Generic Newton polygon for exponential sums in two variables with triangular base

0.48 MB·

by  Rufei Ren

GENERIC NEWTON POLYGON FOR EXPONENTIAL SUMS IN TWO
VARIABLES WITH TRIANGULAR BASE
RUFEIREN
7
1
Abstract. Letpbeaprimenumber. Everytwo-variablepolynomialfpx1,x2qoverafinitefieldof
0
characteristicpdefinesanArtin–Schreier–WitttowerofsurfaceswhoseGaloisgroupisisomorphic
2
to Zp. Our goal of this paper is to study the Newton polygon of the L-functions associated to a

n finitecharacterofZp andagenericpolynomialwhoseconvexhullisafixedtriangle∆. Wedenote
a thispolygonbyGNPp∆q. WeprovealowerboundofGNPp∆q,whichwecalltheimprovedHodge
J polygon IHPp∆q, and we conjecture that GNPp∆q and IHPp∆q are the same. We show that if
GNPp∆qandIHPp∆qcoincideatacertainpoint,thentheycoincideatinfinitelymanypoints.
6
When ∆ is an isosceles right triangle with vertices p0,0q, p0,dq and pd,0q such that d is not

divisiblebypandthattheresidueofpmodulodissmallrelativetod,weprovethatGNPp∆qand
]
T IHPp∆qcoincide atinfinitelymanypoints. Asacorollary,wededuce that the slopesofGNPp∆q
N roughlyformanarithmeticprogressionwithincreasingmultiplicities.
.
h
t
a
m
Contents
[

1. Introduction 1
2
v 2. Dwork trace formula 4
4 3. Improved Hodge polygon for a triangle ∆ 9
5
4. The case when ∆ is an isosceles right triangle I. 22
2
5. The case when ∆ is an isosceles right triangle II. 30
0
0 References 46
.
1
0
7
1
: 1. Introduction
v
i We shall state our main results and their motivation after recalling the notion of L-
X
functions for Witt coverings. Let p be a prime number. Let
r
a
fpx ,x q :“ a xPxxPy
1 2 P 1 2
PPZ2
ÿě0
be a two-variable polynomial in F rx ,x s and write
p 1 2
fˆpx ,x q :“ aˆ xPxxPy
1 2 P 1 2
PPZ2
ÿě0
for its Teichmu¨ller lift, whereaˆ denotes the Teichmu¨ller lift of a . We use F pfq to denote
P P p
the extension of F generated by all coefficients of f and set npfq :“ rF pfq : F s. The
p p p
convex hull of the set of points p0,0qY P | a ‰ 0 is called the polytope of f and denoted
P
by ∆ .
f
(
Date:January9,2017.
2010 Mathematics Subject Classification. 11T23(primary),11L0711F3313F35(secondary).
Keywords and phrases. Artin–Schreier–Witttowers,T-adicexponential sums,SlopesofNewtonpolygon,T-adic
NewtonpolygonforArtin–Schreier–Witttowers,Eigencurves.
1
2 RUFEIREN
Let pG q2 be the two-dimensional torus over F . The main subject of our study is
m pnpfq
the L-function associated to finite characters χ : Z Ñ Cˆ of conductor pmχ given by
p p
1
L˚pχ,sq :“ ,
f 1´χ Tr pfˆpxˆqq sdegpxq
xP|pźGmq2| Qpnpfqdegpxq{Qp
where |pG q2| is the set of closed points of`pG q2 and xˆ is the Te˘ichmu¨ller lift of a closed
m m
point x in pG q2. The characteristic power series C˚pχ,sq is a product of reciprocals of
m f
L-functions:
8
(1.1) C˚pχ,sq “ L˚pχ,pjnpfqsq´pj`1q.
f f
j“0
ź
We can alternatively express L˚pχ,sq in terms of C˚pχ,sq as
f f
C˚pχ,sqC˚pχ,p2npfqsq ´1
L˚pχ,sq “ f f .
