On $L$-functions of certain exponential sums

0.42 MB·English

by  Jun Zhang

On L-functions of certain exponential sums
ByJunZhangatTianjinandWeiduanFengatShanghai
3
1
0 Abstract. LetF denotethefinitefieldoforderq (apowerofaprimep). Westudythe
q
2 p-adic valuations for zeros of L-functions associated with exponential sums of the following

n familyofLaurentpolynomials
a
J
1 1 1

0 f(x1,x2,··· ,xn+1) = a1xn+1(x1+ )+···+anxn+1(xn+ )+an+1xn+1+
x x x
1 1 n n+1

] where ai ∈ F∗q, i = 1,2,··· ,n+1. When n = 2, the estimate of the associated exponen-
T
tialsumappearsinIwaniec’swork,andAdolphsonandSperbergavecomplexabsolutevalues
N
for zeros of the corresponding L-function. Using the decomposition theory of Wan, we de-
.
h terminethegenericNewtonpolygon(q-adicvaluesofthereciprocalzeros)oftheL-function.
t
a Working on the chain level version of Dwork’s trace formula and using Wan’s decomposition
m
theory,weareabletogiveanexplicitHassepolynomialforthegenericNewtonpolygoninlow
[ dimensions,i.e.,n ≤ 3.

1 MSC2010Codes: 11S40,11T23,11L07
v
1
3
0 1. Introduction
2
.
1 L-functionshavebeenapowerfultooltoinvestigateexponentialsumsinnumbertheory.
0
Toestimateanexponentialsum,peopleareinterestedinthezerosandpolesofthecorrespond-
3
1 ingL-function. Mathematiciansstudythenumberofthesezerosandpoles,thecomplexabso-
: lutevalues,l-adicabsolutevaluesofthemforprimesl (cid:54)= p,andp-adicabsolutevaluesofthem,
v
i especiallyforsomeinterestingvarietiesandexponentialsums[6,7,13–15]. Deligne’stheorem
X
givesthegeneralinformationforcomplexabsolutevaluesofthezerosandpolesofL-function.
r
a Forl-adicabsolutevalues,itiswell-knownthatifl (cid:54)= pthenallthezerosandpoleshavel-adic
absolutevalue1. However,forp-adicabsolutevalues,itisstillverymysterious[9–12,19,20],
especiallyinhigherdimensions.
Inthispaper,weconsiderthefollowingfamilyofLaurentpolynomials
(1.1)
1 1 1
f(x ,x ,··· ,x ) = a x (x + )+···+a x (x + )+a x +
1 2 n+1 1 n+1 1 n n+1 n n+1 n+1
x x x
1 n n+1
The research of Jun Zhang is supported by the National Key Basic Research Program of China (Grant
No. 2013CB834204), and the National Natural Science Foundation of China (Nos. 61171082, 10990011 and
60872025).
ZhangandFeng,L-func.ofcert.exp.sums 1
wherea ∈ F∗, i = 1,2,··· ,n+1. Theexponentialsumassociatedtof isdefinedtobe
i q
(cid:88)
S∗(f) = ζTrkf(x1,...,xn),
k p
x1,...,xn∈F∗qk
where ζ is a fixed primitive p-th root of unity in the complex numbers and Tr denotes the
p k
tracemapfromthek-thextendedfieldF totheprimefieldF . Whenn = 2,theestimateof
qk p
exponentialsumS∗(f)isvitalinanalyticnumbertheory. ItappearsinIwaniec’sworkonsmall
k
eigenvalues of the Laplace-Beltrami operator acting on automorphic functions with respect to
thegroupΓ (p).
