Pu i y o exponen ial sums on An
An onio Rojas-Le´on
Abs ac
We gi e a pu i y esul o wo kinds o exponen ial sums o he ype Px∈knψ( (x)),
whe e kis a ini e ield o cha ac e is ic pand ψ:k→C?is a non- i ial addi i e
cha ac e . In he i s case ∈k[x1, . . . , xn] is a polynomial o deg ee di isible by pwhose
highes deg ee homogeneous o m de ines a non-singula p ojec i e hype su ace, and in
he second one is a polynomial o deg ee p ime o pwhose highes deg ee homogeneous
o m de ines a p ojec i e hype su ace wi h isola ed singula i ies.
1. In oduc ion
Le kbe a ini e ield o cha ac e is ic pand ca dinali y q, and le ∈k[x1, . . . , xn] be a polynomial
o deg ee d. Pick a non- i ial addi i e cha ac e ψ:k→C?, and conside he sum Px∈knψ( (x)).
In [Del74] Deligne p o ed, as a co olla y o his p oo o he Riemann hypo hesis o p ojec i e
a ie ies o e ini e ields, he ollowing es ima e:
Theo em 1.([Del74], Th´eo `eme 8.4) Suppose ha
i) The highes deg ee homogeneous o m do de ines a nonsingula hype su ace in Pn−1
¯
k.
ii) dis p ime o p.
Then we ha e he es ima e
¯
¯
¯
¯
¯
X
x∈kn
ψ( (x))¯
¯
¯
¯
¯
⩽(d−1)n·qn/2.
Mo eo e , he showed ha he sum is pu e o weigh nand ank (d−1)n. In pa icula , he e
a e (d−1)ncomplex algeb aic numbe s α1, . . . , α(d−1)n, all pu e o weigh n(meaning ha all hei
conjuga es o e Qha e absolu e alue qn/2) such ha , o e e y in ege m⩾1, i kmdeno es he
deg ee mex ension o kin a ixed algeb aic closu e ¯
k, we ha e
(−1)nX
x∈kn
m
ψ(T acekm/k( (x))) =
(d−1)n
X
i=1
αm
i.
Wha can we say in he case whe e pdi ides d? By pe e si y a gumen s (c . [KL85], [Ka 93],
[Ka 04]) we know ha he sum is pu e o almos all ∈k[x1, . . . , xn]. Mo e p ecisely, i we add a
su icien ly gene al linea o m o (one ha is con ained in a sui able Za iski dense open subse
Uo he dual a ine space ˆ
An
kdepending on ψand q), he sum becomes pu e o weigh n. Howe e ,
hese esul s do no gi e us any in o ma ion abou he sum associa ed o a pa icula . On he
o he hand, in [AS00b] Adolphson and Spe be show, using p-adic me hods, ha i sa is ies
ce ain egula i y hypo heses he L- unc ion associa ed o he exponen ial sum (o i s in e se) is
2000 Ma hema ics Subjec Classi ica ion 11L03,11L07
Keywo ds: exponen ial sums, l-adic cohomology
Pa ially suppo ed by MTM2004-07203-C02-01 and FEDER
An onio Rojas-Le´
on
a polynomial. In his a icle we will use hese esul s o gi e a e sion o Theo em 1 o he case
whe e pdi ides d.
Fix a p ime `6=pand an isomo phism ι:¯
Q`→Cso ha we can speak abou absolu e
alues o elemen s o ¯
Q`and weigh s wi hou ambigui y. F om now on we will assume ha such an
isomo phism has been chosen, wi hou making any u he e e ence o i . Thus, o e e y α∈¯
Q`,
|α|will always mean |ι(α)|. We will also use his isomo phism o iden i y he se s o C?- alued
cha ac e s and o ¯
Q?
`- alued cha ac e s o any ini e g oup. Conside he lisse A in-Sch eie ¯
Q`-
shea Lψon A1
kassocia ed o he non- i ial addi i e cha ac e ψ:k→C?(c . [Del77], 1.7). Fo
e e y ini e ex ension k0/k and e e y ∈A1(k0) = k0, he ace o he geome ic F obenius elemen
in Gal(¯
k/k0) ac ing on he s alk o Lψa a geome ic poin ¯
o e is ψ(T acek0/k( )). In pa icula ,
since ψ akes i s alues among he oo s o uni y, Lψis pu e o weigh 0.
