L- unc ions o symme ic powe s o he gene alized Ai y amily o
exponen ial sums
C. Douglas Haessig∗An onio Rojas-Le´on†
Decembe 7, 2010
Abs ac
This pape looks a he L- unc ion o he k- h symme ic powe o he Q`-shea Ai o e he a ine line
A1
Fqassocia ed o he gene alized Ai y amily o exponen ial sums. Using `-adic echniques, we compu e he
deg ee o his a ional unc ion as well as he local ac o s a in ini y. Using p-adic echniques, we s udy he
q-adic New on polygon o hese L- unc ion.
1 In oduc ion
In his pape we s udy he L- unc ion a ached o he k- h symme ic powe o he Q`-shea Ai associa ed o
he gene alized Ai y amily o exponen ial sums. Symme ic powe s appea in he p oo s o many a i hme ic
p oblems. Fo ins ance, Deligne’s p oo [6] o he Ramanujan-Pe e sson conjec u e elies on he cons uc ion o
a Galois module coming om he k- h symme ic powe o a ce ain `-adic shea . The Sa o-Ta e conjec u e [5]
[16] [27] elies on he analy ic con inua ion o he L- unc ion a ached o he k- h symme ic powe o an `-adic
ep esen a ion coming om an ellip ic cu e. Ano he equidis ibu ion esul conce ning Kloos e man angles was
p o en by Adolphson [4] using esul s o Robba’s [23] on he L- unc ion o he k- h symme ic powe o he `-adic
Kloos e man shea Kl2. Symme ic powe s also a ise in he p oo o Dwo k’s conjec u e [29] [30] [31]. To begin,
le us ecall he gene al se up o an L- unc ion o an `-adic ep esen a ion.
Le Fqbe he ini e ield o qelemen s and cha ac e is ic p. Le Ybe a smoo h, geome ically connec ed, open
a ie y de ined o e Fq; o ins ance, ake Y o be a ine s-space As
Fqo he o us Gs
m. Deno e i s unc ion ield
by K, and i s co esponding absolu e Galois g oup by GK:= Gal(Ksep/K). Le Vbe a ini e dimensional ec o
space o e a ini e ex ension ield o Q`, whe e `6=p. Le ρ:GK→GL(V) be a con inuous `-adic ep esen a ion
un ami ied on Y, and le Fbe he co esponding lisse shea on Y. De ine he L- unc ion o ρon Yby
L(Y, ρ, T ) := Y
x∈|Y|
1
de (1 −ρ(F obx)Tdeg(x)).(1)
By he Le sche z ace o mula, his is a a ional unc ion whose ze os and poles may be desc ibed using ´e ale
cohomology wi h compac suppo :
L(Y, ρ, T ) =
2dim(Y)
Y
i=0
de (1 −F obqT|Hi
c(Y⊗Fq,F))(−1)i+1
Gi en such a ep esen a ion, we may cons uc new L- unc ions ia ope a ions such as enso , symme ic, o
ex e io p oduc s. Na u al ques ions abou hese new L- unc ions conce n he de e mina ion o hei deg ees
(Eule cha ac e is ic) and desc ibing a ious p ope ies abou hei ze os and poles. In his pape , we will ocus
on he symme ic powe s o a pa icula amily o exponen ial sums called he gene alized Ai y amily. O he
amilies whose symme ic powe s ha e been in es iga ed a e he Legend e amily o ellip ic cu es [3] [10] and
he hype Kloos e man amily [12] [13] [23]. We no e ha he o me seems o ha e been mo i a ed by Dwo k’s
p-adic in e es in he Ramanujan-Pe e sson conjec u e.
The gene alized Ai y amily is de ined as ollows. Le be a polynomial o e Fqo deg ee dwi h p-d. Le
ψbe a non i ial addi i e cha ac e on Fq. Fo each ¯
∈Fqde ine i s deg ee by deg(¯
) := [Fq(¯
) : Fq]. I is
well-known ha he associa ed L- unc ion o he sequence o exponen ial sums
Sm(¯
) := X
x∈Fqmdeg(¯
)
ψ◦T Fqmdeg(¯
)/Fq( (x) + ¯
x) o m= 1,2,3, . . .
∗Pa ially suppo ed by NSF g an DMS-0901542
†Pa ially suppo ed by PO8-FQM-03894 (Jun a de Andaluc´ıa), MTM2007-66929 and FEDER
1
is a polynomial o deg ee d−1:
L( , A1,¯
;T) := exp ∞
X
m=1
Sm(¯
)Tm
m!= (1 −π1(¯
)T)· · · (1 −πd−1(¯
)T).
As we will desc ibe la e , he ela i e cohomology o his amily may be ep esen ed `-adically as a lisse shea o
ank d−1 o e A1 ia Fou ie ans o m. Le us deno e his shea by Ai . The L- unc ion o he k- h symme ic
powe o Ai akes he o m:
Mk( , T):=L(A1,Symk(Ai ), T) := Y
∈|A1|Y
a1+···+ad−1=k
(1 −π1( )a1· · · πd−1( )ad−1Tdeg( ))−1,
whe e |A1|deno es he se o closed poin s on A1. By he Le sche z ace o mula, Mk( , T) is a a ional unc ion.
The `-adic shea Ai was ex ensi ely s udied by N. Ka z in [18], whe e i s monod omy g oup is de e mined and,
as a consequence, an equidis ibu ion esul is ob ained o he exponen ial sums in he amily ([18, Co olla y
20]). F om hese esul s i ollows ha , o p > 2d−1, Mk( , T ) is in ac a polynomial. Fo d= 3, a s udy o
he monod omy g oup may be a oided using Adolphson’s me hod [4].
Ou i s main esul is he compu a ion o he deg ee o Mk( , T) o p>d. The deg ee o he a ional
unc ion Mk( , T) equals he k- h coe icien o a gene a ing se ies which is explici ly gi en in Co olla y 3.4.
Simpli ied o mulas a e gi en in sec ion 5 o some pa icula ly nice alues o and p.
As an example o his heo em, conside he amily gene a ed by (x) = xd. Then he deg ee o Mk(xd, T) may
be desc ibed as ollows. Le ζbe a p imi i e (d−1)- h oo o uni y in Fq. Deno e by Nd−1,k he numbe o (d−1)-
uples (a0, a1, . . . , ad−2) o nonnega i e in ege s such ha a0+a1+· · ·+ad−2=kand a0+a1ζ+· · ·+ad−2ζd−2= 0
in Fq.
Theo em 1.1. Wi h he no a ion de ined abo e, we ha e
deg Mk(xd, T) = 1
d−1k+d−2
d−2−dNd−1,k.
I was conjec u ed in [15] ha Mk(x3, T) is a polynomial o all p > 3 since i was shown, in ha pape , ha
Mk(x3, T) is a polynomial o e e y odd in ege k, and also o e e y ke en wi h k < 2p. Su p isingly, o p= 5,
Mk(x3, T) is no a polynomial o in ini ely many k. This was communica ed o he i s au ho by N. Ka z and
is a consequence o he geome ic monod omy g oup o Aix3being ini e.
Theo em 1.2. Suppose p > 2d−1. Then Mk( , T)is a polynomial which may be ac o ed in o a p oduc
Qk( , T)Pk( , T), whe e Pk( , T)sa is ies he unc ional equa ion
Pk( , T) = cTdeg(Pk)Pk( , 1/qk+1T)wi h |c|=qdeg(Pk)(k+1)/2
and Qk( , T)has ecip ocal oo s o weigh ≤k. Fu he mo e, w i ing (x) = Pd
i=0 cixi, i we assume Fqcon ains
he 2(d−1)- h oo s o −dcd hen an explici desc ip ion o Qk( , T )may be gi en; see Co olla y 4.3.
Las ly, we wish o desc ibe he p-adic beha io o he ecip ocal oo s o Mk( , T). Mo i a ion o such a s udy
comes om Wan’s ecip oci y heo em [28] o he Gou ˆea-Mazu conjec u e [14] on he slopes o modula o ms;
see [3] o he connec ion be ween symme ic powe s o he Legend e c ys al wi h Hecke polynomials. Now, while
we s udy in Sec ion 6 he q-adic New on polygon o he L- unc ion o gene al and k, ou mos p ecise esul s
occu in he cubic case (x) = x3:
Theo em 1.3. Assume p≥7,kis odd, and k < p. W i e
Mk(x3, T) = 1 + c1T+· · · +c T .
Then
o dq(cm)≥1
3(m2+m+km) o m= 0,1,2, . . . , . (2)
Fu he mo e, as a consequence o he unc ional equa ion, he endpoin s o he q-adic New on polygon o Mk(x3, T)
coincide wi h he lowe bound (2). I p= 5, hen he nume a o o Mk(x3, T )sa is ies (2).
We a e hope ul ha he es ic ions kodd and k < p may be emo ed om he heo em (see [15] o de ails).
The e is also eason o belie e ha he lowe bound (2) may be op imal in he sense ha he q-adic New on
polygon will in ac equal his lowe bound unde ce ain condi ions on pand k. As suppo ing e idence we no e
wo ac s. Fi s , as men ioned in he abo e heo em, he endpoin s o he q-adic New on polygon o Mk(x3, T)
2
and he lowe bound coincide. Secondly, he lowe bound has he ollowing symme ic p ope y. Conside he
poin s Pm∈R2de ined by he lowe bound:
Pm:= (m, 1
3(m2+m+km)).
The slope o he line segmen joining Pmand Pm+1 is gi en by sm:= 1
3(2m+ 2 + k). I we se m0:= k−1
2−m,
hen we ha e he symme y
(k+ 1) −sm=sm0.
In o he wo ds, o e e y slope sm he e is a co esponding slope sm0. This is p ecisely a consequence o he
unc ional equa ion o Mk(x3, T). Tha is, i αis a ecip ocal oo o Mk(x3, T) o slope s, hen pk+1/α is
ano he ecip ocal oo whose slope is (k+ 1) −s.
Acknowledgmen s. We would like o hank Nicholas Ka z and S e en Spe be o hei e y help ul commen s.
