Twisted Exponential Sums

0.38 MB·

by  Lei Fu

Twisted Exponential Sums
Lei Fu
7
0 Chern Instituteof Mathematics and LPMC, NankaiUniversity,Tianjin 300071, P. R.China
0
[email protected]
2

n
a
J Abstract

0
3
∗
Letkbeafinitefieldofcharacteristicp,laprimenumberdistincttop,ψ :k →Ql anontrivial
]
T additive character, and χ:k∗n →Q∗ a character on k∗n. Then ψ defines an Artin-Schreier sheaf
l
N
L on the affine line A1, and χ defines a Kummer sheaf K on the n-dimensional torus Tn. Let
. ψ k χ k
h
t f ∈ k[X ,X−1,...,X ,X−1] be a Laurent polynomial. It defines a k-morphism f : Tn → A1.
a 1 1 n n k k
m
In this paper, we calculate the dimensions and weights of Hi(Tn,K ⊗f∗L ) under some non-
c k¯ χ ψ
[
degeneracy conditions on f. Our results can be used to estimate sums of the form
2
v
4 χ (f (x ,...,x ))···χ (f (x ,...,x ))ψ(f(x ,...,x )),
1 1 1 n m m 1 n 1 n
6
1 x1,..X.,xn∈k∗
7
where χ ,...,χ : k∗ → C∗ are multiplicative characters, ψ : k → C∗ is a nontrivial additive
0 1 m
6
character, and f ,...,f ,f are Laurent polynomials.
0 1 m
/
h
t Key words: Toric scheme, perverse sheaf, weight.
a
m
v: Mathematics Subject Classification: 14G15, 14F20, 11L40.
i
X
r
a
0. Introduction
Letk beafinitefieldwithq elementsofcharacteristicp,letχ ,...,χ :k∗ →C∗ benontrivial
1 m
multiplicative characters,let ψ :k →C∗ be a nontrivial additive character, and let
f (X ,...,X ),...,f (X ,...,X ),f(X ,...,X )∈k[X ,X−1,...,X ,X−1]
1 1 n m 1 n 1 n 1 1 n n
1
beLaurentpolynomials. Wemaketheconventionthatχ (0)=0(i=1,...,m). Innumbertheory,
i
we are often lead to study the sum
S = χ (f (x ,...,x ))···χ (f (x ,...,x ))ψ(f(x ,...,x )).
1 1 1 1 n m m 1 n 1 n
x1,..X.,xn∈k∗
For this purpose, let’s consider another sum
S = χ−1(x )···χ−1(x )
2 1 n+1 m n+m
x1,...,Xxn+m∈k∗
ψ(f(x ,...,x )+x f (x ,...,x )+···+x f (x ,...,x )).
1 n n+1 1 1 n n+m m 1 n
We have
S
2
= χ−1(x )ψ(x f (x ,...,x )) ··· χ−1(x )ψ(x f (x ,...,x ))
1 n+1 n+1 1 1 n m n+m n+m m 1 n
x1,..X.,xn∈k∗xn+1...X,xn+m∈k∗(cid:0) (cid:1) (cid:0) (cid:1)
ψ(f(x ,...,x ))
1 n
= χ−1(x )ψ(x f (x ,...,x )) ··· χ−1(x )ψ(x f (x ,...,x ))
 1 n+1 n+1 1 1 n   m n+m n+m m 1 n 
x1,..X.,xn∈k∗ xnX+1∈k∗ xn+Xm∈k∗
 ψ(f(x ,...,x )).   