Le Lψ( )deno e he pull-back ?Lψon An
k. The cohomology g oups wi h compac suppo
Hi
c(An
¯
k,Lψ( )) a e endowed wi h an ac ion o he absolu e Galois g oup Gal(¯
k/k) and, in pa icula ,
o he geome ic F obenius elemen F∈Gal(¯
k/k). By he G o hendieck ace o mula we ha e
X
x∈kn
ψ( (x)) =
2n
X
i=0
(−1)iT ace(F|Hi
c(An
¯
k,Lψ( ))).
Ou i s esul is he ollowing
Theo em 2.Le dbe di isible by p. W i e = d+ d0+ 0, whe e dis he deg ee dhomogeneous
componen o ,d0is he deg ee o − dand d0is he deg ee d0homogeneous componen o .
Suppose ha
a) d0/d > p/(p+ (p−1)2)and d0is p ime o p.
b) The equa ion d= 0 de ines a non-singula hype su ace in Pn−1
¯
k.
c) The hype su ace de ined in Pn−1
¯
kby d0= 0 does no con ain any o he common ze oes o
∂ d
∂x1, . . . , ∂ d
∂xnin Pn−1
¯
k.
Then
1. Hi
c(An
¯
k,Lψ( )) = 0 o i6=n.
2. Hn
c(An
¯
k,Lψ( ))has dimension (d0(d−1)n+ (−1)n(d−d0))/d and is pu e o weigh n.
3. We ha e he es ima e
¯
¯
¯
¯
¯
X
x∈kn
ψ( (x))¯
¯
¯
¯
¯
⩽d0(d−1)n+ (−1)n(d−d0)
d·qn/2.
Fo d0=d−1 ( he gene ic case) he inequali y in (a) holds as long as d⩾3, and we ge
Co olla y 3.Assume d⩾3is di isible by p. Le = d+ d−1+ 0be as abo e. Suppose ha
a) The equa ion d= 0 de ines a non-singula hype su ace in Pn−1
¯
k.
b) The equa ion d−1= 0 de ines a hype su ace in Pn−1
¯
kwhich does no con ain any o he
common ze oes o ∂ d
∂x1, . . . , ∂ d
∂xnin Pn−1
¯
k.
Then
1. Hi
c(An
¯
k,Lψ( )) = 0 o i6=n.
2. Hn
c(An
¯
k,Lψ( ))has dimension ((d−1)n+1 −(−1)n+1)/d and is pu e o weigh n.
2
Pu i y o exponen ial sums on An
3. We ha e he es ima e
¯
¯
¯
¯
¯
X
x∈kn
ψ( (x))¯
¯
¯
¯
¯
⩽(d−1)n+1 −(−1)n+1
d·qn/2.
As usual, (3) is a consequence o he anishing o he cohomology oge he wi h Deligne’s heo em
on weigh s (c . [Del80], Co ollai e 3.3.4).
The second esul deals wi h ano he kind o sum s udied by Adolphson and Spe be in [AS00b]
and is a gene aliza ion o ([Ga 98], Theo em 0.4). Le ∈k[x1, . . . , xn] be a polynomial o deg ee
d, which we will now assume o be p ime o p. We will show
Theo em 4.W i e = d+ d0+ 0as in Theo em 2. Suppose ha
a) d0/d > p/(p+ (p−1)2)and d0is p ime o p.
b) The hype su ace de ined by d= 0 in Pn−1
¯
khas a wo s weigh ed homogeneous isola ed
singula i ies o o al deg ees d1, . . . , dsp ime o p(c . [AS00b], Sec ion 2 o [Ga 98], 0.3 o
he de ini ions).
c) The hype su ace de ined by d0= 0 in Pn−1
¯
kdoes no con ain any o hese singula i ies.
Le µ1, . . . , µsbe he Milno numbe s co esponding o he singula i ies o d= 0. Then
1. Hi
c(An
¯
k,Lψ( )) = 0 o i6=n.
2. Hn
c(An
¯
k,Lψ( ))has dimension (d−1)n−(d−d0)Ps
i=1 µiand is pu e o weigh n.
3. We ha e he es ima e
¯
¯
¯
¯
¯
X
x∈kn
ψ( (x))¯
¯
¯
¯
¯
⩽((d−1)n−(d−d0)
s
X
i=1
µi)·qn/2.