2 Cohomological in e p e a ion o Mk( , T )
In his sec ion we will s udy he gene alized Ai y amily o exponen ial sums om he poin o iew o `-adic
cohomology. We will do so by s udying he shea Ai ha ep esen s his amily on he a ine line A1o e he
gi en ini e ield Fq. We begin by obse ing ha he map Fq→Cgi en by 7→ Px∈Fqψ( (x) + x) is he
Fou ie ans o m wi h espec o ψ, in he classical sense, o he map 7→ ψ( ( )). This will ansla e, in he
cohomological sense, o he ac ha Ai is he Fou ie ans o m, in he shea - heo e ical sense, o he Q`-shea
ha ep esen s he la e map, which is jus he pull-back o he A in-Sch eie shea associa ed o ψ ia he map
gi en by . Le us be mo e p ecise.
The polynomial na u ally de ines a mo phism, also deno ed by :A1
Fq→A1
Fq. Le Lψbe he A in-Sch eie
shea on A1
Fqassocia ed o ψ(c . [7, 1.7]). Fo e e y ini e ex ension Fqmo Fq, e e y ∈A1(Fqm) = Fqmand
e e y geome ic poin ¯
o e , we ha e T ace(F ob |Lψ,¯
) = ψ(T aceFqm/Fq( )), whe e F ob deno es a geome ic
F obenius elemen a . Conside he pullback Lψ( ):= ?Lψ.
By [18, Theo em 17], o d≥2 he Fou ie ans o m wi h espec o ψo Lψ( )(which, in p inciple, is an
elemen o he de i ed ca ego y Db
c(A1,Q`)) is in ac a (shi ed) lisse shea on A1, o ank d−1 and wi h d/(d−1)
as i s single slope a in ini y. I s Swan conduc o is he e o e d. Le us deno e his shea by Ai = R1π !Lψ( (x)+ x),
whe e π :A2→A1is he p ojec ion (x, )7→ . Fo e e y ini e ex ension Fqmo Fq, e e y ∈Fqmand e e y
geome ic poin ¯
o e we ha e, deno ing ψm=ψ◦T aceFqm/Fq:
T ace(F ob |(Ai )¯
) = −X
x∈Fqm
ψm( (x) + x).
The cha ac e is ic polynomial o he ac ion o a geome ic F obenius elemen F ob a on he s alk o Ai a a
geome ic poin o e has he o m
L(Ai , , T) = (1 −π1( )T)· · · (1 −πd−1( )T)
whe e πi( ) is a Weil algeb aic numbe o weigh 1 (i.e. all i s complex conjuga es ha e absolu e alue q1/2) and
Px∈Fqmψm( (x) + x) = −Piπi( )m o all m≥1. I s k- h “symme ic powe ” is gi en by
L(k; Ai , , T) := Y
a1+···+ad−1=k
(1 −π1( )a1· · · πd−1( )ad−1T).
These a e he local ac o s o he L- unc ion o he k- h symme ic powe o Ai , which is gi en by he in ini e
p oduc
Mk( , T) := Y
∈|A1|
L(k; Ai , , Tdeg( ))−1
The Le sche z ace o mula demons a es ha he ze os and poles o Mk( , T) may be desc ibed in e ms o
cohomology:
Mk( , T) =
2
Y
i=0
de (1 −F ob T|Hi
c(A1
Fq,SymkAi ))(−1)i+1 .
3
Since SymkAi is a lisse shea on he a ine line, we ha e H0
c(A1
Fq,SymkAi ) = 0, and he p e ious o mula
simpli ies o
Mk( , T) =
de (1 −F ob T|H1
c(A1
Fq,SymkAi ))
de (1 −F ob T|H2
c(A1
Fq,SymkAi )).
On he o he hand, H2
c(A1
Fq,SymkAi ) is jus he space o co-in a ian s o he shea SymkAi , ega ded as
a ep esen a ion o he undamen al g oup π1(A1
Fq), which is he k- h symme ic powe o Ai ega ded as a
ep esen a ion o he same g oup. This is he same as he space o co-in a ian s o i s monod omy g oup,
which is de ined o be he Za iski closu e o i s image in he g oup o au omo phisms o he gene ic s alk o
Ai , isomo phic o GL(d−1) := GL(d−1,Q`). By [18, Theo em 19], o p > 2d−1 he geome ic monod omy
g oup o Ai is ei he SL(d−1) o de en, o Sp(d−1) o dodd i cd−1= 0 and µp·SL(d−1) o de en o
µp·Sp(d−1) o dodd i cd−16= 0 (whe e (x) = Pd
i=0 cixi). In ei he case, i s k- h symme ic powe is s ill an
i educible ep esen a ion o ank d+k−2
d−2o he monod omy g oup (because i is an i educible ep esen a ion
o i s subg oup SL(d−1) o Sp(d−1)), and in pa icula he space o co-in a ian s anishes. Mo e gene ally, i
was p o en by O. ˇ
Such ([26, P oposi ion 1.6]) ha , o p > 2, ei he Ai has ini e monod omy o i s monod omy
g oup con ains SL(d−1) o Sp(d−1). In o de o ule ou he ini e monod omy case o p≤2d−1 one may
use o ins ance [20, P oposi ion 8.14.3], which implies ha Ai has ini e monod omy i and only i o e e y
elemen ∈Fq he New on polygon o he L- unc ion associa ed o he exponen ial sum Pψ( (x) + x) has a
single slope.
Consequen ly, we ha e he ollowing:
Theo em 2.1. I Ai does no ha e ini e monod omy (e.g. i p > 2d−1), he L- unc ion o he k- h symme ic
powe o Ai is a polynomial:
Mk( , T) = de (1 −F ob T|H1
c(A1
Fq,SymkAi ))
While i is emp ing o belie e ha Mk( , T) is always a polynomial, his is no ue, as men ioned in he
in oduc ion. In ac , he monod omy g oup can be ini e in ce ain cases; o ins ance when p= 5 and (x) = x3,
as p o en in [21]. In such cases, H2
c(A1
Fq,SymkAi ) will be non- i ial o in ini ely many alues o k, and
consequen ly Mk( , T) will ha e a denomina o .
Rema k 2.2.A i hme ic di icul ies o en a ise when he cha ac e is ic pis small compa ed o d, as demons a ed
abo e by he link be ween he ini eness o he monod omy g oup when p≤2d−1 and he New on polygons o
he ib es o he amily. By he unc ional equa ion, i we deno e by NP1( ) he slope o he i s line segmen
o he New on polygon o he ib e hen NP1( )≤1/2 wi h equali y i and only i he New on polygon is a
single line segmen . I p≡1 modulo d, and in pa icula when p=d+ 1, hen by [25, Theo em 3.11] he New on
polygon o e e y ib e equals he q-adic New on polygon o he polynomial Qd−1
i=1 (1 −qi/dT). Thus, NP1( ) = 1/d
and so he monod omy g oup is in ini e when p=d+ 1 >3.
Le [ (x)]xNdeno e he coe icien o xNin (x). Suppose d
2+ 1 < p ≤2d−1 and has coe icien s o e Fp.
By [24, Theo em 2], i [( (x) + x)dp−1
de]xp−16≡ 0 modulo p o some 0 ≤ ≤p−1, hen NP1( )≤p−1
d/(p−1)
o hose . By he assump ion on dand p, no ice ha p−1
d/(p−1) equals ei he 1/(p−1) o 2/(p−1). Hence,
he mond omy g oup is in ini e when such a exis s and p≥7. Thei a gumen may be ex ended as ollows.
Le d>p−1. Fo a polynomial h(x), de ine (h(x))s:= h(x)(h(x)−1) · · · (h(x)−s+ 1). De ine he linea
ope a o U:Fp[x]→Fpby linea ly ex ending he map which sends monomials xn o 0 i (p−1) -nand 1
o he wise. Le cs:= U(( (x) + x)s)∈Fp. Suppose c1≡ · · · ≡ ck−1≡0 modulo pand ck6≡ 0 mod p o some
, hen NP1( )≤k
p−1. Hence, i his happens o some k < (p−1)/2 hen he mond omy g oup is in ini e. Fo
example, o d>p−1 and (x) = xd+xp−1 hen c1= 1 and hence he monod omy g oup is in ini e o p≥5.
Las ly, we men ion he case when d= 4, p= 7 and ∈Fq[x] is no o he o m (x+a)4+bx +c. Then by
[17, Theo em 4.6] he monod omy o Ai is in ini e.
3 Compu a ion o he deg ee o he L- unc ion
We will now s udy he deg ee o Mk( , T) when p > d. F om he o mula abo e we ha e
deg(Mk( , T)) = dim(H1
c(A1
Fq,SymkAi )) −dim(H2
c(A1
Fq,SymkAi )) = −χc(A1
Fq,SymkAi ),
4
whe e χcdeno es he Eule cha ac e is ic wi h compac suppo s. Using he G o hendieck-N´e on-Ogg-Sha a e ic
o mula, we ha e hen
deg(Mk( , T)) = Swan∞(SymkAi )− ank(SymkAi )
= Swan∞(SymkAi )−k+d−2
d−2.(3)
In o de o compu e he Swan conduc o o SymkAi we ha e o s udy he shea Ai as a ep esen a ion o he
ine ia g oup I∞o A1
Fqa in ini y. Since Lψ( )is lisse on A1, as a ep esen a ion o he decomposi ion g oup a
in ini y we ha e Ai ∼
=F∞,∞(Lψ( )), whe e F∞,∞is he local Fou ie ans o m as de ined in [22].
Recen ly, Fu [11] and, independen ly, Abbes and Sai o [1] ha e gi en an explici desc ip ion o he di e en
local Fou ie ans o ms o a wide class o `-adic shea es. We will mainly be using he desc ip ion gi en in
[1], which wo ks o e an a bi a y (no necessa ily algeb aically closed) pe ec base ield, and he e o e gi es an
explici o mula o Ai as a ep esen a ion o he decomposi ion g oup D∞.
I S(∞)is he henseliza ion o he local ing o P1
Fqa in ini y wi h uni o mize 1/ , he iple (Lψ( ( )), , − 0( ))
is a Legend e iple in he sense o [1, De ini ion 2.16]. The e o e by [1, Theo em 3.9] we conclude ha , as a
ep esen a ion o D∞, Ai is isomo phic o
(− 0)?(Lψ( ( )) ⊗ Lψ(− 0( )) ⊗ Lρ(1
2 00( )) ⊗ Q) = (− 0)?(Lψ( ( )− 0( )) ⊗ Lρ(1
2 00( )) ⊗ Q)
whe e ρis he unique cha ac e I∞→Q?