1 n
For i=1,...,m and x ,...,x ∈k∗, if f (x ,...,x )=0, we have
1 n i 1 n
χ−1(x )ψ(x f (x ,...,x ))=0;
i n+i n+i i 1 n
xnX+i∈k∗
if f (x ,...,x )6=0, we have
i 1 n
x
χ−1(x )ψ(x f (x ,...,x )) = χ−1 ψ(x)
i n+i n+i i 1 n i f (x ,...,x )
xnX+i∈k∗ xX∈k∗ (cid:18) i 1 n (cid:19)
= χ (f (x ,...,x ))G(χ ,ψ),
i i 1 n i
where
G(χ ,ψ)= χ−1(x)ψ(x)
i i
x∈k∗
X
is the Gauss sum. So in any case, we have
χ−1(x )ψ(x f (x ,...,x ))=χ (f (x ,...,x ))G(χ ,ψ).
i n+i n+i i 1 n i i 1 n i
xnX+i∈k∗
2
Hence
S = χ (f (x ,...,x ))G(χ ,ψ)···χ (f (x ,...,x ))G(χ ,ψ)ψ(f(x ,...,x ))
2 1 1 1 n 1 m m 1 n m 1 n
x1,..X.,xn∈k∗
= G(χ ,ψ)···G(χ ,ψ)S .
1 m 1
As the Gauss sums are well-understood, the study of S is reduced to the study of S .
1 2
In this paper, we use l-adic cohomology theory to study sums of the form
χ (x )···χ (x )ψ(f(x ,...,x )),
1 1 n n 1 n
xXi∈k∗
where χ ,...,χ are multiplicative characters (nontrivial or trivial). Note that S is of this form.
1 n 2
Our results complete those in [DL], where the caseof trivialχ is treated. We follow the approach
i
initiated by Denef and Loeser.
We first associate geometric objects to the above data. The Kummer covering
[q−1]:Tn →Tn, x7→xq−1
k k
on the torus Tn =Speck[X ,X−1,...,X ,X−1] defines a Tn(k)-torsor
k 1 1 n n k
1→Tn(k)→Tn [q→−1]Tn →1,
k k k
where Tn(k) = Hom (Speck,Tn) is the group of k-rational points in Tn. Fix a prime number l
k k k k
distinct to p. Let χ : Tn(k) = k∗n → Q∗ be a character. Pushing-forward the above torsor by
k l
χ−1, we get a lisse Q -sheaf K on Tn of rank 1. We call K the Kummer sheaf associated to χ.
l χ k χ
For any rational point x in Tn(k′) = Hom (Speck′,Tn) with value in a finite extension k′ of k,
k k k
we have
Tr(Fx,(Kχ)x¯)=χ(Normk′/k(x)),
where F is the geometric Frobenius element at x.
x
The Artin-Schreier covering
P :A1 →A1, x7→xq −x
k k
defines an A1(k)-torsor
k
0→A1(k)→A1 →P A1 →0,
k k k
3
whereA1(k)=Hom (Speck,A1)isthegroupofk-rationalpointsinA1. Letψ :A1(k)=k →Q∗
k k k k k l
beanadditivecharacter. Pushing-forwardthistorsorbyψ−1,wegetalisseQ -sheafL ofrank1on
l ψ
A1,whichwecalltheArtin-Schreiersheaf. ForanyrationalpointxinA1(k′)=Hom (Speck′,A1)
k k k k
with value in a finite extension k′ of k, we have
Tr(Fx,(Lψ)x¯)=ψ(Trk′/k(x)),
where F is the geometric Frobenius element at x.
x
Let
f = a Xi ∈k[X ,X−1,...,X ,X−1]
i 1 1 n n
i∈Zn
X
be a Laurent polynomial. The Newton polyhedron ∆ (f) of f at ∞ is the convex hull in Rn of
∞
the set {i∈Zn|a 6=0}∪{0}.We say f is non-degenerate with respect to ∆ (f) if for any face τ
i ∞
of ∆ (f) not containing 0, the locus of
∞
∂f ∂f
τ τ
=···= =0
∂X ∂X
1 n
in Tn is empty, where
k
f = a Xi.