2. A cohomological anishing esul
In his sec ion we will begin he p oo o Theo em 2. We will i s use he me hod o pencils o show
he anishing o Hi
c(An
¯
k,Lψ( )) o i > n + 1. This equi es s udying he ibe s o he map , so he
i s hing we need o do is ind a sui able compac i ica ion o . Un o una ely, he compac i ica ion
de ined in [Ka 99] by embedding Anas a dense open subse o he subscheme o Pn×A1gi en by
he anishing o F−λXd
0no longe wo ks in his case. The eason is ha we a e compac i ying a
map o deg ee di isible by p, and his may in oduce some wild ami ica ion a in ini y in he highe
di ec images o he cons an shea wi h espec o he compac i ied map.
The e o e, ins ead o di ec ly compac i ying , he idea is o i s w i e as he composi ion
o a closed embedding o Anin An×A1(gi en by he g aph o ) ollowed by he p ojec ion, and
hen compac i y he p ojec ion es ic ed o he image o An. Since we a e compac i ying a map o
deg ee 1, we do no un in o any p oblems caused by wild ami ica ion. Howe e , one disad an age
o his compac i ica ion is ha he ibe a in ini y will always ha e a singula poin , so we will only
be able o deduce he anishing o he cohomology g oups o i > n + 1.
P oposi ion 5.Suppose ha he equa ion d= 0 de ines a non-singula hype su ace in Pn−1
¯
k.
Then Hi
c(An
¯
k,Lψ( )) = 0 o i > n + 1.
P oo . De ine Z o be he hype su ace in Pn+1
k(whe e we ake coo dina es X0, . . . , Xn, T) de ined
by he anishing o F−TXd−1
0, whe e Fis he homogeniza ion o wi h espec o he a iable
X0(i.e. F(X0, . . . , Xn) = Xd
0· (X1/X0, . . . , Xn/X0)). The a ine space An
kis na u ally an open
subscheme o Z(jus by embedding i in An+1
kusing he g aph o , and hen iden i ying An+1
kwi h
Pn+1
kminus he hype plane X0= 0).
3
An onio Rojas-Le´
on
Nex , we de ine he incidence a ie y ˜
Zas a di iso o Z×P1
k, gi en (wi h coo dina es X0, . . . , Xn, T
o he i s ac o and λ0, λ1 o he second one) by he ze o locus o λ0T−λ1X0. Thus
˜
Z(¯
k) = {((x0, . . . , xn, ),(λ0, λ1)) ∈Z(¯
k)×P1(¯
k) : λ0 =λ1x0}.
Le ˜
:˜
Z→P1
kbe he es ic ion o ˜
Zo he canonical p ojec ion π2:Z×P1
k→P1
k. I is a p ope
map, being he composi e o a closed imme sion and a p ope p ojec ion (since Zis p ojec i e).
The open subse An
k,→Zcan be embedded as an open subscheme o ˜
Zin he ob ious way.
Namely, we iden i y he poin x∈An(¯
k) wi h (x, (x)) ∈˜
Z(¯
k). In his way we ge a commu a i e
diag am
An
k−−−−→ ˜
Z


y

y
˜
A1
k−−−−→ P1
k
whe e he ho izon al a ows a e open embeddings. The image o An
kin ˜
Zcan be desc ibed as he
se o (x, λ)∈˜
Zsuch ha x6∈ Z∩ {X0= 0}.
Be o e going any u he we need o show ha ˜
is a la map.
Lemma 6.The map ˜
:˜
Z→P1
kis la .
P oo . By ([Ha 77], P oposi ion III.9.9) i su ices o show ha all geome ic ibe s o ˜
ha e he
same Hilbe polynomial. The ibe o e a ini e poin λ∈A1(¯
k) is easily seen o be he comple e
in e sec ion o he deg ee dhype su ace F−λXd
0= 0 and he hype plane T−λX0= 0. Simila ly,
he ibe o e in ini y is he comple e in e sec ion o he hype su ace F= 0 and he hype plane
X0= 0. Since he Hilbe polynomial o a comple e in e sec ion only depends on i s mul ideg ee,
we conclude ha i is he same o all geome ic ibe s o ˜
.
We ex end by ze o he shea Lψ o he whole P1
k, and ake i s pull-back by ˜
o ˜
Z, which we
will also deno e by Lψ( ). This is compa ible wi h he p e ious no a ion, since i s es ic ion o An
k
is jus he pull-back o Lψby .
Lemma 7.The e is a quasi-isomo phism
RΓc(An
¯
k,Lψ( ))∼
→RΓc(˜
Z⊗¯
k, Lψ( )).