0
TounderstandthesequenceS∗(f) ∈ Q(ζ )(1 ≤ k < ∞)ofalgebraicintegers,westudy
k p
theL-functionassociatedtoS∗(f)
k
(cid:32)(cid:88)∞ Tk(cid:33)
L∗(f,T) = exp S∗(f) .
k k
k=1
By the theorem of Adolphson and Sperber [1], the L-function L∗(f,T)(−1)n for non-
degenerate f is a polynomial of degree (n + 1)!Vol(∆), where ∆ = ∆(f) is the Newton
polyhedronoff definedexplicitlylater. AstheoriginisaninteriorpointoftheNewtonpoly-
hedron∆,bythetheoremofDenefandLoeser[5],wehave
Theorem1.1. Assumethatf isnon-degenerate,thatis,
(cid:89)
(2c a +2c a +···+2c a +a ) (cid:54)= 0.
1 1 2 2 n n n+1
(c1,c2,···,cn)∈{±1}n
Then,theL-functionL∗(f,T)associatedtotheexponentialsumS∗(f)ispureofweightn+1,
k
i.e.,
(n+1)!Vol(∆(f))
L∗(f,T)(−1)n = (cid:89) (1−α T)
i
i=1
withthecomplexabsolutevalue|α | = q(n+1)/2.
i
On the other hand, for each l-adic absolute value |·| with prime l (cid:54)= p, the reciprocal
l
zerosα arel-adicunits: |α | = 1.Sowenextstudythep-adicslopesofthereciprocalzeros
i i l
ofL∗(f,T)(−1)n ofsuchafamilyofexponentialsumsfortheremainingprimep. Whenn = 1,
suchnon-degenerateLaurentpolynomialsf arealwaysordinaryandtheHodgepolygonisvery
clear. So,weonlyconsidern ≥ 2. Oneofourmainresultsinthispaperis
Theorem1.2. Assumethatf oftheform(1.1)isnon-degenerate.
(i)ThepolynomialL∗(f,T)(−1)n hasdegree2n+1.
(ii)Thereexistsanon-zeropolynomialh (∆)(a ,a ,··· ,a ) ∈ F [a ,a ,··· ,a ]
p 1 2 n+1 p 1 2 n+1
such that iff has coefficientsa ,a ,··· ,a ∈ F∗ verifyingh (∆)(a ,a ,··· ,a ) (cid:54)= 0,
1 2 n+1 q p 1 2 n+1
thenforeachk = 0,1,··· ,n+1,thenumberofreciprocalzerosofL∗(f,T)(−1)n withq-adic
slopek is
(cid:18) (cid:19)
n+1
,
k
andforanyrationalnumberk (cid:54)∈ {0,1,...,n+1},thereisnoreciprocalzeroofL∗(f,T)(−1)n
withq-adicslopek.
2 ZhangandFeng,L-func.ofcert.exp.sums
We identify a Laurent polynomial f of the form (1.1) with the vector of its coefficients
a = (a ,a ,··· ,a ),astheyareone-to-onecorrespondenttoeachother. LetM (∆) ⊆ An+1
1 2 n+1 p
betheopensubsetconsistingallnon-degenerateLaurentpolynomialswithNewtonpolyhedron
∆,explicitly
(cid:89)
M (∆) = {a ∈ An+1|a ···a (±2a +±2a +···+±2a +a ) (cid:54)= 0}.
p 1 n+1 1 2 n n+1
ByTheorem1.2(ii),wehavedeterminedp-adicslopesofallreciprocalzerosofL∗(f,T)(−1)n
for “almost all” f’s in M (∆) except a Zariski closed subset defined by the polynomial
p
h (∆)(a ,a ,··· ,a ). Thepolynomialh (∆)iscalleda HassepolynomialoftheNewton
p 1 2 n+1 p
polyhedron∆. Next,wetrytogiveanexplicitHassepolynomial. Wewillseethatitisalready
quite complicated to explicitly determine the Hasse polynomial for ∆ = ∆(f) we consider,
a priori for more general polyhedrons. For low dimensions, we obtain the following explicit
formulaeofHassepolynomialsoftheNewtonpolyhedron∆oftheLaurentpolynomialsf of
theform(1.1). WorkingonthechainlevelversionofDwork’straceformula,weprove
Theorem1.3. Whenn = 2,aHassepolynomialcanbetakentobe
h (∆)(a ,a ,a ) = (cid:88) 1 a2va2uap−1−2u−2v.