τ i
i∈τ
X
This is equivalent to saying that the morphism
f :Tn =SpecA[X ,X−1,...,X ,X−1]→A1 =Speck[T]
τ k 1 1 n n k
defined by the k-algebra homomorphism
k[T]→k[X ,X−1,...,X ,X−1], T 7→f
1 1 n n τ
is smooth.
The first main result of this paper is the following theorem.
Theorem 0.1. Let f : Tn → A1 be a k-morphism defined by a Laurent polynomial f ∈
k k
k[X ,X−1,...,X ,X−1] that is non-degenerate with respect to ∆ (f) and let K be a Kum-
1 1 n n ∞ χ
mer sheaf on Tn. Suppose dim(∆ (f))=n. Then
k ∞
(i) Hi(Tn,K ⊗f∗L )=0 for i6=n.
c k¯ χ ψ
(ii) dim(Hn(Tn,K ⊗f∗L ))=n!vol(∆ (f)).
c k¯ χ ψ ∞
4
(iii) If 0 is an interior point of ∆ (f), then Hn(Tn,K ⊗f∗L ) is pure of weight n.
∞ c k¯ χ ψ
Herethe conclusionof(iii) meansthatforanyeigenvalueλofthe geometricFrobenius element
F inGal(k¯/k)actingonHn(Tn,K ⊗f∗L ),λisanalgebraicnumber,andallthegaloisconjugates
c k¯ χ ψ
of λ have archimedean absolute value qn2.
Note that we have
[q−1] [q−1]∗f∗L ∼= (K ⊗f∗L ),
∗ ψ χ ψ
χ
M
where [q−1]:Tn →Tn is the Kummer covering,and in the direct sum on the right-hand side, χ
k k
goes over the set of all characters χ : Tn(k) → Q . The composition f ◦[q−1] is defined by the
k l
Laurent polynomial
f′(X ,...,X )=f(Xq−1,...,Xq−1).
1 n 1 n
Note that f′ is also non-degenerate with respect to its Newton polyhedron at ∞. We have
Hi(Tn,f′∗L ) ∼= Hi(Tn,[q−1] [q−1]∗f∗L )
c k¯ ψ c k¯ ∗ ψ
∼= Hi(Tn,K ⊗f∗L ).
c k¯ χ ψ
χ
M
So Hi(Tn,K ⊗f∗L ) aredirectfactorsof Hi(Tn,f′∗L ). Hence Theorem0.1(i) and(iii) follow
c k¯ χ ψ c k¯ ψ
directly from the main theorem 1.3 in [DL] applied to f′. Using [I1] 2.1, one can show
χ (Tn,K ⊗f∗L )=χ (Tn,f∗L ),
c k¯ χ ψ c k¯ ψ
where χ denotes the Euler characteristic for the cohomology with compact support. Hence The-
c
orem 0.1 (ii) can also be deduced from [DL] 1.3.
In this paper, we give a proof of Theorem 0.1 independent of the main theorem of [DL]. On
the other hand, oursecondmain Theorem0.4on the weightsof Hn(Tn,K ⊗f∗L ) doesn’t seem
c k¯ χ ψ
to follow from the corresponding theorem in [DL].
Corollary 0.2. Letf ∈k[X ,X−1,...,X ,X−1]be aLaurentpolynomialthatis non-degenerate
1 1 n n
with respect to ∆ (f) and suppose dim(∆ (f)) = n. Then for any multiplicative characters
∞ ∞
χ ,...,χ :k∗ →C∗ and any nontrivial additive character ψ :k →C∗, we have
1 n
n
| χ1(x1)···χn(xn)ψ(f(x1,...,xn))|≤n!vol(∆∞(f))q2.
xXi∈k∗
5
Proof. Note that the values of χ and ψ are algebraic integers. In particular, we may consider
i
them to have values in Q . Let χ:k∗n →Q∗ be the character defined by
l l
χ(x)=χ (x )...χ (x )
1 1 n n
for any x=(x ,...,x )∈k∗n. We then have
1 n
χ (x )···χ (x )ψ(f(x ,...,x ))= Tr(F ,(K ⊗f∗L ) ).