`o o de 2, Lρ he co esponding Kumme shea and Qis he pull-back
o he cha ac e Gal(Fq/Fq)→Q`mapping he geome ic F obenius o he quad a ic Gauss sum g(ψ, ρ) :=
−P ∈F?
qψ( )ρ( ).
W i e ( ) = Pd
i=0 ci i. Fo simplici y, om now on we will assume ha Fqcon ains he 2(d−1)- h oo s o
−dcd(which can always be achie ed by a ini e ex ension o he base ield). Following [11, P oposi ion 3.1] we
can ind an in e ible powe se ies Pi≥0 i −i∈Fq[[ −1]] wi h d−1
0=−dcdsuch ha u( ) := Pi≥0 i −iis a
solu ion o 0( ) + u( )d−1= 0 ( he o he solu ions being ζu( ) o e e y (d−1)- h oo o uni y ζ). The map
φ: 1/ 7→ 1/u( ) de ines an au omo phism S(∞)→S(∞), and by cons uc ion − 0= [d−1] ◦φ, whe e [d−1] is
he (d−1)- h powe map. So Ai is isomo phic o
[d−1]?φ?(Lψ( ( )− 0( )) ⊗ Lρ(1
2 00( )) ⊗ Q) = [d−1]?(φ−1)?(Lψ( ( )− 0( )) ⊗ Lρ(1
2 00( )) ⊗ Q)
= [d−1]?(Lψ( ( ( ))+ ( ) d−1)⊗ Lρ(1
2 00( ( ))) ⊗ Q)
= [d−1]?(Lψ( ( ( ))+ ( ) d−1)⊗ Lρ(1
2 00( ( ))))⊗ Q
since [d−1]?Q=Q, whe e ( ) := φ−1( ) = Pi≥0si −i.
Le g( ) be he polynomial o deg ee dob ained om ( ( )) + ( ) d−1by emo ing he e ms wi h neg-
a i e powe s o . I is impo an o no ice ha he coe icien s o ga e polynomials in he coe icien s o .
Mo e p ecisely, i we w i e g( ) = Pbi i, he coe icien biis a polynomial in he coe icien s ai, ai+1, . . . , ad
o . Since Lψ(h( )) is i ial as a ep esen a ion o D∞ o any h( )∈ −1Fq[[ −1]], we ha e an isomo phism
Lψ( ( ( ))+ ( ) d−1)∼
=Lψ(g( )) as ep esen a ions o D∞.
On he o he hand, om 0( ( )) + d−1= 0 we ge 00( ( )) 0( ) + (d−1) d−2= 0, so Lρ(1
2 00( ( ))) =
Lρ(−d−1
2 0( ) d−2). Since 0( ) = Pi≥0(1−i)si −i=s0(1+Pi≥2(1−i)si
s0 −i) and 1+Pi≥2(1−i)si
s0 −iis a squa e
in Fq[[ −1]], we ha e Lρ(−d−1
2 0( ) d−2)=Lρ(−d−1
2s0 d−2)=Lρ(d(d−1)
2cd(s0 )d−2)(since sd−1
0=−1/dcd). So we inally
ge
Ai ∼
=[d−1]?(Lψ(g( )) ⊗ Lρd(s0 ))⊗ Lρ(d(d−1)cd/2) ⊗ Q.(4)
We can now easily compu e he Swan conduc o a in ini y o i s symme ic powe s. By [19, 1.13.1],
Swan∞SymkAi =1
d−1Swan∞[d−1]?SymkAi =1
d−1Swan∞Symk[d−1]?Ai
Lemma 3.1. Le ζbe a p imi i e (d−1)- h oo o uni y i Fq,Id−1
∞ he unique closed subg oup o I∞o index
d−1. As a ep esen a ion o Id−1
∞, he es ic ion [d−1]?Ai o Ai is isomo phic o he di ec sum
d−2
M
i=0
Lψ(g(ζi )) ⊗ Lρd(s0ζi )∼
=
d−2
M
i=0
Lψ(g(ζi )) ⊗ Lρd( )
.
5

P oo . Since (ζi)?Lψ(g)=Lψ(g(ζi )), (ζi)?Lρd(s0 )=Lρd(s0ζi )and [d−1] ◦ζi= [d−1] o e e y i, we ha e
[d−1]?(Lψ(g(ζi )) ⊗Lρd(s0ζi )) = [d−1]?(Lψ(g( )) ⊗Lρd(s0 )), and he e o e by F obenius ecip oci y HomId−1
∞([d−
1]?Ai ,Lψ(g(ζi )) ⊗ Lρd(s0ζi )) = HomI∞(Ai ,[d−1]?(Lψ(g(ζi )) ⊗ Lρd(s0ζi ))) = HomI∞(Ai ,Ai )∼
=Q`since
he la e is an i educible ep esen a ion o I∞. So o e e y i,Lψ(g(ζi )) ⊗ Lρd(s0ζi )is a sub ep esen a ion o
[d−1]?Ai .
Now Lψ(g(ζi )) ⊗Lρd(s0ζi )and Lψ(g(ζj )) ⊗ Lρd(s0ζj )a e isomo phic i and only i Lψ(g(ζi )) and Lψ(g(ζj )) a e,
i and only i g(ζi )−g(ζj ) = hp−h o some h∈Fq[ ]. Since p>d, his can only happen i g(ζi ) = g(ζj ).
Compa ing he highes deg ee coe icien s we conclude ha ζiand ζjmus be equal. The e o e he di ec sum o
he Lψ(g(ζi )) ⊗ Lρd(s0ζi ) o i= 0, . . . , d −2 injec s in o [d−1]?Ai and we conclude ha i mus be isomo phic
o i , since hey ha e he same ank.
Consequen ly, we ha e an isomo phism o Q`[I∞]-modules
Symk[d−1]?Ai ∼
=M
a0+a1+···+ad−2=k
Lψ(Pd−2
i=0 aig(ζi )) ⊗ Lρdk( ).
Fo e e y ini e subse I⊂Zand e e y in ege k≥0 de ine
Sd−1(k, I) := {(a0, . . . , ad−2)∈Zd−1
≥0|a0+a1+· · · +ad−2=k, a0+a1ζi+· · · +ad−2ζi(d−2) = 0 o e e y i∈I}
I is clea om he de ini ion ha Sd−1(k, I) = Sd−1(k, I0) i φ(I) = φ(I0), whe e φ:Z→Z/(d−1)Zis
educ ion modulo d−1. Also, Sd−1(k, I) = ∅i pdoes no di ide kand I∩(d−1)Z6=∅. The numbe o elemen s
in Sd−1(k, I) can be con enien ly exp essed in e ms o a gene a ing unc ion:
Lemma 3.2. Le Fd−1(I;T) := P∞
k=0 #Sd−1(k, I)Tk. Then
Fd−1(I;T) = 1
q#IX
γ∈(Fq)I
d−2
Y
j=0
(1 −ψ(X
i∈I
γiζji)T)−1
whe e ψis any non- i ial addi i e cha ac e o Fq.
P oo . F om he de ini ion,
Fd−1(I;T) = X
(a0,...,ad−2)∈Zd−1
≥0Y
i∈I
δ(a0+a1ζi+· · · +ad−2ζi(d−2))Ta0+a1+···+ad−2
whe e δ(a) = 1 i a= 0, 0 o he wise. Equi alen ly, δ(a) = 1
qPγ∈Fqψ(γa). So we ge
Fd−1(I;T) = X
(a0,...,ad−2)∈Zd−1
≥0Y
i∈I
1
qX
γi∈Fq
ψ(γi(a0+a1ζi+· · · +ad−2ζi(d−2)))Ta0+a1+···+ad−2
=X
(a0,...,ad−2)∈Zd−1
≥0X
γ∈(Fq)I
1
q#I Y
i∈I
ψ(γia0)!Ta0 Y
i∈I
ψ(γia1ζi)!Ta1· · · Y
i∈I
ψ(γiad−2ζ(d−2)i)!Tad−2
=1
q#IX
γ∈(Fq)IX
(a0,...,ad−2)∈Zd−1
≥0
ψ(X
i∈I
γi)a0Ta0ψ(X
i∈I
γiζi)a1Ta1· · · ψ(X
i∈I
γiζ(d−2)i)ad−2Tad−2
=1
q#IX
γ∈(Fq)I
X
a0∈Z≥0
ψ(X
i∈I
γi)a0Ta0

X
a1∈Z≥0
ψ(X
i∈I
γiζi)a1Ta1
· · · 
X
ad−2∈Z≥0
ψ(X
i∈I
γiζ(d−2)i)ad−2Tad−2

=1
q#IX
γ∈(Fq)I
d−2
Y
j=0
(1 −ψ(X
i∈I
γiζji)T)−1.
W i e g( ) = Pd
j=0 bj j, and le J={1≤j≤d|bj6= 0}and J≥j:= J∩ {j, j + 1, . . . , d} o e e y j∈
{1, . . . , d, d + 1}. We ha e
Swan∞Symk[d−1]?Ai =X
a0+a1+···+ad−2=k
Swan∞Lψ(Pd−2
i=0 aig(ζi )) ⊗ Lρdk( )
=X
a0+a1+···+ad−2=k
deg(
d−2
X
i=0
aig(ζi ))
6
and d−2
X
i=0
aig(ζi ) =
d−2
X
i=0
ai
d
X
j=0
bjζij j=
d
X
j=0
(bj
d−2
X
i=0
ζij) j
so i s deg ee is he g ea es jsuch ha bjPd−2
i=0 ζij 6= 0. The e o e we ge
(d−1)Swan∞SymkAi = Swan∞Symk[d−1]?Ai
=X
j∈J
j·(#Sd−1(k, J≥j+1)−#Sd−1(k, J≥j))
=dk+d−2
d−2−X
j∈J
h(j)·#Sd−1(k, J≥j)
whe e h(j) := j−sup(J−J≥j) is he “gap” be ween he j e m and he nex lowe deg ee e m in g( ). Taking
he co esponding gene a ing unc ion we ge he o mula
Co olla y 3.3. Le G( ;T) := P∞
k=0(Swan∞SymkAi )Tk, hen
G( ;T) = d
(d−1)(1 −T)d−1−1
d−1X
j∈J
h(j)·Fd−1(J≥j;T)
Using he p e ious o mula o he deg ee, we deduce
Co olla y 3.4. The deg ee o Mk( ;T)is he k- h coe icien o he powe se ies expansion o
1
(d−1)(1 −T)d−1−1
d−1X
j∈J
h(j)·Fd−1(J≥j;T).