p 1 2 3 (u!v!(p−1−2u−2v)!)2 1 2 3
0≤u+v≤p−1
2
u,v∈Z
For n = 3, it has already been a bit more complicated than the case n = 2. Using the
chain level version of Dwork’s trace formula, we can easily give the condition when Newton
polygonandHodgepolygoncoincideatthefirstbreakpointjustaswetreatinthecasen = 2.
However, to find out when they meet at the second break point with the same method above,
itneedsustocomputethedeterminantofamatrixofsize33×33whoseentriesareallpoly-
nomials. In fact, it even requires a while to write down the matrix, let alone to compute the
determinant. Todealwiththisproblem,weusetheboundarydecompositiontheoremofWanto
dividethe“characteristicpowerseries”det(I−TA (f))into“interiorpieces”andthenhandle
1
thempiecebypiece.
Theorem1.4. Whenn = 3,aHassepolynomialcanbetakentobe
h (∆)(a) = T(a) (cid:88) 1 a2ua2va2wap−1−2u−2v−2w,
p (u!v!w!(p−1−2u−2v−2w)!)2 1 2 3 4
0≤u+v+w≤p−1
2
u,v,w∈Z
wherea = (a ,a ,a ,a ) ∈ M (∆)andT(a)isexplicitlypresentedbyFormula(3.5)which
1 2 3 4 p
isessentiallythedeterminantofa4×4matrixwhoseentriesareallpolynomials.
Therestofthispaperisorganizedasfollows. Tomakethispaperself-contained,wefirst
recallsomebaicconceptsandresultsaboutL-functions,NewtonpolygonandHodgepolygon
of L-functions. Then we review some powerful tools to study L-functions, such as Dwork’s
p-adic method and Wan’s decomposition theory. Finally, we use these methods to investigate
L-functionsofthefamilyofLaurentpolynomialsweconsiderin(1.1).
ZhangandFeng,L-func.ofcert.exp.sums 3
2. Preliminaries
2.1. ExponentialsumsandL-functions. LetF bethefinitefieldofq elementswith
q
characteristic p. For each positive integer k, let F be the finite extension of F of degree
qk q
k. Let ζ be a fixed primitive p-th root of unity in the complex numbers. For any Laurent
p
polynomialf(x ,...,x ) ∈ F [x ,x−1,...,x ,x−1],weformtheexponentialsum
1 n q 1 1 n n
(cid:88)
S∗(f) = ζTrkf(x1,...,xn),
k p
x1,...,xn∈F∗qk
where F∗ denotes the multiplicative group of non-zero elements in F and Tr denotes
qk qk k
the trace map from F to the prime field F . To understand the sequence S∗(f) ∈ Q(ζ )
qk p k p
(1 ≤ k < ∞)ofalgebraicintegers,weformthegeneratingfunctionofS∗(f)
k
(cid:32)(cid:88)∞ Tk(cid:33)
L∗(f,T) = exp S∗(f) ,
k k
k=1
which is called the L-function of the exponential sum S∗(f). The study of L∗(f,T) has fun-
k
damental importance in number theory. For example, it connects with the zeta functions over
finitefields. Consider
U (F ) = {x ,...,x ∈ F∗ | f(x ,...,x ) = 0}
f q 1 n q 1 n
theaffinetorichypersurfacedefinedbyaLaurentpolynomial
f(x ,...,x ) ∈ F [x ,x−1,...,x ,x−1].