f C˚pχ,pnpfqsq2
f
´ ¯
Therefore, C˚pχ,sq and L˚pχ,sq determine each other.
f f
Definition 1.1. From [LWei], we know that
2p2pmχ´1qAreap∆fq
L˚pχ,sq´1 :“ v si
f i
i“0
ÿ
is a polynomial of degree 2p2pmχ´1qAreap∆fq in Zprζpmχsrss, where ζpmχ is a primitive pmχ-
th root of unity. We call the lower convex hull of the set of points i,pmχ´1pp´1qv pv q
pnpfq i
the normalized Newton polygon of L˚pχ,sq´1, which is denoted by NPpf,χq . Here,
f ` L´1 ˘
v p´q is the p-adic valuation normalized so that v ppnpfqq “ 1. Similarly, we write
pnpfq pnpfq
NPpf,χq for the normalized Newton polygon of C˚pχ,sq.
C f
In [DWX], Davis, Wan and Xiao studied the p-adic Newton slopes of L˚pχ,sq when
f
fpxq is a one-variable polynomial whose degree d is coprime to p. They concluded that, for
each character χ : Z Ñ Cˆ of relatively large conductor, NPpf,χq depends only on its
p p L´1
conductor. We briefly introduce their proof as follows.
They proved a lower bound of NPpf,χq when χ is the so-called universal character
C
and an upper bound by the Poincar´e duality of roots of L˚pχ ,sq for a particular character
f 1
χ of conductor p. The lower bound is called the Hodge polygon in their paper. Then they
1
verifiedthattheupperboundcoincideswiththelowerboundatx “ kdforanynon-negative
integer k. Since the Newton polygon of C˚pχ,sq is confined between these two bounds, it
f
also passes through their intersections. See more details in [DWX].
We also mention here that the aforementioned proof strongly inspired the proof of
spectral halo conjecture by Liu, Wan, and Xiao in [LWX]; we refer to [RWXY] for the
discussionontheanalogy ofthetwo proofs. Motivated bytheattempt ofextendingspectral
halo type results beyond the case of modular forms, it is natural to ask whether one can
generalize the main results of [DWX] to more general cases of exponential sums and Artin–
Schreier–Witt towers. For example, in a joint work with Wan, Xiao, and Yu, we examined
the case when the Galois group of the Artin–Schreier–Witt tower is canonically isomorphic
to Z .
pℓ
In this paper, we mainly deal with the generic Newton polygon of L-functions for two-
variable polynomials. We want to apply the methods in [DWX] to this case. Therefore,
it is crucial for us to give a lower bound and an upper bound for C˚pχ,sq. However, the
f
GENERIC NEWTON POLYGON FOR EXPONENTIAL SUMS IN TWO VARIABLES WITH TRIANGULAR BASE3
Hodge polygon provided by Liu and Wan in [LWan] is no longer optimal, and is in general
strictly lower than the upper bound we obtain by Poincare duality. Our main contribution
in this paper is to find an improved lower bound for NPpf,χq , which we call the improved
C
Hodge bound IHPp∆q. We conjecture that our improved Hodge polygon is optimal, and is
equal to the generic Newton polygon, that is the lowest Newton polygon for all polynomials
f with the same convex hull.
When ∆ is an isosceles right triangle with vertices p0,0q, pd,0q and p0,dq, we will
f
give an equivalent condition to verify the coincidence of improved Hodge polygon with the
Newton polygon (at infinitely many points), and we will show that this condition is met for
a generic polynomial with convex hull ∆ .
f
We now turn to stating our main results more rigorously.
Notation 1.2. For a two-dimensional convex polytope ∆ which contains p0,0q, we denote
its cone by
Conep∆q:“ P P R2 kP P ∆ for some k ą 0 ,
and put ! ˇ )
Mp∆q:“ˇConep∆qXZ2
to be the set of lattice points in Conep∆q.