Co olla y 3.5. Fo e e y J⊂ {1, . . . , d −1}, le Pd(J)be he subspace o he a ine space Pdo polynomials o
deg ee do e ksuch ha bj= 0 i and only i j∈J. The se s {Pd(J)|J⊆ {1, . . . , d −1}} de ine a s a i ica ion
o Pdsuch ha he deg ee o Mk( ;T)is cons an in each s a um.
4 The i ial ac o
Suppose p > d and he monod omy o Ai is no ini e. We will now s udy he weigh s o he ( ecip ocal) oo s
o he polynomial Mk( , T). Le us i s conside he easie case whe e dis e en, and he e o e Ai is isomo phic
o [d−1]?Lψ(g( )) ⊗ Lρ(d(d−1)cd/2) ⊗ Q as a ep esen a ion o D∞. Le Dd−1
∞= Gal(Fq((1/ ))/Fq((1/ 1/(d−1)))),
deno e by α:Dd−1
∞→Q?
` he cha ac e co esponding o he shea Lψ(g), and le b∈I∞be a gene a o o
he cyclic g oup D∞/Dd−1
∞∼
=I∞/Id−1
∞. By he explici desc ip ion o induced ep esen a ions, he e is a basis
{ 0, . . . , d−2}o he unde lying ec o space Vsuch ha a· 0=α(a) 0 o e e y a∈Id−1
∞and b· i= i+1 o
i= 0, . . . , d −3. Then b· d−2=bd−1· 0=α(bd−1) 0. Replacing bby a−1b, whe e a∈Id−1
∞is an elemen such
ha α(a)d−1=α(bd−1) (which is always possible since he alues o αa e he p- h oo s o uni y and d−1 is
p ime o psince p > d) we may assume wi hou loss o gene ali y ha α(bd−1) = 1.
Fu he mo e, o any a∈Id−1
∞we ha e a· i= (abi)· 0= (bib−iabi)· 0=bi·α(b−iabi) 0=α(b−iabi) i. So
he es ic ion o Ai o Dd−1
∞is he di ec sum o he cha ac e s a7→ αi(a) := α(b−iabi). Bu we al eady know
ha i is he di ec sum o he cha ac e s associa ed o he shea es Lψ(g(ζi )) ⊗ Lρ(d(d−1)cd/2) ⊗ Q, so hese wo
se s o cha ac e s a e iden ical. Replacing bby a sui able powe o i sel we may assume ha αiis he cha ac e
associa ed o Lψ(g(ζi )) ⊗ Lρ(d(d−1)cd/2) ⊗ Q. In pa icula , Qd−2
i=0 αai
iis geome icaly i ial ( ha is, i ial on
Id−1
∞) i and only i Paig(ζi ) is a cons an in Fq[ ], ha is, i and only i Paiζij = 0 o e e y j∈J.
We u n now o he case dodd. Le χbe a mul iplica i e cha ac e o Fqo o de 2(d−1) (which exis s,
since we a e assuming ha Fqcon ains he 2(d−1)- h oo s o uni y). Then by he p ojec ion o mula Ai is
isomo phic o [d−1]?(Lψ(g( )) ⊗ Lρ(s0 ))⊗ Lρ(d(d−1)cd/2) ⊗ Q ∼
=([d−1]?Lψ(g( )))⊗ Lχ(s0 )⊗ Lρ(d(d−1)cd/2) ⊗ Q.
Le αi:Dd−1
∞→Q?
`( espec i ely β:D∞→Q?
`) be he cha ac e co esponding o he shea Lψ(g(ζi )) ( esp.
Lχ(s0 )). P oceeding as in he de en case, we ind a gene a o b∈I∞o D∞/Dd−1
∞and a basis { 0, . . . , d−2}o
Vsuch ha a· i=αi(a)β(a) i o a∈Dd−1
∞and b· i=β(b) i+1 o i= 0, . . . , d −3, b· d−2=β(b) 0. In his
case, Qd−2
i=0 αai
iβaiis i ial on Id−1
∞i and only i Paig(ζi ) is a cons an in Fq[ ] and Paiis e en (since αihas
o de pand β es ic ed o Id−1
∞has o de 2).
7
We can now compu e he dimension o he in a ian subspace o he ac ion o I∞on SymkAi , in e y much
he same way i is done o he Kloos e man shea in [12, Lemma 2.1]. I s unde lying ec o space is SymkV. An
elemen wis gi en by a linea combina ion
w=X
a0+···+ad−2=k
ca0···ad−2 a0
0· · · ad−2
d−2.
In he de en case we ha e
a·X
a0+···+ad−2=k
ca0···ad−2 a0
0· · · ad−2
d−2=X
a0+···+ad−2=k
ca0···ad−2(αa0
0· · · αad−2
d−2)(a) a0
0· · · ad−2
d−2
o a∈Id−1
∞and
b·X
a0+···+ad−2=k
ca0···ad−2 a0
0· · · ad−2
d−2=X
a0+···+ad−2=k
ca0···ad−2 a0
1 a1
2· · · ad−2
0.
So wis ixed by I∞i and only i he cha ac e αa0
0· · · αad−2
d−2is i ial whene e ca0···ad−26= 0 and ca0···ad−2=
cad−2a0···ad−3 o all a0, . . . , ad−2. A basis o he in a ian subspace is hus gi en by all dis inc sums o he o m
(se ing d−1+l:= l o all l≥0):
d−2
X
j=0
a0
j a1
j+1 · · · ad−2
j+d−2
o all a0, . . . , ad−2such ha αa0
0· · · αad−2
d−2is i ial, ha is, such ha Paiζij = 0 in Fq o e e y j∈J.
In he dodd case we ge
g·X
a0+···+ad−2=k
ca0···ad−2 a0
0· · · ad−2
d−2=X
a0+···+ad−2=k
ca0···ad−2(αa0
0· · · αad−2
d−2)(g)βk(g) a0
0· · · ad−2
d−2
o g∈Id−1
∞and
h·X
a0+···+ad−2=k
ca0···ad−2 a0
0· · · ad−2
d−2=X
a0+···+ad−2=k
ca0···ad−2β(h)k a0
1 a1
2· · · ad−2
0.
So wis ixed by I∞i and only i he cha ac e αa0
0· · · αad−2
d−2βko Id−1
∞is i ial whene e ca0···ad−26= 0 and
ca0···ad−2=cad−2a0···ad−3β(h)k o all a0, . . . , ad−2. Since all αi’s ha e o de pand he es ic ion o β o Id−1
∞has
o de 2, αa0
0· · · αad−2
d−2βkis i ial i and only i bo h αa0
0· · · αad−2
d−2and βka e i ial as cha ac e s o Id−1
∞, ha is,
i and only i Paiζij = 0 in Fq o e e y j∈Jand kis e en. In pa icula , he e a e no non-ze o in a ian s o
I∞i kis odd. I kis e en, a gene a ing se o he in a ian subspace is gi en by all dis inc sums o he o m
d−2
X
j=0
β(h)jk a0
j a1
j+1 · · · ad−2
j+d−2
o all a0, . . . , ad−2such ha Paiζij = 0 in Fq o e e y j∈J. Le be he size o he o bi o (a0, . . . , ad−2)
unde he ac ion o Z/(d−1)Zby cyclic pe mu a ions. I 6=d−1, we can w i e
d−2
X
j=0
β(h)jk a0
j a1
j+1 · · · ad−2
j+d−2=
−1
X
j=0
β(h)jk(1 + β(h) k +· · · +β(h)(d−1
−1) k) a0
j a1
j+1 · · · ad−2
j+d−2.
No ice ha kmus be a mul iple o d−1
, since k=Pd−2
i=0 ai=d−1
P −1
i=0 ai. I k
d−1is odd we ha e
1 + β(h) k +· · · +β(h)(d−1
−1) k =1−β(h)(d−1)k
1−β(h) k = 0,
so he abo e sum anishes. On he o he hand, i k
d−1is e en i is clea ha he elemen
d−2
X
j=0
β(h)jk a0
j a1
j+1 · · · ad−2
j+d−2=d−1
−1
X
j=0
β(h)jk a0
j a1
j+1 · · · ad−2
j+d−2
is non-ze o, and o di e en o bi s co espond di e en elemen s. To summa ize, we ha e
8
P oposi ion 4.1. Le Td−1(k, J)be he se o o bi s o he ac ion o Z/(d−1)Zon he se Sd−1(k, J)by cyclic
pe mu a ions, and le Ud−1(k, J)be he subse o o bi s such ha k
d−1is e en, whe e is hei ca dinali y. I dis
e en, he in a ian subspace o he ep esen a ion SymkAi o I∞has dimension #Td−1(k, J). I dis odd and k
is e en, i has dimension #Ud−1(k, J). I dand ka e odd, he ep esen a ion has no non-ze o in a ian s.
The sequences #Td−1(k, J) and #Ud−1(k, J) can also be desc ibed by means o gene a ing unc ions. By
Bu nside’s lemma, he dimension o he in a ian subspace o de en is gi en by
#Td−1(k, J) = 1
d−1
d−1
X
=1
#{(a0, a1, . . . , ad−2)|ai=ai+ mod d−1}=1
d−1X
|d−1
φ(d−1
)#S (k
d−1, J)
whe e S (k, J) = ∅i kis no an in ege and φis Eule ’s o ien unc ion. So he gene a ing unc ion o he
sequence {#Td−1(k, J)|k≥0}is
Gd−1(J;T) : =
∞
X
k=0
#Td−1(k, J)Tk
=
∞
X
k=0
1
d−1TkX
|d−1
φ(d−1
)#S (k
d−1, J)
=1
d−1X
|d−1
φ(d−1
)X
d−1
|k
#S (k
d−1, J)Tk
=1
d−1X
|d−1
φ(d−1
)
∞
X
s=0
#S (s, J)Td−1
s
=1
d−1X
|d−1
φ(d−1
)F (J;Td−1
)
Nex , suppose ha dis odd, and le (a0, . . . , ad−2)∈Sd−1(k, J). Le be he numbe o elemen s in i s o bi .