1 n q 1 1 n n
Let#U (F )denotethenumberofsolutionsoff in(F∗ )n. Itszetafunctionisgivenby
f qk qk
(cid:32)(cid:88)∞ Tk(cid:33)
Z(U ,T) = exp #U (F ) .
f f qk k
k=1
Itcanbeeasilyshownthat
(2.1) qk#U (F ) = (qk −1)n+S∗(x f),
f qk k 0
andwehave
Z(U ,qT) = Z(Gn,T)L∗(x f,T).
f m 0
Thusweseethatinordertostudythezetafunction,itsufficestostudytheL-functionL∗(x f,T).
0
Alsothe studyof exponentialsumsand theassociatedL-functionshas importantapplications
inanalyticnumbertheory,andsomeappliedmathematicssuchascodingtheory,cryptography,
etc.
ByatheoremofDwork-Bombieri-Grothendieck,thefollowinggeneratingL-functionis
arationalfunction:
(2.2) L∗(f,T) = exp(cid:32)(cid:88)∞ S∗(f)Tk(cid:33) = (cid:81)di=11(1−αiT),
k k (cid:81)d2 (1−β T)
k=1 j=1 j
4 ZhangandFeng,L-func.ofcert.exp.sums
where zeros α (1 ≤ i ≤ d ) and poles β (1 ≤ j ≤ d ) are non-zero algebraic integers.
i 1 j 2
Equivalently,foreachpositiveintegerk,wehavetheformula
(cid:88)d2 (cid:88)d1
(2.3) S∗(f) = βk − αk.
k j i
j=1 i=1
Thus,ourfundamentalquestionaboutthesumsS∗(f)isreducedtounderstandingtherecipro-
k
calzerosα (1 ≤ i ≤ d )andβ (1 ≤ j ≤ d ).
i 1 j 2
Withoutanysmoothnessconditionoff, onedoesnotevenknowexactlythenumberd
1
of zeros and the number d of poles, although good upper bounds are available, see [3]. On
2
the other hand, Deligne’s theorem on the Riemann hypothesis [4] gives the following general
informationaboutthenatureofthezerosandpoles. Forthecomplexabsolutevalue|·|,itsays
|α | = qui/2, |β | = qvj/2, u ∈ Z∩[0,2n], v ∈ Z∩[0,2n]
i j i j
whereZ∩[0,2n]denotesthesetofintegersintheinterval[0,2n]. Furthermore,eachα (resp.
i
each β ) and its Galois conjugates over Q have the same complex absolute value. For each
j
l-adicabsolutevalue|·| withprimel (cid:54)= p,theα andβ arel-adicunits:
l i j
|α | = |β | = 1.
i l j l
Fortheremainingprimep,Deligne’sintegralitytheoremimpliesthat
|α | = q−ri, |β | = q−sj, r ∈ Q∩[0,n],s ∈ Q∩[0,n],
i p j p i j
where the p-adic absolute value is normalized such that |q| = 1/q. Strictly speaking, in
p
defining the p-adic absolute value, we have tacitly chosen an embedding of the field Q¯ of
algebraic numbers into an algebraic closure of the p-adic number field Q . Note that each α
p i
(resp. each β ) and its Galois conjugates over Q may have different p-adic absolute values.
j
ThepreciseversionofvarioustypesofRiemannhypothesisfortheL-functionin(2.2)isthen
to determine the important arithmetic invariants {u ,v ,r ,s }. The integer u (resp. v ) is
i j i j i j
called the weight of the algebraic integer α (resp. β ). The rational number r (resp. s ) is
i j i j
called the slope of the algebraic integer α (resp. β ) defined with respect to q. Without any
i j
smoothnessconditiononf,notmuchmoreisknownabouttheseweightsandtheslopes,since
one does not even know exactly the number d of zeros and the number d of poles. Under a
1 2
suitable smoothness condition, a great deal more is known about the weights {u ,v } and the
i j
slopes{r ,s },seeAdolphson-Sperber[1],Denef-Loesser[5]andWan[16,18].
i j
Toinvestigatetheslopes{r ,s },Newtonpolygonwasintroduced.