P oo . To simpli y he no a ion, we will iden i y each homogeneous o m wi h he p ojec i e hype -
su ace de ined by i s anishing. I is clea ha ˜
Z1:= (Z∩T∩X0)×P1
kis con ained in ˜
Zas a closed
subscheme. Le ˜
Z0be i s complemen . The es ic ion o ˜
o ˜
Z1is jus he second p ojec ion. F om
he decomposi ion
˜
Z0
j
,→˜
Zi
←-˜
Z1
we ge an exac sequence o shea es
0→j!j?Lψ( )→ Lψ( )→i?i?Lψ( )→0
om which we ge a dis inguished iangle in Db(¯
Q`− ec o spaces)
RΓc(˜
Z0⊗¯
k, Lψ( ))→RΓc(˜
Z⊗¯
k, Lψ( ))→RΓc(˜
Z1⊗¯
k, Lψ( ))→
Now in ˜
Z1∼
=(Z∩T∩X0)×P1
k he shea Lψ( )is jus he ex e nal enso p oduc ¯
Q`£Lψ. The e o e
by he K¨unne h o mula we ha e
RΓc(˜
Z1⊗¯
k, Lψ( )) = RΓc((Z∩T∩X0)⊗¯
k, ¯
Q`)⊗RΓc(P1
¯
k,Lψ) = 0
4
Pu i y o exponen ial sums on An
since RΓc(P1
¯
k,Lψ) = RΓc(A1
¯
k,Lψ) = 0 (c . [Del77], Th´eo `eme 2.7*). Hence we ge a quasi-isomo phism
RΓc(˜
Z0⊗¯
k, Lψ( ))∼
→RΓc(˜
Z⊗¯
k, Lψ( )).
The image o he open imme sion h:An
k,→˜
Z0is he se o (x, λ)∈˜
Zsuch ha x6∈ Z∩X0. I s
complemen in ˜
Z0is he se o (x, λ)∈˜
Zsuch ha x∈Z∩X0and x6∈ Z∩T, so i maps o he poin
a in ini y unde ˜
. Since he s alk o Lψa in ini y is ze o, we ha e an equali y h!h?Lψ( )=Lψ( ),
and he e o e a quasi-isomo phism
RΓc(An
¯
k,Lψ( ))∼
→RΓc(˜
Z0⊗¯
k, Lψ( ))∼
→RΓc(˜
Z⊗¯
k, Lψ( )).
We will also deno e by ˜
:˜
Z⊗¯
k→P1
¯
k he map deduced om ˜
:˜
Z→P1
kby ex ension o scala s
o ¯
k. Since ˜
is p ope , we ha e (by composi ion o de i ed unc o s)
RΓc(˜
Z⊗¯
k, Lψ( )) = RΓc(P1
¯
k,R˜
?Lψ( )).
On he o he hand, by he p ojec ion o mula we ha e
R˜
?Lψ( )= R ˜
?(¯
Q`⊗˜
?Lψ) = R ˜
?¯
Q`⊗ Lψ
so P oposi ion 5 is equi alen o
P oposi ion 8.Unde he p e ious hypo heses he cohomology g oup Hi
c(P1
¯
k,R˜
?¯
Q`⊗Lψ) anishes
o i > n + 1.
The e o e we will p o e P oposi ion 8 ins ead.
P oposi ion 9.The shea es Ri˜
?¯
Q`on P1
¯
ka e lisse o i⩾n+ 1. Fo i=ni is he ex ension o
a lisse shea by a punc ual shea .
P oo . The ibe o ˜
a a poin λ∈A1(¯
k) is de ined in Pn+1
¯
k(wi h he usual coo dina es X0, . . . , Xn, T)
by he homogeneous ideal (F−TXd−1
0, T −λX0) = (F−λXd
0, T −λX0). I s in e sec ion wi h he
hype plane X0= 0 is hen de ined by he ideal (F, X0, T ), and is he e o e isomo phic o he hype -
su ace de ined in Pn−1
¯
kby d= 0, which is non-singula by hypo hesis. The e o e, he ibe i sel
has a wo s isola ed singula i ies. On he o he hand, he ibe a λ=∞is de ined in Pn+1
¯
kby he
ideal (F, X0). This is he p ojec i e cone o e he hype su ace de ined in Pn−1
¯
kby d= 0, so i has
only one singula poin ( he e ex).