Moreover, we write T p∆q (resp. T1p∆q) for the set consisting of all points in Mp∆q
k k
with weight w (See Definition 2.12) strictly less than k (resp. less than or equal to k), and
denote its cardinality by p∆q (resp. 1p∆q).
xk xk
Notation 1.3. For integers a and b, we denote by a%b the residue of a modulo b.
Definition 1.4. The generic Newton polygon of ∆ is defined by
GNPp∆q :“ inf NPpf,χq ,
L´1
χ:Zp{pmχZpÑCˆp
∆f“∆ ´ ¯
where χ : Z Ñ Cˆ runs over all finite characters, and f runs over all polynomials in
p p
F rx ,x s such that ∆ “ ∆. The following are our main results.
p 1 2 f
Theorem 1.5. Let ∆ be a right isosceles triangle with vertices p0,0q,p0,dq,pd,0q, where
d is a positive integer not divisible by p. Let p be the residue of p modulo d. Suppose
0
d ě 24p2p2`p q. Then the generic Newton polygon GNPp∆q passes through points p p∆q`
0 0 k
x
i,h p∆q`kiq for any k ě 0 and 0 ď iď kd`1, where
k
pkd`1qkd pp´1qpk´1qkpk`1qd2
p∆q“ and h p∆q “ `k tpwpPqu.
k k
x 2 3
PPÿT1p∆q
The points p ,h p∆qq are vertices for the improved Hodge polygon IHPp∆q (see Def-
k k
x
inition 2.18 and Proposition 3.16). So the essential content of the proof is to show that
the generic Newton polygon GNPp∆q also passes through these points. The proof of Theo-
rem 1.5 consists of two parts: first we show that, for a fixed polynomial f with convex hull
∆ , if IHPp∆q coincides with the corresponding Newton polygon NPpf,χ q at , then
f 1 C 1
x
these two polygons agree at all points x “ p∆q`i for k ě 1 and 0 ď i ď kd`1. This
k
x
is proved in Theorem 3.1, which in fact holds with less constraints on ∆. Next, we prove
that, for a generic polynomial f, NPpf,χ q agrees with IHPp∆q at x “ p∆q. For this,
1 C 1
x
we look at the leading term of v for the universal polynomial f with convex hull ∆
1p∆q univ
and show that this term is non-zxero when dě 24p2p2`p q. This is proved in Theorem 4.2,
0 0
which in fact holds under a wearker condition on p0.
From [LWei, Theorem 1.4], for a finite character χ of conductor pmχ, we know that
L˚pχ,sq´1 has degree of p2pmχ´1qd2.
f
4 RUFEIREN
Theorem 1.6. Under the hypotheses of Theorem 1.5, if we put pα ,...,α q to be
1 p2pmχ´1qd2
the sequence of pnpfq-adic Newton slopes of L˚pχ,sq´1 (in non-decreasing order), then for
f
first p2pmχ´1qd2`ppmχ´1qd-th slopes we have
2
αx1i`1,...,αxi`1 P ppmi´1, pmi`χ1´1q for i“ 0,1,...,pmχ´1´1,
$’αxi`1,...,αx1i “ pmχi´1 for i“ 0,1,2,...,pmχ´1´1,
’&αxpmχ´1`1,...,αx1pmχ´1´2 “ 1.
’
In fact, p’%oints p kp∆q,hpTkqq are vertices of the improved Hodge polygon (see Defini-
x
tion 2.18 and Proposition 3.16).
We do not know if Theorem 1.5 still holds for polytopes which are not right isosceles
triangle. However, for an arbitrary multi-variable polynomial f in F rxs, we are still able
p
to get an improved Hodge polygon for NPpf,χq. Especially, when f is a two-variable
polynomial, it is expected that the slopes of the improved Hodge polygon form certain
generalized arithmetic progression. We plan to address this in a forthcoming paper.”