i j
2.2. NewtonpolygonandHodgepolygon. Let
J
(cid:88)
f(x ,...,x ) = a xVj, a (cid:54)= 0,
1 n j j
j=1
be a Laurent polynomial in F [x ,x−1,...,x ,x−1]. Each V = (v ,...,v ) is a lattice
q 1 1 n n j 1j nj
pointinZnandthepowerxVj meanstheproductxv11j ···xvnnj. Let∆(f)betheconvexclosure
inRn generatedbytheoriginandthelatticepointsV (1 ≤ j ≤ J). ThisiscalledtheNewton
j
ZhangandFeng,L-func.ofcert.exp.sums 5
polyhedronoff. Ifδ isasubsetof∆(f), wedefinetherestrictionoff toδ tobetheLaurent
polynomial
(cid:88)
fδ = a xVj
j
Vj∈δ
Definition 2.1. The Laurent polynomial f is called non-degenerate if for each closed
face δ of ∆(f) of arbitrary dimension which does not contain the origin, the n partial deriva-
tives
∂fδ ∂fδ
{ ,..., }
∂x ∂x
1 n
hasnocommonzeroswithx ···x (cid:54)= 0overthealgebraicclosureofF .
1 n q
Assumenowthatf isnon-degenerate,thentheL-functionL∗(f,T)(−1)n−1 isapolyno-
mial of degree n!V(f) by a theorem of Adolphson-Sperber [1] proved using p-adic methods,
where V(f) denotes the volume of ∆(f). The complex absolute values (or the weights) of
then!V(f)zeroscanbedeterminedexplicitlybyatheoremofDenef-Loeser[5]provedusing
l-adic methods. They depend only on ∆, not on the specific f and p as long as f is non-
degenerate with ∆(f) = ∆. Hence, the weights have no variation as f and p varies. As
indicated above, the l-adic absolute values of the zeros are always 1 for each prime l (cid:54)= p.
Thus, there remains the intriguing question of determining the p-adic absolute values (or the
slopes)ofthezeros. Thisisthep-adicRiemannhypothesisfortheL-functionL∗(f,T)(−1)n−1.
Equivalently,thequestionistodeterminetheNewtonpolygonofthepolynomial
n!V(f)
L∗(f,T)(−1)n−1 = (cid:88) A (f)Ti, A (f) ∈ Z[ζ ].
i i p
i=0
TheNewtonpolygonofL∗(f,T)(−1)n−1,denotedbyNP(f),isdefinedtobethelowerconvex
closureinR2 ofthefollowingpoints
(k,ord A (f)), k = 0,1,...,n!V(f).
q k
And a point in {(k,ord A (f)) | k = 1,2,...,n!V(f)−1} is called a break point of the
q k
Newtonpolygoniftheleftsegmentandtherightsegmenthavedifferentslopes
Let N (∆) be the parameter space of f over F with fixed ∆(f) = ∆. Let M (∆) be
p p p
the set of non-degenerate f over F with fixed ∆(f) = ∆. It is a Zariski open smooth affine
p
subsetofN (∆). Itisnon-emptyifpislargeenough,sayp > n!V(∆). ThusM (∆)isagain
p p
asmoothaffinevarietydefinedoverF . TheGrothendieckspecializationtheorem[17]implies
p
thatasf varies,thelowestNewtonpolygon
GNP(∆,p) = inf NP(f)
f∈Mp(∆)
exitsandisattainedforallf insomeZariskiopendensesubsetofM (∆). Thelowestpolygon
p
canthenbecalledthegenericNewtonpolygon,denotedbyGNP(∆,p).
A general property is that the Newton polygon lies on or above a certain topological or
combinatoriallowerbound,calledtheHodgepolygonHP(∆)whichwedescribebellow.
Let ∆ denote the n-dimensional integral polyhedron ∆(f) in Rn containing the origin.