1 1 n n 1 n x χ ψ x¯
xXi∈k∗ x∈XTnk(k)
By the Grothendieck trace formula ([SGA 41], [Rapport] Th´eor`eme 3.2), we have
2
2n
Tr(F ,(K ⊗f∗L ) )= (−1)iTr(F,Hi(Tn,K ⊗f∗L )).
x χ ψ x¯ c k¯ χ ψ
x∈XTnk(k) Xi=0
By [D] 3.3.1, for any eigenvalue λ of F acting on Hn(Tn,K ⊗f∗L ), λ is an algebraic number
c k¯ χ ψ
and all its galois conjugates have archimedean absolute value ≤qn2. Combined with Theorem 0.1
(i) and (ii), we get
2n
| (−1)iTr(F,Hci(Tnk¯,Kχ⊗f∗Lψ))|≤n!vol(∆∞(f))qn2.
i=0
X
So we have
n
| χ1(x1)···χn(xn)ψ(f(x1,...,xn))|≤n!vol(∆∞(f))q2.
xXi∈k∗
Combining Corollary 0.2 with the discussion at the beginning, and using the fact that Gauss
sums have absolute value q21, we get the following.
Corollary0.3. Letf,f ,...,f ∈k[X ,X−1,...,X ,X−1]beLaurentpolynomials,χ ,...,χ :
1 m 1 1 n n 1 m
k∗ → C∗ nontrivial multiplicative characters, and ψ : k → C∗ a nontrivial additive character.
Suppose the Laurent polynomial
F(x ,...,x ,x ,...,x )=f(x ,...,x )+x f (x ,...,x )+...+x f (x ,...,x )
1 n n+1 n+m 1 n n+1 1 1 n n+m m 1 n
is non-degenerate with respect to ∆ (F), and dim(∆ (F))=m+n. Then we have
∞ ∞
| χ (f (x ,...,x ))···χ (f (x ,...,x ))ψ(f(x ,...,x ))|≤(n+m)!vol(∆ (F))qn/2.
1 1 1 n m m 1 n 1 n ∞
xXi∈k∗
6
Under the assumption of Theorem 0.1, let
E(Tn,f,χ)= e Tw,
k w
w∈Z
X
where e is the number of eigenvalues counted with multiplicities of the geometric Frobenius
w
elementF inGal(k¯/k)actingonHn(Tn,K ⊗f∗L )withweightw. Ournextgoalistodetermine
c k¯ χ ψ
E(Tn,f,χ).
k
Foranyconvexpolyhedralconeδ inRn with0beingaface,definetheconvexpolytopepoly(δ)
to be the intersection of δ with a hyperplane in Rn which does not contain 0 and intersects each
one dimensional face of δ. Note that poly(δ) is defined only up to combinatorial equivalence. For
any convex polytope ∆ in Rn and any face τ of ∆, define cone (τ) to be the cone generated by
∆
u′−u (u′ ∈∆, u∈τ), and define cone◦(τ) to be the image of cone (τ) in Rn/span(τ −τ). Note
∆ ∆
that 0 is a face of cone◦ (τ). We define polynomials α(δ) and β(∆) in one variable T inductively
∆
by the following formulas:
α({0}) = 1,
β(∆) = (T2−1)dim(∆)+ (T2−1)dim(τ)α(cone◦(τ)),
∆
τ faceXof ∆,τ6=∆
α(δ) = trunc ((1−T2)β(poly(δ))),
≤dim(δ)−1
where trunc (·) denotes taking the degree ≤d part of a polynomial. These polynomials are first
≤d
introducedbyStanley[S].Notethatα(δ)andβ(∆)onlyinvolveevenpowersofT,andtheydepend
only on the combinatorial types of δ and ∆. If δ is a simplicial cone, that is, if δ is generated by
linearly independent vectors, then using induction on dim(δ), one can verify α(δ)=1.