By ([SGA7I], Expos´e I, Co . 4.3) we deduce ha o e e y λ∈P1(¯
k) he Iλ-in a ian special-
iza ion map (Ri˜
?¯
Q`)λ→(Ri˜
?¯
Q`)¯η(whe e ¯ηis a geome ic gene ic poin o P1
¯
kand Iλ he ine ia
g oup a λ) is an isomo phism o i > n and su jec i e o i=n. As a consequence, Ri˜
?¯
Q`is lisse
a λ o i > n. Fo i=nwe ha e an exac sequence (c . [Ka 99], Theo em 13)
0→(punc ual shea ) →Rn˜
?¯
Q`→j?j?Rn˜
?¯
Q`→0
whe e jis he inclusion o an open subse o P1
¯
kon which Rn˜
?¯
Q`is lisse. Bu since he specializa ion
map (Rn˜
?¯
Q`)λ→(Rn˜
?¯
Q`)¯ηis su jec i e and Iλ-equi a ian , he ac ion o Iλon (Rn˜
?¯
Q`)¯ηis
i ial. As a consequence, he shea j?j?Rn˜
?¯
Q`is lisse a λ.
P oposi ion 10.The cohomology g oup Ha
c(P1
¯
k,Rb˜
?¯
Q`⊗ Lψ) anishes o :
i) a > 2, all b
ii) b > n, all a
iii) b=n,a > 0.
5

An onio Rojas-Le´
on
P oo . Pa (1) is clea o cohomological dimension easons. Fo b>n, he shea Rb˜
?¯
Q`is lisse
on P1
¯
kby P oposi ion 9. Since P1
¯
kis simply connec ed, i mus be cons an . Then, i ¯ηis a geome ic
gene ic poin o P1
¯
k, we ge
RΓc(P1
¯
k,Rb˜
?¯
Q`⊗ Lψ) = (Rb˜
?¯
Q`)¯η⊗RΓc(P1
¯
k,Lψ) = 0
since RΓc(P1
¯
k,Lψ) = 0. This p o es (2).
To p o e (3), le j:V ,→P1
¯
kbe as in P oposi ion 9, whe e Vis a dense open se on which
Rn˜
?¯
Q`is lisse, and le H=j?j?Rn˜
?¯
Q`. Then His lisse on P1
¯
kby P oposi ion 9, so exac ly as
abo e we ge RΓc(P1
¯
k,H ⊗ Lψ) = 0. F om he exac sequence
0→ I (= punc ual shea ) →Rn˜
?¯
Q`→ H → 0
we ge , a e enso ing wi h Lψ,
0→ I ⊗ Lψ→Rn˜
?¯
Q`⊗ Lψ→ H ⊗ Lψ→0.
Now I⊗Lψis punc ual, so Hi
c(P1
¯
k,I⊗Lψ) = 0 o i > 0. F om he long exac sequence o cohomology
associa ed o he exac sequence abo e we ge isomo phisms
Ha
c(P1
¯
k,Rn˜
?¯
Q`⊗ Lψ)∼
→Ha
c(P1
¯
k,H ⊗ Lψ) = 0
o a > 0. This p o es (3).
We can now comple e he p oo o P oposi ion 8. We ha e a spec al sequence
Ha
c(P1
¯
k,Rb˜
?¯
Q`⊗ Lψ)⇒Ha+b
c(P1
¯
k,R˜
?¯
Q`⊗ Lψ).
Suppose a+b > n + 1. Then ei he
-a > 2, so Ha
c(P1
¯
k,Rb˜
?¯
Q`⊗ Lψ) = 0 by pa (1) o P oposi ion 10,
-b > n, so Ha
c(P1
¯
k,Rb˜
?¯
Q`⊗ Lψ) = 0 by pa (2) o P oposi ion 10 o
-a= 2 and b=n, so Ha
c(P1
¯
k,Rb˜
?¯
Q`⊗ Lψ) = 0 by pa (3) o P oposi ion 10.
The e o e, he spec al sequence implies ha Hi
c(P1
¯
k,R˜
?¯
Q`⊗ Lψ) anishes o i > n + 1.