Let C(∆) be the cone in Rn generated by ∆. Then C(∆) is the union of all rays emanating
6 ZhangandFeng,L-func.ofcert.exp.sums
from the origin and passing through ∆. If c is a real number, we define c∆ = {cx | x ∈ ∆}.
For a point u ∈ Rn, the weight ω(u) is defined to be the smallest non-negative real number c
suchthatu ∈ c∆. Ifsuchcdoesnotexist,wedefineω(u) = ∞.
Itisclearthatω(u)isfiniteifandonlyifu ∈ C(∆). Ifu ∈ C(∆)isnottheorigin, the
rayemanatingfromtheoriginandpassingthroughuintersects∆inafaceδ ofcodimension1
thatdoesnotcontaintheorigin. Thechoiceofthedesired1-codimensionalfaceδ isingeneral
not unique unless the intersection point is in the interior of δ. Let (cid:80)n e X = 1 be the
i=1 i i
equationofthehyperplaneδ inRn,wherethecoefficientse areuniquelydeterminedrational
i
numbers not all zero. Then, by standard arguments in linear programming, one finds that the
weightfunctionω(u)canbecomputedusingtheformula:
n
(cid:88)
(2.4) ω(u) = e u
i i
i=1
where(u ,··· ,u ) = udenotesthecoordinatesofu.
1 n
Let D(δ) be the least common denominator of the rational numbers e (1 ≤ i ≤ n). It
i
followsfrom(2.4)thatforalatticepointuinC(δ),wehave
1
(2.5) ω(u) ∈ Z ,
≥0
D(δ)
where Z denotes the set of non-negative integers. It is easy to show that there are lattice
≥0
points u ∈ C(δ) such that the denominator of ω(u) is exactly D(δ). Let D(∆) be the least
commonmultipleofalltheD(δ)’s:
D(∆) = lcm D(δ),
δ
whereδ runsoverallthe1-codimensionalfacesof∆whichdonotcontaintheorigin. Thenby
(2.5),wededuce
1
(2.6) ω(Zn) ∈ Z ∪{∞}.
≥0
D(∆)
The integer D = D(∆) is called the denominator of ∆. It is the smallest positive integer for
which (2.6) holds. But be careful that there may not have a lattice point u ∈ C(∆) such that
thedenominatorofω(u)isexactlyD(∆).
Foranintegerk,let
(cid:26) (cid:27)
k
W (k) = (cid:93) u ∈ Zn | ω(u) =
∆
D
bethenumberoflatticepointsinZn withweightk/D. Thisisafinitenumberforeachk. The
Hodgenumbersaredefinedtobe
n (cid:18) (cid:19)
(cid:88) n
H (k) = (−1)i W (k−iD), k ∈ Z .
∆ ∆ ≥0
i
i=0
Hodge number H (k) is the number of lattice points of weight k/D in a certain fundamen-
∆
tal domain corresponding to a basis of the p-adic cohomology space used to compute the L-
function. Thus,H (k)isanon-negativeintegerforeachk ∈ Z . Furthermore,bycohomol-
∆ ≥0
ogytheory,
H (k) = 0, for k > nD
∆
ZhangandFeng,L-func.ofcert.exp.sums 7
and
nD
(cid:88)
H (k) = n!V(∆).
∆
k=0
Definition 2.2. The Hodge polygon HP(∆) of ∆ is defined to be the lower convex
polygoninR2 withvertices
(cid:32) k k (cid:33)
(cid:88) 1 (cid:88)
H (k), kH (k) , k = 0,1,2,...,nD.
∆ ∆
D
m=0 m=0
Thatis,thepolygonHP(∆)isthepolygonstartingfromtheoriginandhasasideofslopek/D
with horizontal length H (k) for each integer 0 ≤ k ≤ nD. For k = 1,2,...,nD −1, the
∆
point
(cid:32) k k (cid:33)
(cid:88) 1 (cid:88)
H (k), kH (k)
∆ ∆
D
m=0 m=0
iscalledabreakpointoftheHodgepolygonifH (k+1) (cid:54)= 0.