Then P −1
i=0 ai=k
d−1. We wan o coun he numbe o o bi s such ha his alue is e en. Since k=k
d−1·d−1
,
i he la ges powe o 2 ha di ides d−1 is smalle han he la ges powe o 2 di iding k,k
d−1mus always be
e en. Suppose ha he la ges powe o 2 ha di ides k, 2α(k), di ides d−1. Then k
d−1is odd i and only i 2α(k)
di ides d−1
, i and only i di ides d−1
2α(k). The e o e #Ud−1(k, J)=#Td−1(k, J) i 2α(k)does no di ide d−1
and #Td−1(k, J)−#Td−1
2α(k)(k
2α(k), J) i i does. The gene a ing unc ion is hen
∞
X
k=0
#Ud−1(k, J)Tk=
∞
X
k=0
#Td−1(k, J)Tk−X
j≥1;2j|d−1X
lodd
#Td−1
2j(l, J)T2jl
=Gd−1(J;T)−X
j≥1;2j|d−1
Hd−1
2j(J;T2j)
whe e
H (J;T) := 1
2(G (J;T)−G (J;−T)).
Le F∈Dd−1
∞⊂D∞be a geome ic F obenius elemen , and w=Pd−2
j=0 a0
j a1
j+1 · · · ad−2
j+d−2( esp. w=
Pd−2
j=0 β(h)jk a0
j a1
j+1 · · · ad−2
j+d−2) a gene a o o he I∞-in a ian subspace o SymkV.Fac s on a0
j a1
j+1 · · · ad−2
j+d−2
ia he cha ac e co esponding o Lψ(Paig(ζj+i )) ⊗L⊗k
ρ(d(d−1)cd/2) ⊗Q⊗k( esp. Lψ(Paig(ζj+i )) ⊗Lρ(Q(s0ζj+i )ai)⊗
L⊗k
ρ(d(d−1)cd/2) ⊗ Q⊗k). Since Paig(ζj+i ) mus be a cons an polynomial, we ha e Lψ(Paig(ζj+i )) ∼
=Lψ(kb0).
Addi ionally, i dis odd and ke en, Lρ(Q(s0 )ai)=Lρ(s0 )kis i ial. We conclude:
P oposi ion 4.2. A F obenius geome ic elemen a in ini y ac s on he I∞-in a ian subspace o SymkAi by
mul iplica ion by ψ(kb0)ρ(d(d−1)cd/2)kg(ψ, ρ)k.
As an immedia e consequence we ge
Co olla y 4.3. The local L- unc ion o SymkAi a in ini y de (1 −F ob T|(SymkAi )I∞)is gi en by (1 −
ψ(kb0)ρ(d(d−1)cd/2)kg(ψ, ρ)kT)#Td−1(k,J)i dis e en, (1 −ψ(kb0)ρ(d(d−1)cd/2)kg(ψ, ρ)kT)#Ud−1(k,J)i dis
odd and kis e en, and 1i dand ka e odd.
9
The main esul in Sec ion 6.4 below will be o educe he hypo hesis (7) o a simila hypo hesis which we
belie e is a ainable. I is expec ed ha his simila hypo hesis will hold unde a he gene al condi ions on he
p ime p, he deg ee d, and he symme ic powe k. Howe e , i is also expec ed ha his simila hypo hesis will
ail jus as o en, ye (7) will s ill hold. The condi ions unde which he weake hypo hesis is alid is cu en ly
unde in es iga ion.
The es o his sec ion is de o ed o he p oo o Theo em 6.1, whose a gumen closely ollows ha o Dwo k’s
[9, §7] and Adolphson-Spe be ’s [2]. The p oo es s on ela ing he New on polygon o ¯
β o ano he ope a o ,
¯
β1, whose New on polygon is much easie o es ima e due o he Dwo k decomposi ion o Ngi en in (7). The
eason is ha Dwo k decomposi ion allows us o wo k on he chain le el, whe e he ope a o β1ac s in an easily
unde s ood way. Once es ima es on β1a e ound on he chain le el Dwo k decomposi ion p o ides es ima es in
homology o ¯
β1.
Fo b0≤band b≤p/(p−1), de ine α1:K(b0, b)→ K(b0/p, b) by
α1: = τ−1◦ψx◦F( , x)
=1
G( p, x)◦τ−1◦ψx◦G( , x).
No ice ha α1◦D( ) = pD( p)◦α1, and so α1induces a map
¯α1( ) : H1, (b0, b)→ H1, p(b0/p, b).
On K(b0, b), since
α( ) = ψa
x◦Fa( , x)
=ψa
x◦Fτa−1( pa−1, xpa−1)· · · Fτ( p, xp)F( , x)
=τ−1◦ψx◦F( pa−1, x)◦ · · · ◦ τ−1◦ψx◦F( p, x)◦τ−1◦ψx◦F( , x)
=α1( pa−1)◦ · · · ◦ α1( p)◦α1( ),
i ollows ha
¯α( ) = ¯α1( pa−1)◦ · · · ◦ ¯α1( p)◦¯α1( ).(8)
We also ha e he p ope y ha
ψ ◦¯α1( p) = ¯α1( )◦ψ .(9)
Consequen ly, wi h β1:= ψ ◦Symk(¯α1( )) : H(k)
1, (b0, b)→ H(k)
1, (b0, b), we ha e he ela ion
βa
1=ψa
◦Symk¯α1( pa−1)◦ · · · ◦ ¯α1( p)◦¯α1( )
=ψa
◦Symk(¯α( ))
=β
whe e we ha e used (9) o he i s equali y and (8) o he second. Since de Qq(π)(1 −¯
βT |H1,k)∈Qp(π), we
ha e ha
de Qq(π)(1 −¯
βT |H1,k)a=No mQq(π)/Qp(π)de Qq(π)(1 −¯
βT |H1,k)
=de Qp(π)(1 −¯
βT |H1,k).
Thus,
de Qq(π)(1 −¯
βTa|H1,k)a=de Qp(π)(1 −¯
βTa|H1,k)
=de Qp(π)(1 −¯
βa
1Ta|H1,k)
=Y
ζa=1
de Qp(π)(1 −ζ¯
β1T|H1,k).(10)
Coun ing mul iplici ies, le mideno e he numbe o ecip ocal oo s o de Qp(π)(1 −¯
β1T|H1,k) which ha e
slope si; no e, we say λhas slope sii o dp(λ) = si. Then, om (10), de Qq(π)(1 −¯
βTa|H1,k)ahas ami
ecip ocal oo s o slope si, and so de Qq(π)(1 −¯
βTa|H1,k) has mi ecip ocal oo s wi h slope si. We conclude
ha de Qq(π)(1 −¯
βT |H1,k) has mi/a ecip ocal oo s o slope asi.
Nex , o he q-adic alua ion o dq(·) := 1
ao dp(·), we will say a oo λhas q-adic slope sii o dq(λ) = si.
Obse e ha he abo e pa ag aph has demons a ed ha de Qq(π)(1 −¯
βT |H1,k) has mi ecip ocal oo s wi h
16

q-adic slope sii and only i de Qp(π)(1 −¯
β1T|H1,k) has ami ecip ocal oo s wi h p-adic slope si. In e ms o
New on polygons, his means he e ices o he q-adic New on polygon o de Qq(π)(1 −¯
βT |H1,k) a e
(0,0) and n
X
i=1
mi,
n
X
i=1
misi!n= 1,2, . . . , dimQq(π)(H1,k)
i and only i he e ices o he p-adic New on polygon o de Qp(π)(1 −¯
β1T|H1,k) a e
(0,0) and n
X
i=1
ami,
n
X
i=1
amisi!n= 1,2, . . . , dimQp(π)(H1,k).
Using his ela ion, any lowe bound o he p-adic New on polygon o he la e may be ans o med o a lowe
bound o he q-adic New on polygon o he o me by di iding he coo dina es o he e ices by a. Le us now
concen a e on a lowe bound o he p-adic New on polygon o de Qp(π)(1 −¯
β1T|H1,k).
De ine K(b0, b)•:= xK(b0, b), and le Vbe he L(b0)-span o he se {x, x2, . . . , xd−1}in K(b0, b)•. De ine
K(b0, b;ρ)•:= K(b0, b)•∩ K(b0, b;ρ) and V(b0, b;ρ) := V ∩ K(b0, b;ρ). By ou hypo hesis on he p ime p, we will
p o e in he ollowing sec ion he Dwo k decomposi ion
K(b0, b; 0)•⊂ V(b0, b; 0) ⊕D( )K(b0, b;e),(11)
whe e e:= b−1
p−1. Consequen ly, K(b0, b)•=V ⊕D( )K(b0, b), so we may iden i y H1wi h V, a ee L(b0)-module
o ank d−1 wi h basis {x, x2, . . . , xd−1}. Consequen ly, H(k)
1is a ee L(b0)-module wi h basis {ei1
1· · · eid−1
d−1:= ei},
whe e i= (i1, . . . , id−1), each ijis a nonnega i e in ege sa is ying i1+· · · +id−1=k, and ej:= xj.
Se b=p
p−1. Fo i= 1, . . . , d −1, xi∈ K(b
p,b
p;−b
pw0(i))•and so F( , x)xi∈ K(b
p,b
p;−b
pw0(i))•. Thus,
α1(xi)∈ K(b
p, b;−b
pw0(i))•. By (11) we may w i e his as
α1(xi) = Ai,1x+· · · +Ai,d−1xd−1mod(D( p)K(b
p, b)) (12)
whe e Ai,j ∈L(b
p;b
p(pw0(j)−w0(i))).
Le S(i1, . . . , id−1) deno e he se o nonnega i e in ege s (l( )
s)1≤s, ≤d−1 ha sa is y he sys em
l(1)
1+· · · +l(1)
d−1=i1
.
.
..
.
.
l(d−1)
1+· · · +l(d−1)
d−1=id−1.
and T(j1, . . . , jd−1) deno e he se o nonnega i e in ege s (l( )
s)1≤s, ≤d−1 ha sa is y he sys em
l(1)
1+· · · +l(d−1)
1=j1
.
.
..
.
.
l(1)
d−1+· · · +l(d−1)
d−1=jd−1.