The Newton polygon for exponential sums was explicitly computed in the “ordinary”
case by Adolphson–Sperber [AS], Berndt–Evans [BE], and Wan [W] in many special cases,
and in general (namely the T-adic setup) by Liu–Wan [LWan]. For the ∆ we considered in
Theorem 1.5, the ordinary condition amounts to requiring p ” 1 pmod dq. Blache, Ferard,
and Zhu in [BFZ] proved a lower bound for the Newton polygon of one-variable Laurent
polynomial over F of degree pd ,d q, which is called a Hodge-Stickelberger polygon. They
q 1 2
also showed that when p approaches to infinite, the Newton polygon coincides the Hodge-
Stickelberger polygon.
Going beyond the ordinary case, there has been many researches on understanding
the generic Newton polygon of L pχ,sq when f is a polynomial of a single variable. The
f
first results are due to Zhu [Z1] and Scholten–Zhu [SZ], when p is large enough. In [BF],
Blache and Ferard worked on the generic Newton polygon associated to characters of large
conductors. In[OY],OuyangandYangstudiedtheone-variablepolynomialfpxq“ xd`a x.
1
A similar result can be found in [OZ], where Ouyang and Zhang studied the family of
polynomials of the form fpxq“ xd`a xd´1.
d´1
Our Theorem 1.5 maybe considered as the firststep beyond the ordinary case when the
base polynomial is multivariable. A similar result is obtained by Zhu in [Z2] independently
which shows that GNPp∆ q and IHPp∆ q coincide for characters of Z of conductor p.
f f p
Acknowledgments. The author would like to thank his advisor Liang Xiao for the extra-
ordinary supportin this paper and also thank Douglass Haessig, HuiJuneZhu, and Daqing
Wan for helpful discussion.
2. Dwork trace formula
Let p be an odd prime and let fpx ,x q :“ a xPxxPy be a two-variable polyno-
1 2 PPZ2 P 1 2
ě0
mialinF rx ,x s. DenoteF pfqtobethefinitefieldgeneratedbythecoefficientsoff,which
p 1 2 p ř
we call the coefficient field of f. The convex hull of the set of points tp0,0quY P | a ‰ 0
P
is called the polytope of f and denoted by ∆ .
f
(
Ourdiscussionwillfocusonafixedf untilProposition4.4. WeputF “ F pfqandn “
q p
rF : F s. Let aˆ P Z be the Teichmu¨ller lift of a . We call fˆpx ,x q :“ aˆ xPxxPy
q p P q P 1 2 PPZ2 P 1 2
ě0
the Teichmu¨ller lift of fpxq.
ř
For convenience, we putv p´q (resp. v p´q) be the p-adic valuation normalized so that
p q
v ppq“ 1 (resp. v pqq“ 1).
p q
GENERIC NEWTON POLYGON FOR EXPONENTIAL SUMS IN TWO VARIABLES WITH TRIANGULAR BASE5
2.1. T-adic exponential sums.
Notation 2.1. We recall that the Artin–Hasse exponential series is defined by
8 πpi
(2.1) Epπq “ exp “ 1´πi ´µpiq{i P 1`π`π2Z rrπss.
pi p
i“0 p∤i, iě1
`ÿ ˘ ź ` ˘
Putting Epπq “ T `1 gives an isomorphism Z JπK – Z JTK.
p p
Definition 2.2. For each power series in Z JTK, say gpTq, we define its T-adic valuation
q
as the largest k such that g P TkZ JTK and denote it by v pgq.
q T
Definition 2.3. For each k ě 1, the T-adic exponential sum of f over Fˆ is
qk
Sf˚pk,Tq :“ p1`TqTrQqk{Qppfˆpxˆ1,xˆ2qq P ZprrTss.
px1,x2ÿqPpFˆqkq2
Definition 2.4. The T-adic L-function of f is defined by
8 sk
L˚pT,sq “ exp S˚pk,Tq
f f k
´kÿ“1 ¯
and its corresponding T-adic characteristic power series is defined by
8 sk
(2.2) C˚pT,sq :“ exp ´pqk ´1q´2S˚pk,Tq
f f k
´kÿ“1 ¯
8
“ u pTqsk P Z JT.sK,
k p
k“0
ÿ
We put u pTq “ u Tj P Z rrTss.