∆
ThelowerboundofAdolphsonandSperber[1]saysthatiff ∈ M (∆),thenNP(f) ≥
p
HP(∆) and they have the same endpoint. The Laurent polynomial f is called ordinary if
NP(f) =HP(∆(f)). CombiningwiththedefinitionofthegenericNewtonpolygon,wededuce
Proposition2.1. Foreveryprimepandeveryf ∈ M (∆),wehavetheinequalities
p
NP(f) ≥ GNP(∆,p) ≥ HP(∆).
Let H (∆) be the moduli space of those f ∈ M (∆) such that ∆(f) = ∆, f is non-
p p
degenerate with respect to ∆ and NP(f) =HP(∆). In Dwork’s terminology, H (∆) is called
p
theHassedomainofthegenericLaurentpolynomialf definedoverF with∆(f)containedin
p
∆, and it is a Zariski-open subset of M (∆) (possibly empty). Moreover the complement of
p
H (∆)inM (∆)isanaffinevarietydefinedbyapolynomialinthevariablesa (coefficients
p p j
off),calledtheHassepolynomialanddenotedbyh (∆). VerylittleaboutHassepolynomials
p
isknown. ItisverydifficulttocomputeHassepolynomialingeneral. Inthispaper,westudya
familyofLaurentpolynomialsanddeterminetheHassepolynomialsinlowdimensions.
2.3. Diagonallocaltheory. ALaurentpolynomialf iscalleddiagonaliff hasexactly
n non-constant terms and ∆ = ∆(f) is n-dimensional (i.e., a simplex). In this case, the L-
functioncanbecomputedexplicitlyusingGausssums. Let
n
(cid:88)
(2.7) f(x) = a xVj, a ∈ F∗,
j j q
j=1
where 0,V ,...,V are the vertices of an n-dimensional integral simplex ∆ in Rn. Let its
1 n
vertexmatrixbethenon-singularn×nmatrix
M = (V ,...,V ),
1 n
whereeachV iswrittenasacolumnvector.
j
8 ZhangandFeng,L-func.ofcert.exp.sums
Proposition2.2. Forf in(2.7),f isnon-degenerateifandonlyifpisrelativelyprime
todet(M).
Proof. Notethat∆(f)hasonlyonefaceofdimensionn−1notcontainingtheorigin.
Forthisface,letyj = ajxVj,Wehave
n n
∂f (cid:88) (cid:88)
(2.8) x = V (a xVj) = V y , (1 ≤ i ≤ n).
i ji j ji j
∂x
i
j=1 j=1
Thenpartialderivatives ∂f ,..., ∂f havenocommonzeroswithx ···x (cid:54)= 0isequivalent
∂x1 ∂xn 1 n
tothenlinearequationsofy (1 ≤ j ≤ n)havenocommonzerosin(2.8),whichisequivalent
j
tothatpisrelativelyprimetodet(M).
For any other face δ of dimension m < n−1, by a orthogonal transformation, we can
assumeδ isonthehyperplanex = ... = x = 0,whichreducetotheabovesituation.
m+1 n
Considerthesolutionsofthefollowinglinearsystem
 
r
1
 . 
(2.9) M  ..  ≡ 0(mod1), ri rational, 0 ≤ ri < 1.
 
r
n
LetS(∆)bethesetofsolutionsr of(2.9). ItiseasytoseethatS(∆)isafiniteabeliangroup
anditsorderispreciselygivenby
|det(M)| = n!V(∆).
LetS (∆)betheprimetoppartofS(∆). Itisanabeliansubgroupoforderequaltotheprime
p
topfactorofdet(M). Inparticular,S (∆) = S(∆)ifpisrelativelyprimetodet(M),i.e.,f
p
isnon-degenerate.