The k- h symme ic powe o ¯α1ac s on he basis {ei}as ollows:
Symk(¯α1( ))ei1
1· · · eid−1
d−1
= (¯α1( )e1)i1· · · (¯α1( )ed−1)id−1
=

X
j=1
A1,jej

i1
· · · 

X
j=1
Ad−1,jej

id−1
=X
(l( )
s)∈S(i1,...,id−1)
Z>0Al(1)
1
1,1· · · Al(1)
d−1
1,d−1· · · Al(d−1)
1
d−1,1· · · Al(d−1)
d−1
d−1,d−1el(1)
1+···+l(d−1)
1
1· · · el(1)
d−1+···+l(d−1)
d−1
d−1
=X
j:=(j1,...,jd−1)∈Zd−1
≥0
j1+···+jd−1=k
B(i,j)ej1
1· · · ejd−1
d−1,
17
whe e
B(i;j) := X
(l( )
s)∈S(i1,...,id−1)∩T(j1,...,jd−1)
(Z>0)Al(1)
1
1,1· · · Al(1)
d−1
1,d−1· · · Al(d−1)
1
d−1,1· · · Al(d−1)
d−1
d−1,d−1
and “Z>0” is some de e minable nonze o posi i e in ege . I ollows ha B(i;j)∈L(b
p;b
p(pw0(j)−w0(i))), and
so
Symk(¯α1( )) : H(k)
1(b
p,b
p; 0) → H(k)
1(b
p, b; 0).(13)
Recall, M:= H(k)
1, and N:= H(k)
1, . We a e supposing ha he e exis s a ee Zq[π]-submodule Vo Nwi h
basis Γ := { nei|(n;i)∈A}such ha
N(b, b; 0) ⊂ V(b, b; 0) ⊕∂M(b, b;) (14)
o some ∈R. Now, Γ ep esen s a basis o H1,k o e Qq(π), bu we need o unde s and he F edholm de e minan
o ¯
β1on H1,k iewed as a ec o space o e Qp(π). To do his ecall ha Qq(π) is an un ami ied ex ension ield
o Qp(π). We ha e deno ed by Zq[π] he ing o in ege s o Qq(π) wi h uni o mize πand esidue ield Fq, and
Zp[π] he ing o in ege s o Qp(π) wi h uni o mize πand esidue ield Fp. Le {¯η1, . . . , ¯ηa}be a basis o Fqo e
Fp, and le {η1, . . . , ηa}be a li ing o his basis o an in eg al basis o Qq(π) o e Qp(π).
Lemma 6.3 (Dwo k).The basis {ηi}has he p ope y o p-adic di ec ness; ha is, o any g∈Qq(π), w i ing
g=h1η1+· · · +haηawi h hi∈Qp(π), hen
o dp(g) = min
i=1,...,a{o dp(hi)}.
P oo . Wi hou loss o gene ali y, we may assume ha o dp(g) = 0. Se −c:= (p−1) min{o dp(hi)} ∈ Z.
Suppose c > 0. Fo any ξ∈Zq[π], deno e by ¯
ξi s image in he esidue ield Fq. Using his no a ion, we see ha
0 = (πcg) = (πch1)¯η1+· · · + (πcha)¯ηamod(π).
Since {¯ηi}is a basis o Fq, we mus ha e πchi= 0 in Fq o e e y i. Hence, πchi∈πZq[π], and so hi∈π1−cZq[π]
o e e y i. Thus, o each iwe ha e
o dp(hi)≥1−c
p−1=1
p−1+ min
j=1,...,a{o dp(hj)}.
Howe e , since his is no possible we mus ha e cnonnega i e. Thus, −c≥0 which means min{o dp(hi)} ≥ 0 =
o dp(g). Since we easily ha e o dp(g)≥min{o dp(hi)}, we mus ha e equali y, p o ing he lemma.
Since nei∈ M(b
p,b
p;−b
pW(n, i)), we ha e by (13) ha Symk(α1( ))( nei)∈ N(b
p, b;−b
pW(n, i)) and so
β1( nei)∈ N(b, b;−b
pW(n, i)). By Dwo k decomposi ion (14), his means
β1( nei) = X
(m,j)∈Γ
C(n, i;m, j) mejmod(∂M)
wi h
o dp(C(n, i;m, j)) ≥b
p(pW(m, j)−W(n, i)).
F om he lemma abo e, i B∈Zq[π] sa is ies o dp(B)≥ρ, hen w i ing B=B1η1+· · · +Baηa he coe icien s
sa is y o dp(Bi)≥ρ. Thus, we may w i e
C(n, i;m, j) =
a
X
=1
C(n, i;m, j) η
wi h C(n, i;m, j) ∈Zp[π] and o dp(C(n, i;m, j) )≥b
p(pW(m, j)−W(n, i)). Now, a basis o H1,k o e he ield
Qp(π) is gi en by Γ0:= {ηj nei|j= 1, . . . , a, (n, i)∈A}. Thus, o ηj nei∈Γ0, we ha e
β1(ηj nei) = τ−1(ηj)X
(m,j)∈Γ
C(n, i;m, j) mejmod(∂M)
=τ−1(ηj)X
(m,j)∈Γ
a
X
=1
C(n, i;m, j) η ej(15)
18
W i ing
τ−1(ηj)η =
a
X
s=1
bs,j, ηswi h bs,j, ∈Zp[π],
hen (15) becomes
¯
β1(ηj nei) = X
(m,j)∈Γ
a
X
s=1
D(j, n, i;s, m, j)ηs mej
whe e
D(j, n, i;s, m, j) :=
a
X
=1
C(n, i;m, j) bs,j, .
I ollows ha
o dp(D(j, n, i;s, m, j)) ≥b
p(pW(m, j)−W(n, i)).
W i ing de Qp(π)(1 −¯
β1T|H1,k) = P∞
m=0 cmTm hen
cm= (−1)mXX
σ∈Sm
sgn(σ)
m
Y
l=1
D(jl, nl,i(l);jσ(l), nσ(l),i(σ(l))),
whe e Smis he pe mu a ion g oup on {1, . . . , m}and he ou e summa ion uns o e all se s consis ing o m
dis inc elemen s o he o m (j, n, i) whe e ηj nei∈Γ0. I ollows ha
o dp(cm)≥min (m
X
l=1
W(nl,i(l)))
whe e he minimum uns o e all se s consis ing o mdis inc elemen s o he o m (j, n, i) whe e ηj nei∈Γ0.
Le N:= #{ nei∈Γ|W(n, i) = N/d}. Then he e a e a Nnumbe o elemen s ηj nei∈Γ0wi h weigh
W(n, i) = N/d. Thus, i
m=
R
X
N=0
a N
hen
o dp(cm)≥
R
X
N=0
a NN
d.
In o he wo ds, he p-adic New on polygon o de Qp(π)(1 −¯
β1T|H1,k) lies on o abo e he lowe con ex hull o
he poin s R
X
N=0
a N,
R
X
N=0
a N
N
d!R= 0,1, . . . , dimQp(π)H1,k
Thus, he q-adic New on polygon o de Qq(π)(1 −βT |H1,k) lies on o abo e he lowe con ex hull o he poin s
R
X
N=0
N,
R
X
N=0
N
N
d!R= 0,1, . . . , dimQq(π)H1,k.
This inishes he p oo o Theo em 6.1.
6.3 Rela i e Dwo k homology
In eg al o he p oo o Theo em 6.1 was he Dwo k decomposi ion o ela i e homology gi en on (11). The main
esul o his sec ion is o p o ide a p oo o his esul . This p oo will closely ollow a gumen s o Dwo k’s [9,
§7] and Adolphson-Spe be ’s [2]. We begin by ecalling ha
V(b0, b;ρ) := L(b0)x⊕ · · · ⊕ L(b0)xd−1∩ K(b0, b;ρ)
V(b0, b) := [
ρ∈R
V(b0, b;ρ).
Theo em 6.4. Suppose (p, d)=1. Le band b0be eal numbe s sa is ying b0≤band 1
p−1≤b≤p
p−1. Se
e:= b−1
p−1. Then
19
1. K(b0, b)•=V(b0, b)⊕D( )K(b0, b),
2. K(b0, b; 0)•⊂ V(b0, b; 0) ⊕D( )K(b0, b;e),
3. D( )is injec i e i b > 1
p−1,
4. i g∈ K(b0, b)•is di isible by nand we w i e g=ξ+D( )ζwi h ξ∈ V(b0, b)and ζ∈ K(b0, b), hen n
di ides ξand ζ.
The p oo o his heo em will consis o a se ies o lemmas which will comp ise he es o his sec ion.
Lemma 6.5. Suppose b0≤b. Then
K(b0, b; 0)•⊂ V(b0, b; 0) + πˆ
xK(b0, b;e).
Fu he mo e, i ξ∈ K(b0, b)•is di isible by n hen when we w i e ξ=ζ+(πˆ
x)νwi h ζ∈ V(b0, b)and ν∈ K(b0, b),
hen ndi ides bo h ζand ν.
P oo . W i e ˆ
( , x) = xd+Pd−1
j=0 ˆajxj+ x. Now, wi h
πˆ
x( , x) = π(dxd+
d−1
X
j=1
jˆajxj+ x)
we may w i e, o m≥d,
xm=πdxd1
πdxm−d+π

d−1
X
j=1
jˆajxj+ x
1
πdxm−d−1
πdxm−d
=πˆ
x( , x)1
πdxm−d−
d−1
X
j=1
j
dˆajxm+j−d−1
d xm+1−d.
No ice ha he las igh -hand sum consis s o e ms in xo deg ee s ic ly smalle han xm. This is ou educ ion
o mula o xm, educing all monomials xm o some linea combina ion o {x, x2, . . . , xd−1}.
Nex , conside Bnm nxm∈ K(b0, b; 0) wi h Bnm ∈Zq[π]. The educ ion o mula akes he o m
Bnm nxm=πˆ
x( , x)Bnm
πd nxm−d−
d−1
X
j=1
jBnm
dˆaj nxm+j−d−Bnm
d n+1xm+1−d.(16)
Obse e ha , since m≥d, i is immedia e ha Bnm
πd nxm−d∈ K(b0, b;e) while he o he e ms jBnm
dˆaj nxm+j−d
and Bnm
d n+1xm+1−dlie in K(b0, b; 0) since b≥b0. I e a ing he ecu si e equa ion (16), we ob ain
Bnm nxm∈ V(b0, b; 0) + πˆ
x( , x)K(b0, b;e).(17)
Nex , le ξ=Pn,m≥0Bnm nxm∈ K(b0, b; 0)•. Fo each N∈Z≥0we may w i e ξ=ζ(N)+Pn≥0η(n,N)whe e
η(n,N):=
N
X
m=0
Bnm nxmand ζ(N):= X
n≥0X
m≥N+1
Bnm nxm.