k k,j p
Moreover, they determine each other by relations:
8
´1
(2.3) C˚pT,sq “ L˚pT,qjsqj`1
f f
´jź“0 ¯
and
C˚pT,sqC˚pT,q2sq ´1
(2.4) L˚pT,sq “ f f .
f C˚pT,qsq2
f
´ ¯
It is clear that for a finite character χ :Z Ñ Cˆ, we have
p p
L˚pχ,sq “ L˚pT,sq and C˚pχ,sq “ C˚pT,sq ,
f f T“χp1q´1 f f T“χp1q´1
where L˚pχ,sq and C˚pχ,sq areˇdefined in the introduction. ˇ
f f ˇ ˇ
Notation 2.5. Recall that we put Epπq “ T `1. We put
E px ,x q:“ Epaˆ πxPxxPyq
f 1 2 P 1 2
PPZ2
źě0
(2.5)
“ e pTqxPxxPy P Z JTKJx ,x K.
P 1 2 q 1 2
PPZ2
ÿě0
6 RUFEIREN
2.2. Dwork’s trace formula. Recall that ∆ is the convex hull of fpx ,x q and Mp∆ q
f 1 2 f
is defined in Notation 1.2 as a set consisting of all the lattice points in the Conep∆ q. Let
f
D be the smallest positive integer such that wpMp∆ qq Ă 1Z.
f D
Definition 2.6. We fix a D-th root T1{D of T. Define
B “ b pT1{Dx qPxpT1{Dx qPy b P Z JT1{DK,v pb q Ñ `8,when wpPq Ñ 8 .
P 1 2 P q T P
!PPMÿp∆fq ˇ )
ˇ
Let ψ denote the operator on B sucˇh that
p
ψ b xPxxPy :“ b xPxxPy.
p P 1 2 ppPq 1 2
´PPMÿp∆fq ¯ PPMÿp∆fq
Recall that n “ rF :F s.
q p
Definition 2.7. Define
(2.6) ψ :“ σ´1 ˝ψ ˝E px ,x q :B ÝÑ B,
Frob p f 1 2
and its n-th iterate
n´1
ψn “ ψn˝ EσFirobpxpi,xpiq,
p f 1 2
i“0
ź
whereσ representsthearithmetic Frobeniusactingonthecoefficients, andforany g P B
Frob
we have E px ,x qpgq :“ E px ,x q¨g .
f 1 2 f 1 2
One can easily check that
ψ ˝E px ,x q xPxxPy “ e pTqxQxxQy,
p f 1 2 1 2 pQ´P 1 2
` ˘ QPMÿp∆fq
where e pTq is defined in (2.5).
pQ´P
Theorem 2.8 (Dwork Trace Formula). For every integer k ą0, we have
pqk ´1q´2Sf˚pk,πq“ TrB{Zqrrπss ψnk .
Proof. This was proved by [LWei, Lemma 4.7]. ` ˘ (cid:3)
One can see [W] for a a thorough treatment of the universal Dwork trace formula.
Proposition 2.9 (Analytic trace formula). The theorem above has an equivalent multi-
plicative form:
(2.7) C˚pT,sq “det I ´sψn | B{Z rrπss .
f q
Proof. Also see [LWei, Theorem 4.8]. ` ˘ (cid:3)
Definition 2.10. The normalized Newton polygon of C˚pT,sq, denoted by NPpf,Tq , is
f C
the lower convex hull of the set of points i, vTpuiq .
n
!´ ¯)
Notation 2.11. Inthispaper,wefix∆tobeatrianglewithverticesatp0,0q, P1 :“ pa1,b1q
and P2 :“ pa2,b2q.
Definition 2.12. For each lattice point P in Z2, assume that Q is the intersection of the
lines OP and P1P2. Then we call
ÝÝÑ
OP
wpPq :“ ÝÝÑ
OQ
the weight of P.