Let m be an integer relatively prime to the order of S (∆), then multiplication by m
p
induces an automorphism of the finite abelian group S (∆). The map is called the m-map of
p
S (∆)denotedbyr (cid:55)→ {mr},where
p
{mr} = ({mr },...,{mr })
1 n
and {mr } denotes the fractional part of the real number mr . For each element r ∈ S (∆),
i i p
letd(m,r)bethesmallestpositiveintegersuchthatmultiplicationbymd(m,r) actstriviallyon
r,i.e.
(md(m,r)−1)r ∈ Zn.
Let S (m,d) be the set of r ∈ S(∆) such that d(m,r) = d, We have the disjoint m-degree
p
decomposition
(cid:91)
S (∆) = S (m,d).
p p
d≥1
Letχ : F∗ → C∗ beamultiplicativecharacterandlet
q
(cid:88)
G (q) = − χ(a)−kζTr(a) (0 ≤ k ≤ q−2)
k p
a∈F∗
q
betheGausssums. Thenwehave
ZhangandFeng,L-func.ofcert.exp.sums 9
Theorem2.1([18]).
L∗(f/F ,T)(−1)n−1 = (cid:89) (cid:89) (cid:32)1−Td(cid:89)n χ(a )ri(qd−1)G (qd)(cid:33)d1 ,
q i ri(qd−1)
d≥1r∈Sp(q,d) i=1
wherer = (r ,...,r ).
1 n
TheStickelbergertheoremforGausssumsis
Theorem 2.2 ([8]). Let 0 ≤ k ≤ q −2. Let σ (k) be the sum of the p-digits of k in
p
its base p expansion. That is, σ (k) = k + k + k + ..., k = k + k p + k p2 + ...,
p 0 1 2 0 1 2
0 ≤ k ≤ p−1. Then,
i
σ (k)
p
ord G (q) = .
p k
p−1
ByTheorems2.1and2.2,withacalculation,wehavetheordinarycriterionforadiagonal
Laurentpolynomialf.
Theorem 2.3 ([18]). Let d (p) be the largest invariant factor of S (∆). Let d be the
n p n
largestinvariantfactorofS(∆). Ifp ≡ 1(modd (p)),thenthediagonalLaurentpolynomial
n
in(2.7)isordinaryatp. Inparticular,ifp ≡ 1(modd ),thenthediagonalLaurentpolynomial
n
in(2.7)isordinaryatp.
Inordertostudythe(generically)ordinarypropertyofL-functionsanddetermineaHasse
polynomialh (∆),wearegoingtobrieflyreviewDwork’straceformula,Wan’sdescenttheo-
p
remandWan’sdecompostiontheoryforL-function.
2.4. Dwork’s trace formula. Let Q be the field of p-adic numbers. Let Ω be the
p
completion of an algebraic closure of Q . Let q = pa for some positive integer a. Denote
p
by ord the additive valuation on Ω normalized by ord p = 1, and denote by ord the additive
q
valuation on Ω normalized by ord q = 1. Let K denote the unramified extension of Q in Ω
q p
ofdegreea. LetΩ = Q (ζ ),whereζ isaprimitivep-throotofunity. ThenΩ isthetotally
1 p p p 1
ramified extension of Q of degree p−1. Let Ω be the compositum of Ω and K. Then Ω
p a 1 a
is an unramified extension of Ω of degree a. The residue fields of rings of algebraic integers
1
ofΩ andK arebothF ,andtheresiduefieldsofringsofalgebraicintegersofΩ andQ are
a q 1 p
bothF . Letπ beafixedelementinΩ satisfying
p 1
(cid:88)∞ πpm 1
= 0, ord π = .
pm p p−1
m=0
Then,π isauniformizerofΩ = Q (ζ )andwehave
1 p p
Ω = Q (π).
1 p
TheFrobeniusautomorphismx (cid:55)→ xp ofGal(F /F )liftstoageneratorτ ofGal(K/Q )such
q p p
thatτ(π) = π. Ifζ isa(q−1)-strootofunityinΩ ,thenτ(ζ) = ζp.
a