Obse e ha xN+1 |ζ(N) o e e y N, and n|η(n,N). By (17), we may w i e η(n,N)=ν(n,N)
1+ (πˆ
x)ν(n,N)
2
wi h ν(n,N)
1∈V(b0, b; 0) and ν(n,N)
2∈ K(b0, b;e), bo h wi h he p ope y ha hey a e di isible by n. Hence,
Pn≥0ν(n,N)
1and Pn≥0ν(n,N)
2p oduce well-de ined elemen s o V(b0, b; 0) and K(b0, b;e), espec i ely. Le us
deno e hese elemen s by ν(N)
1and ν(N)
2. We ha e hus cons uc ed sequences o elemen s {ν(N)
1}N≥1in V(b0, b; 0)
and {ν(N)
2}N≥1in K(b0, b;e) which sa is y
ξ=ζ(N)+ν(N)
1+ (πˆ
x)ν(N)
2
Now, in he opology o coe icien -wise con e gence (i.e. he (π, , x)-adic opology), Zq[π][[ , x]] is compac .
Thus, K(b0, b;ρ) is compac o each ρin he induced opology. Hence, we may es ic ou sel es o con e gen
subsequences o {ν(N)
1}N≥1and {ν(N)
2}N≥1wi h limi s ν1∈V(b0, b; 0) and ν2∈ K(b0, b;e), espec i ely. Thus,
ξ= lim
N→∞ ζ(N)+ν(N)
1+ (πˆ
x)ν(N)
2=ν1+ (πˆ
x)ν2,
whe e limN→∞ ζ(N)= 0 since xN+1 |ζ(N) o each N. This p o es he lemma.
20
Lemma 6.6. Le band b0be eal numbe s. Then V(b0, b)∩πˆ
xK(b0, b) = {0}.
P oo . Suppose η:= c1x+· · ·+cd−1xd−1∈ V(b0, b)∩πˆ
xK(b0, b). Le ζ:= P∞
j=0 Bjxj∈ K(b0, b) sa is y η=πˆ
xζ.
Now,
πˆ
x
∞
X
j=0
Bjxj=c1x+c2x2+· · · +cd−1xd−1.
W i ing πˆ
x( , x) = π(dxd+Pd−1
j=1 jˆajxj+ x), we ha e
∞
X
j=0
πdBjxd+j+
∞
X
=0
d−1
X
j=1
πB ˆajxj+ +
∞
X
j=0
πBj xj=1 =c1x+· · · +cd−1xd−1.
Fo j≥0, since he coe icien o xd+jin his equa ion mus anish, we ha e
πdBj+
d+j−2
X
=j+1
πB ˆad+j− +πBd+j−1 = 0
and so
Bj=−1
d
d+j−2
X
=j+1
B ˆad+j− −1
dBd+j−1 . (18)
Using (18) ecu si ely, we see ha Bj→0 (p, )-adically. Hence, Bj= 0 o all j≥0 as desi ed.
Lemma 6.7. Le ξ∈ K(b0, b)and suppose πˆ
xξ∈ K(b0, b;ρ). Then ξ∈ K(b0, b;ρ+e).
P oo . Le ξ=P∞
j=0 Bjxj∈ K(b0, b) and πˆ
xξ=P∞
j=0 Cjxj∈ K(b0, b;ρ). W i ing πˆ
x( , x) = π(dxd+
Pd−1
j=1 jˆaj( )xj), whe e ˆa1( ) := , we ha e
∞
X
j=0
πdBjxd+j+
∞
X
=0
d−1
X
j=1
πB ˆajxj+ =
∞
X
j=0
Cjxj.
F om his, he coe icien o xd+jsa is ies
πdBj+
d+j−1
X
=j+1
πB ˆad+j− =Cd+j
o all j≥0. Rew i ing his, we ha e
Bj=1
πdCd+j−1
d
d+j−1
X
=j+1
B ˆad+j− .
I e a ing his n- imes p oduces
Bj=ζ(j)
n+ξ(j)
1+ξ(j)
2+· · · +ξ(j)
n
whe e
ζ(j)
n:= ±
d+j−1
X
1=j+1
d+ 1−1
X
2= 1+1
· · ·
d+ n−1−1
X
n= n−1+1
1
dnB nˆad+j− 1ˆad+ 1− 2· · · ˆad+ n−1− n
and
ξ(j)
1:= 1
πdCd+j
ξ(j)
2:= 1
d2π
d+j−1
X
1=j+1
Cd+ 1ˆad+j− 1
ξ(j)
3:= 1
d3π
d+j−1
X
1=j+1
d+ 1−1
X
2= 1+1
Cd+ 2ˆad+j− 1ˆad+ 1− 2
.
.
.
ξ(j)
n:= 1
dnπ
d+j−1
X
1=j+1
d+ 1−1
X
2= 1+1
· · ·
d+ n−2−1
X
n−1= n−2+1
Cd+ n−1ˆad+j− 1ˆad+ 1− 2· · · ˆad+ n−2− n−1.
21

Since B n∈L(b0;bw0( n)), ζ(j)
n→0 as n ends o in ini y. Thus, o comple e he lemma, le us show P∞
n=1 ξ(j)
n∈
L(b0;bw0(j) + e).
We know Cd+ n−1∈L(b0;bw0(d+ n−1) + ρ). We wish o show
1
dn−1πCd+ n−1ˆad+j− 1ˆad+ 1− 2· · · ˆad+ n−2− n−1∈L(b0;bw0(j) + e+ρ).(19)
No ice ha (19) ypically has many ˆa e ms equal o 1. These ˆawill no a ec he L(b0;σ) space ha (19) lies in.
I is only when he ˆaequals ha hings change. The wo se case is when all ˆaequal . In his case, i=id +j−i
o i= 1,2, . . . , n −1 making (19) ake he o m
1
dn−1πCd+(n−1)d+j−(n−1) n−1,
which may easily be shown o lie in L(b0;bw0(j) + e+ρ). The gene al case is simila . This concludes he p oo
o he lemma.
Lemma 6.8. Le b0≤band 1
p−1≤b≤p
p−1. Wi h e:= b−1
p−1we ha e
K(b0, b; 0)•⊂ V(b0, b; 0) + D( )K(b0, b;e).
Fu he mo e, i ξ∈ K(b0, b)•is di isible by n, hen when we w i e ξ=η+D( )ζwi h η∈ V(b0, b)and ζ∈ K(b0, b),
hen ηand ζa e also di isible by n.
P oo . Recall, D( ) = x∂
∂x +H( , x) wi h H( , x) := P∞
j=0 πjpjˆ
τj
x( pj, xpj). Now, obse e ha
ˆ
τj
x( pj, xpj) = ˆ
x( , x)pj+phj( , x)
o some polynomial hjwi h coe icien s in Zq[π]. W i e
H( , x) = πˆ
x( , x)Q1( , x) + K1( , x)
whe e
Q1( , x) :=
∞
X
j=0
πjπ−1pjˆ
x( , x)pj−1
K1( , x) :=
∞
X
j=1
πjpj+1hj( , x).
We claim ha Q1,1
Q1, K1∈ K(p
p−1,p
p−1; 0). To see his, no e ha since ˆ
∈ K(p
p−1,p
p−1;−p
p−1), we ha e
ˆ
pj−1
x∈ K(p
p−1,p
p−1;−p
p−1(pj−1)).
Thus,
πjπ−1pjˆ
pj−1
x∈ K(p
p−1,p
p−1; 0)
p o ing he esul o Q1and 1/Q1. Nex , since hjconsis s o e ms coming om he expansion o ˆ
pj
x, we see
ha hj∈ K(p
p−1,p
p−1;−p
p−1pj). Thus πjpj+1hj∈ K(p
p−1,p
p−1; 0) p o ing he esul o K1.
We will i s suppose b > 1
p−1. Le ξ∈ K(b0, b; 0)•. By Lemma 6.5, he e exis s η1∈ V(b0, b; 0) and ζ1∈
K(b0, b;e) such ha ξ=η1+ (πˆ
x)ζ1. Thus,
ξ=η1+ (H−K1)Q−1
1ζ1
=η1+ (Q−1
1K1ζ1−x∂
∂xQ−1
1ζ1
| {z }
=:ν1
) + D( )(Q−1
1ζ1
|{z}
=:ζ0
1
).
No ice ha ν1∈ K(b0, b;e)•and ζ0
1∈ K(b0, b;e). Con inuing his same p ocess, bu now wi h ν1ins ead o ξ, we
a e lead o
ξ= (η1+· · · +ηN) + νN+D( )(ζ0
1+· · · +ζ0
N)
22
whe e ηi∈ V(b0, b; (i−1)e), νN∈ K(b0, b;Ne)•, and ζ0
i∈ K(b0, b;ie). Thus, η:= P∞
i=1 ηi∈ V(b0, b; 0) and
ζ0:= P∞
i=1 ζ0
i∈ K(b0, b;e). Upon aking he limi in he coe icien wise con e gence opology we see ha
ξ=η+D( )ζ0
as desi ed.
We now conside he case when b=1
p−1. Le ξ∈ K(b0, b; 0)•. Fo each N∈Z≥0, we may w i e
ξ=
N
X
n=1
Bnxn+X
n≥N+1
Bnxn.
Le  > 0. Fo 1 ≤n≤N, since bw0(n)≥(b+
w0(N))w0(n)−we see ha
N
X
n=1
Bnxn∈ K(b0, b +
w0(N);−)•.
Since b+
w0(N)>1
p−1, he e exis s η(,N)∈ V(b0, b +
w0(N);−) and ζ(,N)∈ K(b0, b +
w0(N);−+
w0(N)) such
ha N
X
n=1
Bnxn=η(,N)+D( )ζ(,N).
We ha e jus cons uc ed sequences {η(,N)}∞
N=1 ⊂ V(b0, b;−) and {ζ(,N)}∞
N=1 ⊂ K(b0, b;−). Since K(b0, b;ρ)
and V(b0, b;ρ) a e compac in he coe icien wise con e gence opology o each ρ, we may es ic ou sel es o
con e gen subsequences wi h limi s η()∈ V(b0, b;−) and ζ()∈ K(b0, b;−) which sa is y
ξ=η()+D( )ζ().