GENERIC NEWTON POLYGON FOR EXPONENTIAL SUMS IN TWO VARIABLES WITH TRIANGULAR BASE7
The weight function w is linear, i.e. Any two points P and Q in Z2 satisfy
ě0
(2.8) wpP `Qq “ wpPq`wpQq.
Equality (2.8) does not always hold for a general polytope.
We shall frequently work with multisets, i.e. sets of possibly repeating elements. They
are often marked by a superscript star to be distinguished from regular sets, e.g. S‹. The
disjoint union of two multiset S‹ and S1‹ is denoted by S‹ZS1‹ as a multiset.
Definition 2.13. Let S bea subset of Mp∆q. Then we write S‹m (resp. S‹8) for the union
of m (resp. countably infinite) copies of S as a multiset.
Notation 2.14. For any sets S‹ and S‹ in Mp∆q‹8 of the same cardinality, we denote by
1 2
IsopS‹,S‹q the set of all bijections (as multisets) from S‹ to S‹. When S‹ “ S‹ “ S‹, we
1 2 1 2 1 2
denote IsopS‹q :“ IsopS‹,S‹q.
Definition 2.15. For a bijection τ in IsopS‹,S‹q, we define
1 2
(2.9) hpS‹,S‹,τq :“ wppτpPq´Pq .
1 2
PPS‹
ÿ1P T
For any submultiset S11‹ of S‹1, we write τ|S1‹ for the restriction of τ to S11‹. Moreover,
1
the minimum of hpS‹,S‹,τq is denoted by
1 2
(2.10) hpS‹,S‹q :“ min phpS‹,S‹,τqq,
1 2 1 2
τPIsopS‹,S‹q
1 2
where τ varies among all bijections from S‹ to S‹.
1 2
Definition 2.16. We call a bijection from S‹ to S‹ minimal, if it reaches the minimum in
1 2
(2.10). When S‹ “ S‹, we call it a minimal permutation of S‹ and abbreviate hpS‹,S‹,‚q
1 2
(resp. hpS‹,S‹q) to hpS‹,‚q (resp. hpS‹q).
Remark 2.17. If S‹ for i“ 1,2 belongs Mp∆q, we suppress the star from the notation.
i
Definition 2.18. The improved Hodge polygon of ∆, denoted by IHPp∆q, is the lower
convex hull of the set of points ℓ, min hpS‹q , where M pnq represents for the set
ˆ S‹PMℓpnq n ˙ ℓ
consisting of all multi-subsets of!Mp∆q‹n of cardinali)ty nℓ, note M p1q “ M .
ℓ ℓ
we shall prove in Proposition 3.16 later that
min hpS‹q “ n¨ min hpSq,
S‹PMℓpnq SPMℓ
and hence the IHPp∆q is independent of n. In particular, IHPp∆q is the convex hull of the
set of points
ℓ, min hpSq .
ˆ SPMℓ ˙
Notation 2.19. We denote by
m m ¨¨¨ m
0 1 ℓ´1
n n ¨¨¨ n
„ 0 1 ℓ´1 M
the ℓ ˆ ℓ-submatrix formed by elements of a matrix M whose row indices belong to
tm ,m ,...,m u and whose column indices belong to tn ,n ,...,n u.
0 1 ℓ´1 0 1 ℓ´1
Put ∆ “ ∆. Recall that we define T1 in Notation 1.2.
f 1
Lemma 2.20. We have e pTq “ 1 and v pe pTqq ě wpQq for all Q P Mp∆q.
O T Q
P T
8 RUFEIREN
Proof. (1) It follows from the definition of e pTq in (2.5).
O
(2) Let
Spfq:“ tP P T1 |a ‰ 0u “ tQ ,Q ,...,Q u,
1 P 1 2 t
where ta u is the set of coefficients of fpx ,x q and t is the cardinality of Spfq.