3. A sum o Milno numbe s compu a ion
Conside he L- unc ion associa ed o he shea Lψ( )on An
k:
L(T, Lψ( )) = exp
∞
X
m=1
Sm
mTm
whe e
Sm=X
x∈kn
m
ψ(T acekm/k( (x)))
and kmis he ex ension o deg ee mo kin ¯
k. By he G o hendieck ace o mula, we ha e
L(T, Lψ( )) =
2n
Y
i=0
de (1 −T·F|Hi
c(An
¯
k,Lψ( )))(−1)i+1
whe e F∈Gal(¯
k/k) is he geome ic F obenius elemen .
The ollowing esul o Adolphson and Spe be ([AS00b], Theo em 1.11 and P oposi ion 6.5)
gi es an impo an es ic ion on he shape o his L- unc ion:
6
Pu i y o exponen ial sums on An
Theo em 11.W i e = d+ d0+ 0, whe e dis he deg ee dhomogeneous componen o ,
d0is he deg ee o − dand d0is he deg ee d0homogeneous componen o . Suppose ha
d0/d > p/(p+ (p−1)2)and d0is p ime o p. Suppose also ha ∂ d
∂x1, . . . , ∂ d
∂xnha e a ini e numbe o
common ze oes in Pn−1
¯
k(which is au oma ic i he hype su ace d= 0 in Pn−1
¯
kis non-singula ) and
he hype su ace de ined in Pn−1
¯
kby d0= 0 does no con ain any o hem. Then L(T, Lψ( ))(−1)n+1
is a polynomial o deg ee (d−1)n−(d−d0)Ps
i=1 µi, whe e he sum is aken o e he se {P1, . . . , Ps}
o common ze oes o ∂ d
∂x1,..., ∂ d
∂xnin Pn−1
¯
kand µideno es he co esponding Milno numbe
µi= dim¯
kOS,Pi.
He e Sis he ze o-dimensional subscheme o Pn−1
¯
kde ined by he ideal (∂ d
∂x1, . . . , ∂ d
∂xn), and OS,Pii s
local ing a Pi, which is a ini e ¯
k-algeb a.
We will now compu e his sum o Milno numbe s explici ly in he ollowing mo e gene al se ing
Lemma 12.Le F1, . . . , Fn∈¯
k[x1, . . . , xn]be (possibly ze o) homogeneous polynomials o deg ee
d−1. Suppose ha
i) F1, . . . , Fnha e a ini e numbe o common ze oes in Pn−1
¯
k.
ii) We ha e he ela ion
n
X
i=1
xi·Fi= 0.
Le {P1, . . . , Ps}be he se o common ze oes o F1, . . . , Fnin Pn−1
¯
k, and o e e y i= 1, . . . , s le
µibe he co esponding Milno numbe . Then we ha e
s
X
i=1
µi=(d−1)n−(−1)n
d.
P oo . By induc ion on n, we i s p o e i o n= 2. In his case, bo h F1and F2mus be non-ze o
(o he wise, by (2) hey would bo h be ze o, and (1) would no hold). The ela ion x1F1+x2F2= 0
implies ha x1di ides F2and x2di ides F1. Le F1=x2G1and F2=x1G2. Then x1x2(G1+G2) =
0, so G2=−G1. The e o e he subscheme de ined by F1and F2is he one de ined by G1, which is a
polynomial o deg ee d−2. The common ze oes o F1and F2a e hen in one- o-one co espondence
wi h he dis inc linea ac o s o G1, and he Milno numbe s a e he co esponding mul iplici ies.
Thus in his case we ge Ps
i=1 µi=d−2 = ((d−1)2−1)/d.
We assume now ha he lemma is ue o n−1⩾2, and p o e i o n. Choose (α1, . . . , αn−1)∈
¯
kn−1such ha none o he poin s P1, . . . , Psis con ained in he hype plane xn−Pn−1
j=1 αjxj= 0.
We cons uc he polynomials F0
1, . . . , F0
ngi en by
F0
i(x1, . . . , xn−1, xn) = Fi(x1, . . . , xn−1, xn+Pn−1
j=1 αjxj)+
+αiFn(x1, . . . , xn−1, xn+Pn−1
j=1 αjxj) o i= 1, . . . , n −1
F0
n(x1, . . . , xn−1, xn) = Fn(x1, . . . , xn−1, xn+Pn−1
j=1 αjxj)
Then he schemes Sde ined by he ideal (F1, . . . , Fn) and S1de ined by (F0
1, . . . , F0
n) co espond
o each o he ia he au omo phism ϕo Pn−1
¯
kgi en by ϕ(x1, . . . , xn−1, xn)=(x1, . . . , xn−1, xn+
Pn−1
j=1 αjxj). In pa icula he sums o he Milno numbe s a he poin s o Sand S1a e he same.