Let χ : Tn(k) → Q∗ be a character. For a rational convex polytope ∆ in Rn of dimension n,
k l
let T be the set of faces τ of ∆ so that τ 6=∆, 0∈τ, and K ∼=p∗K for a Kummer sheaf K on
χ τ τ τ
the torus T =Speck[Zn∩span(τ −τ)], where
τ
p :Tn =Speck[Zn]→T =Speck[Zn∩span(τ −τ)]
τ k τ
is the morphism defined by the canonical homomorphism
k[Zn∩span(τ −τ)]֒→k[Zn].
7
Define
e(∆,χ)=n!vol(∆)+ (−1)n−dim(τ)(dim(τ))!vol(τ)α(cone◦ (τ))(1)
∆
τ∈T
X
and define a polynomial E(∆,χ) inductively by
E(∆,χ)=e(∆,χ)Tn− (−1)n−dim(τ)E(τ,χ )α(cone◦ (τ)).
τ ∆
τ∈T
X
Our second main result is the following.
Theorem 0.4. Let f : Tn → A1 be a k-morphism defined by a Laurent polynomial f ∈
k k
k[X ,X−1,...,X ,X−1] that is non-degenerate with respect to ∆ (f) and let K be a Kum-
1 1 n n ∞ χ
mer sheaf on Tn. Suppose dim(∆ (f))=n. Let
k ∞
E(Tn,f,χ)= e Tw,
k w
w∈Z
X
where e is the number of eigenvalues counted with multiplicities of the geometric Frobenius
w
element F in Gal(k¯/k) acting on Hn(Tn,K ⊗ f∗L ) with weight w. Then E(Tn,f,χ) is a
c k¯ χ ψ k
polynomial of degree ≤n, and
E(Tn,f,χ) = E(∆ (f),χ),
k ∞
e = e(∆ (f),χ).
n ∞
Corollary 0.5. Let f : Tn → A1 be a k-morphism defined by a Laurent polynomial f ∈
k k
k[X ,X−1,...,X ,X−1] that is non-degenerate with respect to ∆ (f) and let K be a Kummer
1 1 n n ∞ χ
sheaf on Tn. Suppose dim(∆ (f))=n and suppose for any face τ of ∆ (f) of codimension one
k ∞ ∞
containing 0, K is not the inverse image of any Kummer sheaf on T =Speck[Zn∩span(τ −τ)]
χ τ
under the morphism
p :Tn =Speck[Zn]→T =Speck[Zn∩span(τ −τ)].
τ k τ
Then Hn(Tn,K ⊗f∗L ) is pure of weight n.
c k¯ χ ψ
Remark 0.6. Note that this corollary together with Theorem 0.1 is Theorem 4.2 in [AS], except
that Adolphson and Sperber prove the theorem for almost all p.
8
ProofofCorollary0.5. Underourassumption,thesetT offacesτ of∆ (f)sothatτ 6=∆ (f),
∞ ∞
0 ∈ τ, and K ∼= p∗K for a Kummer sheaf K on the torus T = Speck[Zn ∩span(τ −τ)] is
χ τ τ τ τ
empty. Therefore
E(∆ (f),χ)=e(∆ (f),χ)Tn.
∞ ∞
By Theorem 0.4, this implies
E(Tn,f,χ)=e(∆ (f),χ)Tn,
k ∞
and hence Hn(Tn,K ⊗f∗L ) is pure of weight n.
c k¯ χ ψ
For a rational convex polytope ∆ in Rn with dimension n, recall that T is the set of faces
τ of ∆ so that τ 6= ∆, 0 ∈ τ, and K ∼= p∗K for a Kummer sheaf K on the torus T =
χ τ τ τ τ
Speck[Zn∩span(τ −τ)]. Define
V (∆,χ)= vol(τ)
i
τ∈T,Xdim(τ)=i
for 0≤i≤n−1 and define
V (∆,χ)=vol(∆).
n
Let τ be the smallest face of ∆ containing 0. We say ∆ is simplicial at the origin if there are
0
exactly n−dim(τ ) faces of ∆ that have dimension n−1 and contain τ . If ∆ is simplicial at
0 0
the origin and τ is a face containing 0, then the number of faces of ∆ that have dimension k and
n−dim(τ)
contain τ is .
n−k
(cid:18) (cid:19)
The following corollary is Theorem 4.8 in [AS].
