give the
1 1 0 1 0 0 1 0
relations
a Dd.1/; d Da.1/; c.1/ D−v(cid:25)e2−e1b ; c D−v(cid:25)e2−e1b.1/:
1 0 1 0 0 1 1 0
Weseethat A1 isdeterminedby A0 andc0 (cid:17)0 mod (cid:25)e2−e1.
(2)If f isodd,thenfAigaredeterminedby A0 and A0 2SL2.OF0 /. Wehave
p (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) p
F.X0/ D J X1 ; J D 0 −(cid:25)e1 :
F.Y0/ Y1 v(cid:25)e2 0
Applying F(cid:30) D(cid:30)F,wehave JA D A.1/J. As A D A.f C1/,thisequationgives
1 0 1 0p
d D a(cid:28); c D−v(cid:25)e2−e1b(cid:28):
0 0 0 0
As−visanormofsomeelementin O(cid:2) (over O(cid:2) ),wechooseanisomorphism
F0 F
(cid:26)(cid:18) p (cid:19) p (cid:27)
a b
KM ’ (cid:25)e2−e1b(cid:28) a(cid:28) 2SL2.OF0 / :
p p
Thiscompletestheproof.
3.2. Itremainstocomputethelocalterm(cid:22) .K =K /. Wewriteq for N. /,the
0; M
cardinalityoftheresiduefieldk. /. p p p p
When f is even, KM D 00.(cid:25)pe2−e1/ is anopencompact subgroup ofSL2.F /. If
e De ,thepn(cid:22) .K =Kp /D1;ife >e ,then p
1 2 0; M 2 1
p p p
(cid:22) .K0; =KM /D#P1.OF =(cid:25)e2−e1/Dqe2−e1−1.q C1/:
p p p p
Considernowthecasewhere f isodd. Put
(cid:26)(cid:18)p (cid:19) (cid:27)
a b
K.n/VD (cid:25)nb(cid:28) a(cid:28) 2SL2.OF0 / :
p