7
An onio Rojas-Le´
on
Mo eo e , we ha e
Pn
i=1 xi·F0
i(x1, . . . , xn−1, xn) =
=Pn−1
i=1 xi·(Fi(x1, . . . , xn−1, xn+Pn−1
j=1 αjxj)+
+αiFn(x1, . . . , xn−1, xn+Pn−1
j=1 αjxj))+
+xn·Fn(x1, . . . , xn−1, xn+Pn−1
j=1 αjxj) =
=Pn−1
i=1 xi·Fi(x1, . . . , xn−1, xn+Pn−1
j=1 αjxj)+
+(xn+Pn−1
i=1 αixi)·Fn(x1, . . . , xn−1, xn+Pn−1
j=1 αjxj) = 0.
I P= (x1, . . . , xn) is a common ze o o F0
1, . . . , F0
n, hen ϕ(P) = (x1, . . . , xn−1, xn+Pn−1
j=1 αjxj)
is a common ze o o F1, . . . , Fnso, by he choice o he αi,ϕ(P) is no con ained in he hype plane
xn−Pn−1
j=1 αjxj= 0. Hence Pis no con ained in he hype plane xn= 0. The e o e we can assume,
and we will, ha none o he common ze oes o F1, . . . , Fnis con ained in he hype plane xn= 0.
Unde his assump ion, we claim ha F1, . . . , Fn−1 o m a egula sequence in ¯
k[x1, . . . , xn]
(compa e [AS00b], Lemma 5.1). O he wise, he subscheme de ined by hem in Pn−1
¯
kwould ha e an
i educible componen Yo dimension a leas 1. F om (2) we deduce ha Yis con ained in he
hype su ace xnFn= 0. Being i educible, i mus be con ained ei he in xn= 0 o in Fn= 0.
Fu he mo e, since i has dimension ⩾1, i s in e sec ions wi h bo h xn= 0 and Fn= 0 a e non-
emp y. So in ei he case, he in e sec ion o F1, . . . , Fn−1, Fnand xn= 0 would be non-emp y, in
con adic ion wi h he assump ion made abo e.
Deno e by S1 he subscheme o Pn−1
¯
kde ined by (F1, . . . , Fn−1). The suppo o S1is he disjoin
union o he poin s P1, . . . , Ps, which a e con ained in Fn= 0, and he poin s Ps+1, . . . , Ps+ ,
which a e con ained in xn= 0. Le ν1, . . . , νs+ be he co esponding Milno numbe s (i.e. νi=
dim¯
kOS1,Pi). Since F1, . . . , Fn−1 o m a egula sequence o polynomials o deg ee d−1, S1is a
ze o-dimensional comple e in e sec ion o deg ee (d−1)n−1, he e o e
s+
X
i=1
νi= dim¯
kΓ(S1,OS1) = (d−1)n−1.
Fo e e y i= 1, . . . , s,xnis in e ible in he local ing OPn−1,Pi. So om (2) we deduce ha Fn
is con ained in he ideal gene a ed by F1, . . . , Fn−1in his local ing. The e o e
OS,Pi=OPn−1,Pi/(F1, . . . , Fn−1, Fn) =
=OPn−1,Pi/(F1, . . . , Fn−1) = OS1,Pi
and in pa icula νi=µi.
On he o he hand, o i= 1, . . . , ,Fnis in e ible in he local ing OPn−1,Ps+i, so xnis
con ained in he ideal gene a ed by F1, . . . , Fn−1in his local ing. Le Gj=Fj(x1, . . . , xn−1,0),
S2 he subscheme o Pn−2
¯
k(which we iden i y wi h he hype plane xn= 0 in Pn−1
¯
k) de ined by
(G1, . . . , Gn−1). The poin s Q1, . . . , Q o S2a e in one- o-one co espondence wi h Ps+1, . . . , Ps+
ia he inclusion Pn−2(¯
k),→Pn−1(¯
k), and
OS2,Qi=OPn−2,Qi/(G1, . . . , Gn−1) =
=OPn−1,Ps+i/(F1, . . . , Fn−1, xn) =
=OPn−1,Ps+i/(F1, . . . , Fn−1) = OS1,Ps+i,
so he Milno numbe s a e he same.