Corollary 0.7. Let f : Tn → A1 be a k-morphism defined by a Laurent polynomial f ∈
k k
k[X ,X−1,...,X ,X−1] that is non-degenerate with respect to ∆ (f) and let K be a Kummer
1 1 n n ∞ χ
sheaf on Tn. Suppose dim(∆ (f)) = n and suppose ∆ (f) is simplicial at the origin. Let e
k ∞ ∞ w
be the number of eigenvalues counted with multiplicities of the geometric Frobenius element F in
Gal(k¯/k) acting on Hn(Tn,K ⊗f∗L ) with weight w. Then we have
c k¯ χ ψ
w
n−i
e = (−1)w−ii! V (∆ (f),χ).
w n−w i ∞
i=0 (cid:18) (cid:19)
X
Proof. The hypothesis that ∆ (f) is simplicial at the origin implies that
∞
α(cone◦ (τ))=1
∆∞(f)
9
for any face τ of ∆ (f) containing 0. By Theorem 0.4, we have
∞
e = e(∆ (f),χ)
n ∞
= n!vol(∆ (f))+ (−1)n−dim(τ)(dim(τ))!vol(τ)
∞
τ∈T
X
n−1
= n!V (∆ (f),χ)+ (−1)n−ii!V (∆ (f),χ)
n ∞ i ∞
i=0
X
n
= (−1)n−ii!V (∆ (f),χ).
i ∞
i=0
X
This proves our assertion for w=n. We use induction on n. Under our assumption, we have
E(∆ (f),χ)=e(∆ (f),χ)Tn− (−1)n−dim(τ)E(τ,χ ).
∞ ∞ τ
τ∈T
X
For w ≤ n−1, taking the coefficients of Tw on both sides of the above equality and applying
Theorem 0.4 and the induction hypothesis to the pairs (τ,χ ) (τ ∈T), we get
τ
w
dim(τ)−i
e =− (−1)n−dim(τ) (−1)w−ii! vol(τ′).
w dim(τ)−w
τ∈T,w≤dim(τ)≤n−1 i=0 (cid:18) (cid:19)τ′≺τ,dim(τ′)=i,τ′∈T
X X X
So we have
w
dim(τ)−i
e = (−1)w−i+n+1i! (−1)dim(τ) vol(τ′).
w dim(τ)−w
Xi=0 τ′∈T,Xdim(τ′)=i τ′≺τ,w≤Xdim(τ)≤n−1 (cid:18) (cid:19)
By our assumption, for each τ′ containing 0 of dimension i, the number of faces of ∆ (f) that
∞
n−i
have dimension k and contain τ′ is . So we have
n−k
(cid:18) (cid:19)
w n−1
k−i n−i
e = (−1)w−i+n+1i! (−1)k vol(τ′).
w k−w n−k
Xi=0 τ′∈T,Xdim(τ′)=ikX=w (cid:18) (cid:19)(cid:18) (cid:19)
We have
n−1
k−i n−i
(−1)k
k−w n−k
k=w (cid:18) (cid:19)(cid:18) (cid:19)
X
n−1
n−w n−i
= (−1)k
k−w n−w
k=w (cid:18) (cid:19)(cid:18) (cid:19)
X
n−1−w
n−i n−w
= (−1)w (−1)j
n−w j
(cid:18) (cid:19) j=0 (cid:18) (cid:19)
X
n−i
= (−1)n−1 .
n−w
(cid:18) (cid:19)
10