In he coe icien wise con e gence opology, le ing →0+, he e exis s η∈ V(b0, b; 0) and ζ∈ K(b0, b; 0) such ha
ξ= lim
→0+η()+D( )ζ()=η+D( )ζ.
This p o es he i s pa o he lemma.
The second pa ollows om he di isibili y esul in Lemma 6.5 and unning h ough he abo e a gumen .
Lemma 6.9. Le b0≤band 1
p−1≤b≤p
p−1. Then
V(b0, b)∩D( )K(b0, b).
P oo . Le us i s assume b > 1
p−1. Le η∈V(b0, b)∩D( )K(b0, b) be non-ze o, and le ξ∈ K(b0, b) be such ha
D( )ξ=η. Find a eal numbe csuch ha ξ∈ K(b0, b;c) bu ξ6∈ K(b0, b;c+e). We will p o e ha no such c
exis s, con adic ing he exis ence o η.
Since D( ) = x∂
∂x + (πˆ
x)Q1+K1, we ha e
η=D( )ξ= (πˆ
x)Q1ξ+x∂
∂xξ+K1ξ.
Since x∂
∂x ξ+K1ξ∈ K(b0, b;c)•, by Lemma 6.8 he e exis s ν1∈ V(b0, b;c) and ζ1∈ K(b0, b;c+e) such ha
x∂
∂xξ+K1ξ=ν1+D( )ζ1.
Hence,
η= (πˆ
x)Q1ξ+ν1+D( )ζ1
= (πˆ
x)Q1(ξ+ζ1) + ν1+K1ζ1.
Since K1ζ1∈ K(b0, b;c+e), applying Lemma 6.8 again p oduces ν2∈ V(b0, b;c+e) and ζ2∈ K(b0, b;c+ 2e) such
ha K1ζ1=ν2+D( )ζ2. Thus,
η= (πˆ
x)Q1(ξ+ζ1+ζ2)+(ν1+ν2) + K1ζ2.
23
I e a ing his ia induc ion, and aking he limi in he coe icien -wise con e gence opology, we ob ain
η= (πˆ
x)Q1(ξ+
∞
X
i=1
ζi) +
∞
X
i=1
νi.
This means (πˆ
x)Q1(ξ+P∞
i=1 ζi)∈ V(b0, b), and so by Lemma 6.6, ξ=−P∞
i=1 ζi∈ K(b0, b;c) which is impossible.
Suppose now ha b=1
p−1and η∈ V(b0, b)∩D( )K(b0, b). Wi h his choice o b,α1:= τ−1◦ψx◦F( , x) is a
map om K(b0, b) o K(b0, pb). The e o e, since α1◦D( ) = pD( p)◦α1, we see ha
α1(η)∈D( p)K(b0, pb).
Thus, he educ ion o α1(η) equals ze o in H1, p(b0, pb). Now, we may also iew α1as an endomo phism o
K(b0, pb) and so, abusing no a ion, we ob ain a map on homology ¯α1:H1, (b0, pb)→ H1, p(b0, pb). By Lemma
6.10 below, his map ¯α1is in e ible. The e o e, since he educ ion o α1(η) is ze o in H1, p(b0, pb), we mus ha e
he educ ion o ηin H1, (b0, pb) equal o ze o as well. Hence, η∈D( )K(b0, pb). Since D( )K(b0, pb)⊂ K(b0, pb),
we see ha η∈V(b0, pb)∩D( )K(b0, pb). Howe e , his in e sec ion equals {0}by he a gumen abo e since
bp > 1/(p−1), p o ing η= 0 as desi ed.
Lemma 6.10. Le b0≤1/(p−1). Then ¯α1:H1, (b0, p/(p−1)) → H1, p(b0, p/(p−1)) is an isomo phism.
P oo . Since b0≤1/(p−1), i ollows om de ini ion ha α1is a map om K(b0,1/(p−1)) o K(b0, p/(p−1)).
Now, de ine a map α0
1:K(b0, p/(p−1)) → K(b0,1/(p−1)) by
α0
1:= F( , x)−1◦Φx◦τ
whe e Φxis he map x7→ xp. Clea ly, α1◦α0
1=id, he iden i y map on K(b0, p/(p−1)). Hence, we ha e
K(b0, p/(p−1)) = α1α0
1K(b0, p/(p−1)) ⊂α1K(b0,1/(p−1)) ⊂ K(b0, p/(p−1)).
Hence, α1maps K(b0,1/(p−1)) isomo phically on o K(b0, p/(p−1)). A simila a gumen shows α1maps
K(b0,1/(p−1))•isomo phically on o K(b0, p/(p−1))•.
By Lemma 6.8, we know
K(b0,1/(p−1))•⊂ V(b0,1/(p−1)) + D( )K(b0,1/(p−1)).
Applying α1 o his we ob ain
K(b0, p/(p−1))•=α1K(b0,1/(p−1)) ⊂α1V(b0,1/(p−1)) + D( p)K(b0, p/(p−1)).
Since V(b0, p/(p−1)) ⊂ K(b0, p/(p−1))•, we ha e
V(b0, p/(p−1)) ⊂α1V(b0,1/(p−1)) + D( p)K(b0, p/(p−1)).
Now, i ollows om he de ini ion ha V(b0, b1) = V(b0, b2) o any posi i e eal numbe s b1and b2. Thus,
V(b0, p/(p−1)) ⊂α1V(b0, p/(p−1)) + D( p)K(b0, p/(p−1)).
Viewing α1as an endomo phism o K(b0, p/(p−1))•, his shows ¯α1:H1, (b0, p/(p−1)) → H1, p(b0, p/(p−1)) is
su jec i e. Since bo h o hese spaces a e ee L(b0)-modules o ini e ank, ¯α1mus also be injec i e. This inishes
he lemma.
Lemma 6.11. Suppose b > 1
p−1. I ξ∈ K(b0, b)and D( )ξ∈ K(b0, b;ρ), hen ξ∈ K(b0, b;ρ+e).
P oo . Suppose ξ6= 0. Choose c∈Rsuch ha ξ∈ K(b0, b;c) bu ξ6∈ K(b0, b;c+e). Then
(πˆ
x)Q1ξ=D( )ξ−x∂
∂xξ−K1ξ∈ K(b0, b; min{ρ, c}).
Thus, (πˆ
x)ξ∈ K(b0, b; min{ρ, c}) which, by Lemma 6.7, implies ξ∈ K(b0, b; min{ρ, c}+e). By ou choice o c he
lemma ollows.
Co olla y 6.12. Suppose b > 1
p−1. Then D( )is injec i e.
P oo . Suppose he e exis s nonze o ξ∈ K(b0, b) such ha D( )ξ= 0. Then, by Lemma 6.11, since 0 ∈ K(b0, b;ρ)
o e e y ρ, we ha e ξ∈ K(b0, b;ρ+e) o e e y ρ. Hence, ξmus be ze o.
24
6.4 Dwo k decomposi ion o he symme ic powe s o ela i e homology
Theo em 6.1 demons a ed one consequence o Dwo k decomposi ion wi h he ope a o ∂. In his sec ion, we
wish o ake he hypo hesis o Dwo k decomposi ion in ha heo em and educe i o a simila hypo hesis which
eplaces he di e en ial ope a o ∂wi h an easie ope a o LΦdesc ibed below. We expec ha his simila
hypo hesis may be demons a ed o a la ge class o ( , x). Howe e , we also expec ha i will ail jus as o en.
New ideas will be equi ed o handle he la e case.
I is easies o see he main obs uc ions o he heo y i we gene alize a bi . Le dbe a posi i e in ege . We
call a unc ion w:Zs
≥0→1
dZ≥0aweigh unc ion i i sa is ies he ollowing h ee p ope ies:
1. w(0) = 0,
2. w(cu) = cw(u) o e e y c∈Q≥0, and
3. w(u+ )≤w(u) + w( ) o all u, ∈Zs
≥0.
Le w1:Zs
≥0→1
d1Z≥0be a weigh unc ion. Le band b0be posi i e eal numbe s. Fo each ρ∈Rde ine
L(b0;ρ) := 

X
j∈Zs
≥0
Aj j|Aj∈Zq[π], o dp(Aj)≥b0w1(j) + ρ

.
This is a p-adic Banach space wi h no m gi en by kPAj jk:= minjo dp(Aj). De ine
L(b0) := [
ρ∈R
L(b0;ρ).
Le Mbe a ee L(b0)-module wi h basis {e1, . . . , e }. We place a weigh on each basis elemen eias ollows: ix
a posi i e in ege d0and le w0:{1, . . . , } → 1
d0Z≥0be any unc ion. No e, w0is no a weigh unc ion since he
se {1, . . . , }is ini e. De ine
M(b0, b;ρ) := (
X
i=1
Biei|Bi∈L(b0;bw0(i) + ρ))(20)
and
M(b0, b) := M=[
ρ∈R
M(b0, b;ρ).
Fo any subse Uo M, we may de ine U(b0, b;ρ) := U∩ M(b0, b;ρ) and U(b0, b) := U∩ M(b0, b).
Deno e by M(k):= SymkM he k- h symme ic powe o Mo e L(b0). Simila o (20) de ine
M(k)(b0, b;ρ) := 





X
i:=(i1,...,i )∈Z
≥0
i1+···+i =k
Biei|Bi∈L(b0;bw0(i) + ρ)






whe e ei:= ei1
1· · · ei
and
w0(i) := w0(i1, . . . , i ) :=
X
j=1
ijw0(j).
Le Nbe a ee, ini e ank L(b0)-module, and deno e by N(k) he k- h symme ic powe o No e L(b0). Le
Φ : M→N be an L(b0)-module mo phism. De ine he Leibni z ope a o o Φ o be he ope a o LΦ:M(k)→
N(k)de ined by
LΦ(ei1
1· · · ei
) :=
X
m=1
imei1
1· · · eim−1
m· · · ei
Φ(em).
Nex , we men ion a sho echnical lemma which will be use ul.
Lemma 6.13. Suppose Φ : M(b0, b; 0) → N (b0, b;ρ). Then LΦ:M(k)(b0, b; 0) → N (k)(b0, b;ρ).
25