8
Pu i y o exponen ial sums on An
Now G1, . . . , Gn−1 all unde he hypo heses o he lemma, so we can apply he induc ion hy-
po hesis and deduce ha Ps+
i=s+1 νi= ((d−1)n−1−(−1)n−1)/d. The e o e
s
X
i=1
µi=
s
X
i=1
νi=
s+
X
i=1
νi−
s+
X
i=s+1
νi=
= (d−1)n−1−((d−1)n−1−(−1)n−1)/d = ((d−1)n−(−1)n)/d.
Thus, unde he hypo heses o Theo em 11, L(T, Lψ( ))(−1)n+1 is a polynomial o deg ee (d−
1)n−(d−d0)((d−1)n−(−1)n)/d = (d0(d−1)n+ (−1)n(d−d0))/d.
4. End o he p oo o Theo em 2
Pa (3) o he heo em is a di ec consequence o he p e ious wo pa s ia he ace o mula
and Deligne’s heo em. So i su ices o p o e (1) and (2). Fix a posi i e in ege d0< d p ime
o psuch ha d0/d > p/(p+ (p−1)2). Deno e by Pd,d0 he a ine space o all polynomials in
k[x1, . . . , xn] o deg ee ⩽dwhose homogeneous componen o deg ee iis ze o o all d0< i < d.
Le π1:Pd,d0×An
k→ Pd,d0be he p ojec ion and e :Pd,d0×An
k→A1
k he e alua ion map. Le
K∈ Db
c(Pd,d0,¯
Q`) be he objec Rπ1!e ?Lψ[n+ dim Pd,d0].
Lemma 13.The objec Kis pe e se and pu e o weigh n+ dim Pd,d0.
P oo . Fo d0=d−1 (i.e. when Pd,d0is he a ine space o all polynomials o deg ee ⩽d) his is
([Ka 04], Pa (1) o Theo em 3.1.2). We will see ha he same p oo wo ks in gene al.
The e is a na u al ini e map τ:An
k→ˆ
Pd,d0. Namely, o e e y ∈An(¯
k), τ( )∈ˆ
Pd,d0(¯
k) is he
e alua ion map a ,e (−, ) : Pd,d0(¯
k)→¯
k. Since ¯
Q`[n] is pe e se and pu e o weigh non An
k, so
is τ?¯
Q`[n] on ˆ
Pd,d0. I s Fou ie ans o m Tψ(τ?¯
Q`[n]) ∈ Db
c(Pd,d0,¯
Q`) wi h espec o ψis K(by he
e y de ini ion o K). The e o e Kis pe e se and pu e o weigh n+ dim Pd,d0(c . [KL85], Sec ion
2 o [KW01], Sec ion III.8 o he de ini ion and main p ope ies o he Fou ie ans o m).
No ice ha o e e y ini e ex ension k0/k and e e y ∈ Pd,d0(k0), he ace o he geome ic
F obenius elemen in Gal(¯
k/k0) ac ing on he s alk o Ka a geome ic poin o e is he sum
(−1)n+dim Pd,d0X
x∈k0n
ψ(T acek0/k (x)).
Le U⊂ Pd,d0be he maximal dense open se on which Khas lisse cohomology shea es. Then
Hi(K)|U= 0 o i6=−dim Pd,d0and F:= H−dim Pd,d0(K) = Rnπ1!e ?Lψis lisse and pu e o weigh
non U. Thus, o di e en ini e ex ensions k0/k and polynomials ∈U(k0), he exponen ial sums
Px∈k0nψ(T acek0/k (x)) a e pu e o weigh nand he same ank as F.
Le V⊂ Pd,d0( esp. W⊂ Pd,d0) be he dense open se o all polynomials such ha dde ines
a non-singula hype su ace on Pn−1
¯
k( esp. he dense open se o all such ha ∂ d
∂x1,..., ∂ d
∂xnha e
a ini e numbe o common ze oes in Pn−1
¯
kand he hype su ace d0= 0 does no con ain any o
hem). We know ha
i) Fo e e y ∈V(k), we ha e Hi
c(An
¯
k,Lψ( )) = 0 o i6=n, n+1. Fo i > n+1, his is P oposi ion
5. Fo i < n i is jus Poinca ´e duali y, since An
¯
kis smoo h and Lψ( )is lisse.
ii) Fo e e y ∈W(k), he L- unc ion
L(T, Lψ( ))(−1)n+1 =
2n
Y
i=0
de (1 −T·F|Hi
c(An
¯
k,Lψ( )))(−1)n+i
9