P 1 2
Expandingeach Epaˆ πxpQiqxxpQiqyq to be a power series in variables x and x , we get
Qi 1 2 1 2
t t
E px ,x q “ Epaˆ πxpQiqxxpQiqyq“ c paˆ πxpQiqxxpQiqyqji,
f 1 2 Qi 1 2 ~j Qi 1 2
źi“1 ~jPÿZtě0 źi“1
where taˆ u is the set of coefficients of fˆpx ,x q and c belongs to Z .
P 1 2 ~j q
It is not hard to get that
t
e pTq “ c paˆ πqji
Q ~j Qi
t i“1
~j ÿjiQi“Q ź
i“1
! ˇ ř )
ˇ t t
“ ˇ c~j paˆQiqjiπiř“1ji .
~j tÿjiQi“Q ´ źi“1 ¯
i“1
! ˇ ř )
ˇ t
Since wpQ q ď 1 for each Q P Spfqˇ, then for each~j such that j Q “ Q, we have
i i i i
i“1
ř
t t t t
vT c~j paˆQiqjiπiř“1ji “ ji ě jiwpQiq“ wpQq,
´ źi“1 ¯ iÿ“1 iÿ“1
where T “ Epπq ´ 1. Therefore, we immediately get that v pe pTqq ě wpQq. Since
T Q
v pe pTqq is an integer, we have
T Q
v pe pTqq ě wpQq . (cid:3)
T Q
Notation 2.21. We label points in Mp∆q suchPthat MTp∆q “ tP ,P ,...u.
1 2
Proposition 2.22. The normalized Newton polygon NPpf,Tq lies above IHPp∆ q.
C f
Proof. We write N for the standard matrix of ψ ˝E corresponding to the basis
p f
txpP1qxxpP1qy,xpP2qxxpP2qy,¨¨¨u
1 2 1 2
of theBanach spaceB. By[RWXY, Corollary 3.9], weknowthatthestandardmatrix of ψn
corresponding to the same basis is equal to σn´1pNq˝σn´2pNq˝¨¨¨˝N. Then by [RWXY,
Frob Frob
Proposition 4.6], for every ℓ P N we have
(2.11)
n´1
m m ¨¨¨ m
u pTq “ det j`1,0 j`1,1 j`1,ℓ´1 ,
ℓ m m ¨¨¨ m
tPm0,0,Pm0,1ÿ,...,Pm0,ℓ´1uPMℓ ˆjź“0„ j,0 j,1 j,ℓ´1 σFjrobpNq˙
tPm1,0,Pm1,1,...,Pm1,ℓ´1uPMℓ
.
.
.
tPmn´1,0,Pmn´1,1,...,Pmn´1,ℓ´1uPMℓ
where m :“ m for each 0ď i ď ℓ´1.
n,i 0,i
GENERIC NEWTON POLYGON FOR EXPONENTIAL SUMS IN TWO VARIABLES WITH TRIANGULAR BASE9
Then for S “ tP ,P ,...,P u, we have
j mj,0 mj,1 mj,ℓ´1
n´1
m m ¨¨¨ m
v det j`1,0 j`1,1 j`1,ℓ´1
T m m ¨¨¨ m
´ `jź“0„ j,0 j,1 j,ℓ´1 σFjrobpNq˘¯
n´1
“v sgnpτ q σj pe q
T j Frob pτjpPq´P
(2.12) ´jź“0τj“IsoÿpSj`1,Sjq PPźSj`1 ¯
n´1
ě hpS ,S q
j`1 j
j“0
ÿ
n´1
ěhp S‹q,
j
j“0
ě
where S “ S . Therefore, it is easily seen that
n 0
(2.13) v pu pTqq ě min hpS‹q. (cid:3)
T ℓ
S‹PMℓpnq
3. Improved Hodge polygon for a triangle ∆
Recall that ∆ is a triangle with vertices p0,0q, P1 :“ pa1,b1q and P2 :“ pa2,b2q and as
we defined in Notation 1.2, “ p∆q (resp. 1 “ 1p∆q) is the number of lattice points
xk xk xk xk
in Mp∆q whose weight is strictly less than (resp. less than or equal to) k. For the rest of
this paper, we restrict p to be a prime satisfying
2pb a ´b a q
1 2 2 1
p ∤a b ´a b and p ą `1.
2 1 1 2
gcdpa ´a ,b ´b q
1 2 1 2
ThegoalofthissectionistoshowthatifNPpf,χq (SeeDefinition1.1)andIHPp∆qcoincide
C
at a certain point, then they willcoincide at infinitely many points. More precisely, we have
the following.
Theorem 3.1. Let fpx ,x q be a two-variable polynomial with convex hull ∆. Suppose
1 2
that there exists a nontrivial finite character χ : Z Ñ Cˆ and an integer k ą 0 such that
1 p p 1
NPpf,χ q coincides with IHPp∆q at x “ . Then for any finite character χ and positive
integer k1,CIHPp∆q and NPpf,χq coincidexak1t `i for all 0 ď i ď 1 ´ .
C xk k k xk xk
Moreover, the leading coefficients u presp. u q of the -th presp. 1-thq
xk,nhpTkq x1k,nhpT1kq xk xk
terms of the characteristic power series (see (2.2) for precise definition) are Z -units.
p
The proof of this theorem will occupy the rest of this section.
Notation 3.2. Without loss of generality, we can assume that a b ´a b ą 0. We call the
2 1 1 2
parallelogram withvertices O,P ,P and P `P (excluding theupperandright sides)the
1 2 1 2
fundamental parallelogram of ∆, and denote it by (cid:3) , i.e. the shadow region in Figure 1.
∆
We put (cid:3)Int to be the set of lattice points in (cid:3) , which contains a b ´a b points.
∆ ∆ 2 1 1 2
Let Λ be the lattice generated by P1 and P2. For each point P in Conep∆q, we write P%
for its residue in (cid:3) modulo Λ.
∆
Lemma 3.3. The map
η : (cid:3)Int Ñ (cid:3)Int
∆ ∆
P ÞÑ ppPq%
is a permutation.
10 RUFEIREN
y
P1 :“ pa1,b1q
P2 :“ pa2,b2q
x
Figure 1. The fundamental parallelogram.
Proof. since p and a b ´a b are coprime, there exist integers p1 and n such that pp1´1“
2 1 1 2 1
pa b ´a b qn . For a point P “ pP ,P q, we have
2 1 1 2 1 x y
ppp1´1qP “ n1pb2Px´a2PyqP1 `n1p´b1Px `a1PyqP2 P Λ.
It implies that composite
(cid:3)Int ÝPÝÞÑÝÝpPÝ%Ñ (cid:3)Int ÝPÝÞÑÝÝpÝ1PÝ%Ñ (cid:3)Int
∆ ∆ ∆
is the identity map. (cid:3)
The key to proving Theorem 3.1 is to gain precise control of the improved Hodge
polygon in Proposition 2.22. In [W], Wan made use of the following coarser estimate of this
Hodge polygon, for each multiset S‹ of Mp∆q‹8 we have
h pS‹q ě h pS‹q :“ pp´1q wpPq.
1 1
PPS‹
ÿ
It is however important for our method to understand the difference between h pS‹q and
1
hpS‹q (or more generally hpS‹,τq.
For each r P R, we put Rprq :“ rrs´r; and for a permutation τ of S‹, we set
‹
(3.1) U‹pS‹,τq :“ RpwppτpPq´Pqq |P P S‹ ,
! )
(3.2)
‹ ‹
U‹pS‹,τqďa :“ r P U‹pS‹,τq |r ď a and U‹pS‹,τqăa :“ r P U‹pS‹,τq |r ă a
! ) ! )
to measure the distance of these weights to the next integer values.
Write
(3.3) h pS‹,τq :“ r and h pS‹q “ min h pS‹,τq
2 2 2
τPIsopS‹q
rPUÿ‹pS‹,τq ! )
where IsopS‹q is set consisting all permutations of S‹ (see Notation 2.14).